Lesson 1: Exploring Trigonometric Ratios

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1 Lesson 1: Exploring Trigonometric Ratios Common Core Georgia Performance Standards MCC9 12.G.SRT.6 MCC9 12.G.SRT.7 Essential Questions 1. How are the properties of similar triangles used to create trigonometric ratios? 2. What are the relationships between the sides and angles of right triangles? 3. On what variable inputs are the ratios in trigonometry dependent? 4. What is the relationship between the sine and cosine ratios for the two acute angles in a right triangle? WORDS TO KNOW adjacent side cofunction the leg next to an acute angle in a right triangle that is not the hypotenuse a trigonometric function whose ratios have the same values when applied to the two acute angles in the same right triangle. The sine of one acute angle is the cofunction of the cosine of the other acute angle. complementary angles two angles whose sum is 90º cosecant cosine the reciprocal of the sine ratio; the cosecant of = csc = length of hypotenuse length of opposite side a trigonometric function of an acute angle in a right triangle that is the ratio of the length of the side adjacent to the length of the hypotenuse; the cosine of = cos = length of adjacent side length of hypotenuse U2-3

2 Lesson 1: Exploring Trigonometric Ratios cotangent the reciprocal of tangent; the cotangent of = cot = length of adjacent side hypotenuse identity opposite side phi ( ) ratio reciprocal length of opposite side the side opposite the vertex of the 90º angle in a right triangle an equation that is true regardless of what values are chosen for the variables the side across from an angle a Greek letter sometimes used to refer to an unknown angle measure the relation between two quantities; can be expressed in words, fractions, decimals, or as a percentage a number that, when multiplied by the original number, has a product of 1 right triangle a triangle with one angle that measures 90º scale factor a multiple of the lengths of the sides from one figure to the transformed figure. If the scale factor is larger than 1, then the figure is enlarged. If the scale factor is between 0 and 1, then the figure is reduced. secant the reciprocal of cosine; the secant of = sec = similar sine length of hypotenuse length of adjacent side two figures that are the same shape but not necessarily the same size. Corresponding angles must be congruent and sides must have the same ratio. The symbol for representing similarity is. a trigonometric function of an acute angle in a right triangle that is the ratio of the length of the opposite side to the length of the hypotenuse; the sine of = sin = lengthofoppositeside length of hypotenuse U2-4

3 Lesson 1: Exploring Trigonometric Ratios tangent theta ( ) trigonometry a trigonometric function of an acute angle in a right triangle that is the ratio of the length of the opposite side to the length of the adjacent side; the tangent of = tan = lengthofoppositeside length of adjacent side a Greek letter commonly used to refer to unknown angle measures the study of triangles and the relationships between their sides and the angles between these sides Recommended Resources AJ Design Software. Triangle Equations Formulas Calculator. This excellent website provides links to interactive calculators that help users solve for the various attributes of different types of triangles. For the Pythagorean Theorem, users can plug in known values for the legs of a right triangle, and the site will calculate the length of the hypotenuse. Keisan: High Accuracy Calculation. Solar elevation angle (for a day). This website features a solar calculator that will return the angle of elevation of the sun for any time of day at any latitude. The results are shown in a table and a graph. MathIsFun.com. Sine, Cosine and Tangent. Scroll down to the bottom of this site to find a tool for changing an angle and seeing the three main ratios of trigonometry change. Teach Engineering. Hands-on Activity: Stay in Shape. This site provides a real-life example of how the equations in basic geometry and trigonometry are used in navigation. U2-5

4 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Lesson 2.1.1: Defining Trigonometric Ratios Warm-Up Contractors are repairing and painting the tower on the county courthouse. They need to calculate the height of the tower to know what equipment they will need. The diagram below shows how they measured the height of the tower using a laser light, a post, and a tape measure. They put the post in the ground 3 feet in front of a 3-foot-tall tripod that has a laser on it. The height of the post is 8 feet, and the post is perpendicular to the ground. The laser is aimed just above the post so that the light beam hits the top of the tower. The post is 45 feet from the base of the courthouse tower. 5 feet 3 feet 45 feet 3 feet 1. How many triangles are shown in the diagram? Draw and label each one. 2. Which angle do the triangles share in common? continued U2-6

5 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 3. Which angles are congruent? 4. Are the triangles similar? 5. What is the scale factor, k, of the triangles? 6. How can you use the scale factor, k, to find the height, h, of the tower? U2-7

6 Lesson 1: Exploring Trigonometric Ratios Lesson 2.1.1: Defining Trigonometric Ratios Common Core Georgia Performance Standard MCC9 12.G.SRT.6 Warm-Up Debrief Contractors are repairing and painting the tower on the county courthouse. They need to calculate the height of the tower to know what equipment they will need. The diagram below shows how they measured the height of the tower using a laser light, a post, and a tape measure. They put the post in the ground 3 feet in front of a 3-foot-tall tripod that has a laser on it. The height of the post is 8 feet, and the post is perpendicular to the ground. The laser is aimed just above the post so that the light beam hits the top of the tower. The post is 45 feet from the base of the courthouse tower. 5 feet 3 feet 45 feet 3 feet 1. How many triangles are shown in the diagram? Draw and label each one. U2-8

7 Lesson 1: Exploring Trigonometric Ratios B B 5 feet A 3 feet C 45 feet C There are two triangles in the drawing: ABC and 2. Which angle do the triangles share in common? The triangles share A. 3. Which angles are congruent? ABC. Both triangles are right triangles with a 90º angle and a shared angle. Because the triangles have two congruent angles, their third angles must also be congruent. 4. Are the triangles similar? Yes, the triangles are similar by the Angle-Angle Similarity Statement. ABC ABC because they have two congruent angles ( A A and C C ). U2-9

8 Lesson 1: Exploring Trigonometric Ratios 5. What is the scale factor, k, of the similar triangles? length of image side lengthofpreimageside =scalefactor Equation for scale factor AC = AC k Divide the image side by the preimage side. Substitute values and solve. 6. How can you use the scale factor, k, to find the height, h, of the tower? Divide the height of the tower (B C ) by the height of the post (BC ). BC = k BC BC = BC k BC = 5 16= 80 Add 3 feet to account for the height of the laser tripod. h = = 83 feet Connection to the Lesson Students will use the properties of similar triangles to solve real-life problems involving side lengths. Students will use trigonometry to find unknown angles and lengths in right triangles. Students will learn that the applications of trigonometry can be even more powerful than the use of similar triangles or the Pythagorean Theorem. Students will discover the ratios found in right triangles. U2-10

9 Lesson 1: Exploring Trigonometric Ratios Prerequisite Skills This lesson requires the use of the following skills: measuring angles with a protractor understanding how to label angles and sides in triangles converting fractions into decimals solving for one unknown number in a ratio or proportion understanding the properties of similar triangles understanding and applying the properties of dilations using the Pythagorean Theorem understanding how to set up and use ratios converting measurement units within the same system (e.g., from inches to feet) Introduction Navigators and surveyors use the properties of similar right triangles. Designers and builders use right triangles in constructing structures and objects. Cell phones and Global Positioning Systems (GPS) use the mathematical principles of algebra, geometry, and trigonometry. Trigonometry is the study of triangles and the relationships between their sides and the angles between these sides. In this lesson, we will learn about the ratios between angles and side lengths in right triangles. A ratio is the relation between two quantities; it can be expressed in words, fractions, decimals, or as a percentage. Key Concepts Two triangles are similar if they have congruent angles. Remember that two figures are similar when they are the same shape but not necessarily the same size; the symbol for representing similarity is. Recall that the hypotenuse is the side opposite the vertex of the 90º angle in a right triangle. Every right triangle has one 90º angle. If two right triangles each have a second angle that is congruent with the other, the two triangles are similar. Similar triangles have proportional side lengths. The side lengths are related to each other by a scale factor. Examine the proportional relationships between similar triangles ABC and DEF in the diagram that follows. The scale factor is k = 2. Notice how the ratios of corresponding side lengths are the same as the scale factor. U2-11

10 Lesson 1: Exploring Trigonometric Ratios Proportional Relationships in Similar Triangles A b =10 c = 26 D e = 5 f = 13 C a = 24 B F d = 12 E Corresponding sides a b c d e f Side lengths Examine the three ratios of side lengths in ABC. Notice how these ratios are equal to the same ratios in DEF. Corresponding sides a c b c d f e f a d b e Side lengths The ratio of the lengths of two sides of a triangle is the same as the ratio of the corresponding sides of any similar triangle. The three main ratios in a right triangle are the sine, the cosine, and the tangent. These ratios are based on the side lengths relative to one of the acute angles. The sine of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse; the sine of = sin = lengthofoppositeside. length of hypotenuse The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the length of the hypotenuse; the cosine of = cos = lengthofadjacentside. length of hypotenuse U2-12

11 Lesson 1: Exploring Trigonometric Ratios The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side; the tangent of = tan = lengthofoppositeside. length of adjacent side The acute angle that is being used for the ratio can be called the angle of interest. It is commonly marked with the symbol (theta). Theta ( ) is a Greek letter commonly used as an unknown angle measure. Side opposite angle θ Hypotenuse Side adjacent to angle θ Angle θ See the following examples of the ratios for sine, cosine, and tangent. sine of = sin = lengthofoppositeside length of hypotenuse Abbreviation: opposite hypotenuse cosine of = cos = lengthofadjacentside length of hypotenuse tangent of = tan = lengthofoppositeside length of adjacent side Abbreviation: adjacent hypotenuse Abbreviation: opposite adjacent Unknown angle measures can also be written using the Greek letter phi ( ). The three main ratios can also be shown as reciprocals. The reciprocal is a number that when multiplied by the original number the product is 1. The reciprocal of sine is cosecant. The reciprocal of cosine is secant, and the reciprocal of tangent is cotangent. cosecant of = csc = length of hypotenuse length of opposite side length of hypotenuse secant of = sec = length of adjacent side U2-13

12 Lesson 1: Exploring Trigonometric Ratios cotangent of = cot = lengthofadjacentside length of opposite side Each acute angle in a right triangle has different ratios of sine, cosine, and tangent. The length of the hypotenuse remains the same, but the sides that are opposite or adjacent for each acute angle will be different for different angles of interest. The two rays of each acute angle in a right triangle are made up of a leg and the hypotenuse. The leg is called the adjacent side to the angle. Adjacent means next to. In a right triangle, the side of the triangle opposite the angle of interest is called the opposite side. Calculations in trigonometry will vary due to the variations that come from measuring angles and distances. A final calculation in trigonometry is frequently expressed as a decimal. A calculation can be made more accurate by including more decimal places. The context of the problem will determine the number of decimals places to which to round. Examples: A surveyor usually measures tracts of land to the nearest tenth of a foot. A computer manufacturer needs to measure a microchip component to a size smaller than an atom. A carpenter often measures angles in whole degrees. An astronomer measures angles to of a degree or smaller. Common Errors/Misconceptions confusing the differences between the trigonometric ratios forgetting to change the adjacent and opposite sides when working with the two acute angles mistakenly trying to use sine, cosine, and tangent ratios for triangles that are not right triangles mistakenly thinking that trigonometry will always find the exact length of a side or the exact measure of an angle U2-14

13 Lesson 1: Exploring Trigonometric Ratios Guided Practice Example 1 Find the sine, cosine, and tangent ratios for A and B in ABC. Convert the ratios to decimal equivalents. A b = 3 c C a = 4 B 1. Find the length of the hypotenuse using the Pythagorean Theorem. a 2 + b 2 = c 2 Pythagorean Theorem = c 2 Substitute values for a and b = c 2 Simplify. 25 = c 2 ± 25= c 2 c = 5 Since c is a length, use the positive value, c = 5. U2-15

14 Lesson 1: Exploring Trigonometric Ratios 2. Find the sine, cosine, and tangent of A. Set up the ratios using the lengths of the sides and hypotenuse, then convert to decimal form. sin A = opposite 4 hypotenuse cos A = adjacent 3 hypotenuse tan A = opposite 4 adjacent Find the sine, cosine, and tangent of B. Set up the ratios using the lengths of the sides and the hypotenuse, and then convert to decimal form. opposite 3 sin B hypotenuse adjacent 4 cos B hypotenuse opposite 3 tan B adjacent U2-16

15 Lesson 1: Exploring Trigonometric Ratios Example 2 Given the triangle below, set up the three trigonometric ratios of sine, cosine, and tangent for the angle given. Compare these ratios to the trigonometric functions using your calculator. A 53.1 b = 6 c = 10 C a = 8 B 1. Set up the ratios for sin A, cos A, and tan A and calculate the decimal equivalents. 8 sin 53.1º cos 53.1º tan 53.1º U2-17

16 Lesson 1: Exploring Trigonometric Ratios 2. Use your graphing calculator to find sin 53.1º. On a TI-83/84: First, make sure your calculator is in Degree mode. Step 1: Press [MODE]. Step 2: Arrow down to RADIAN. Step 3: Arrow over to DEGREE. Step 4: Press [ENTER]. The word DEGREE should be highlighted inside a black rectangle. Step 5: Press [2ND]. Step 6: Press [MODE] to QUIT. Important: You will not have to change to Degree mode again unless you have changed your calculator to Radian mode. Now you can find the value of sin 53.1º: Step 1: Press [SIN]. Step 2: Enter the angle measurement: Step 3: Press [ ) ]. Step 4: Press [ENTER]. On a TI-Nspire: First, make sure the calculator is in Degree mode. Step 1: From the home screen, select 5: Settings & Status, then 2: Settings, then 2: Graphs & Geometry. Step 2: Press [tab] twice to move to the Geometry Angle field, and then select Degree from the drop-down menu. Step 3: Press [tab] to move to the OK option and select it using the center button on the navigation pad. Next, calculate the sine of 53.1º. Step 1: From the home screen, select a new Calculate window. Step 2: Press [trig] to bring up the menu of trigonometric functions. Use the keypad to select sin, then type 53.1 and press [enter]. sin 53.1º U2-18

17 Lesson 1: Exploring Trigonometric Ratios 3. Use your graphing calculator to find cos 53.1º. On a TI-83/84: Step 1: Press [COS]. Step 2: Enter the angle measurement: Step 3: Press [ ) ]. Step 4: Press [ENTER]. On a TI-Nspire: Step 1: From the home screen, select a new Calculate window. Step 2: Press [trig] to bring up the menu of trigonometric functions. Use the keypad to select cos, then type 53.1 and press [enter]. cos 53.1º Use your graphing calculator to find tan 53.1º. On a TI-83/84: Step 1: Press [TAN]. Step 2: Enter the angle measurement: Step 3: Press [ ) ]. Step 4: Press [ENTER]. On a TI-Nspire: Step 1: From the home screen, select a new Calculate window. Step 2: Press [trig] to bring up the menu of trigonometric functions. Use the keypad to select tan, then type 53.1 and press [enter]. tan 53.1º Compare the calculator values to the ratios. Explain the difference. The values are very close to the ratios of the side lengths. The differences are due to the angle measurement of 53.1º being an approximation. U2-19

18 Lesson 1: Exploring Trigonometric Ratios Example 3 A right triangle has a hypotenuse of 5 and a side length of 2. Find the angle measurements and the unknown side length. Find the sine, cosine, and tangent for both angles. Without drawing another triangle, compare the trigonometric ratios of ABC with those of a triangle that has been dilated by a factor of k = First, draw the triangle with a ruler, and label the side lengths and angles. A b = 2 cm c = 5 cm C a =? cm B 2. Find a by using the Pythagorean Theorem. a 2 + b 2 = c 2 Pythagorean Theorem a = 5 2 Substitute values for b and c. a = 25 Simplify. a 2 = 21 a centimeters U2-20

19 Lesson 1: Exploring Trigonometric Ratios 3. Use a protractor to measure one of the acute angles, and then use that measurement to find the other acute angle. m A 66.5 We know that m C= 90 by the definition of right angles. The measures of the angles of a triangle sum to 180. Subtract m A and m B= 180 m A m C m B= 180 (66.5) (90) m B= m B 23.5 m C from 180 to find m B. 4. Find the sine, cosine, and tangent ratios for both acute angles. Express your answer in decimal form to the nearest thousandth. A B sin 66.5º sin23.5º cos 66.5º cos23.5º tan66.5º tan23.5º U2-21

20 Lesson 1: Exploring Trigonometric Ratios 5. Without drawing a triangle, find the sine, cosine, and tangent for a triangle that has a scale factor of 3 to ABC. Compare the trigonometric ratios for the two triangles. Multiply each side length (a, b, and c) by 3 to find a', b', and c'. a = 3 a = 3 (4.5826) = b = 3 b = 3 (2) = 6 c = 3 c = 3 (5) = 15 Set up the ratios using the side lengths of the dilated triangle sin 66.5º sin23.5º cos 66.5º cos23.5º tan 66.5º tan23.5º The sine, cosine, and tangent do not change in the larger triangle. Similar triangles have identical side length ratios and, therefore, identical trigonometric ratios. U2-22

21 Lesson 1: Exploring Trigonometric Ratios Example 4 What are the secant (sec), cosecant (csc), and cotangent (cot) ratios for an isosceles right triangle? 1. An isosceles right triangle has two angles that measure 45º. The two side lengths have equal lengths of 1 unit. 2. Use the Pythagorean Theorem to find the hypotenuse = c = c 2 2 = c 2 2 = c 3. Substitute the side lengths into the ratios. hypotenuse 2 csc 45º = = opposite 1 hypotenuse 2 sec 45º = = adjacent 1 adjacent cot 45º 1 opposite 1 1 U2-23

22 Lesson 1: Exploring Trigonometric Ratios Example 5 Triangle ABC is a right triangle where m A = 40º and side a = 10 centimeters. What is the sine of A? Check your work with the sin function on your graphing calculator. 1. Draw ABC with a ruler and protractor. A 40 c b C a = 10 cm B 2. Measure the length of hypotenuse c. c 15.6 centimeters Substitute the side lengths into the ratios to determine the sine of A. 10 sin40º Use your calculator to check the answer. Follow the steps outlined in Example 2. sin 40º The two answers are fairly close. The difference is due to the imprecise nature of manually drawing and measuring a triangle. U2-24

23 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.1: Building a Porch Ramp Bill s father is building a ramp to reach the front porch of their house. The ramp will rise from the flat ground to the top of the porch. The porch is 2 feet tall. The ramp will rise at a 10º angle from the ground. How long does the ramp need to be? What is the sine of 10º? U2-25

24 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.1: Building a Porch Ramp Coaching a. Create a drawing of the problem. b. Is there a triangle in this problem? c. Do you have enough information to draw a similar triangle to the drawing? d. Can you make a precise drawing to scale that shows the known length and angles? e. Is your small triangle similar to the large scale of the ramp? f. What is the scale factor between the height of the porch in your drawing and the porch s actual height? g. Which length in your drawing corresponds to the ramp length? h. How do you use the length in the drawing and the scale factor to find the length of the ramp? i. What is the ratio for the sine of 10º? U2-26

25 Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.1: Building a Porch Ramp Coaching Sample Responses a. Create a drawing of the problem. Porch 2 feet Ramp b. Is there a triangle in this problem? Yes; the triangle is formed by the ramp (the hypotenuse), the height of the porch, and the flat ground. c. Do you have enough information to draw a similar triangle? Yes. There is information for two angles: an angle of 10º is given, and the angle the ground makes with the porch is 90º. This means you can draw a similar triangle to the triangle created by the ramp. The third angle is 90º 10º = 80º. d. Can you make a precise drawing to scale that shows the known length and angle? Yes. You can use a protractor to create a 10º angle and a 90º angle, and a ruler to draw a side length that corresponds in units to the porch height (suggested scale: 1/2 inch = 1 foot). Scale factors of 1 inch : 1 foot will require paper that is at least 12 inches long. e. Is your small triangle similar to the large scale of the ramp? Yes. This is a scaled drawing that includes the 10º angle, the 90º angle, and the height of the porch. f. What is the scale factor between the height of the porch in your drawing and the porch s actual height? The scale factor will vary depending on how large the triangle is in the drawing. If students use the ratio 1/2 : 1, the scale factor is 1 : 24, making k = 24. g. Which length in your drawing corresponds to the ramp length? The length is the hypotenuse on the drawing. 10 U2-27

26 Lesson 1: Exploring Trigonometric Ratios h. How do you use the length in the drawing and the scale factor to find the length of the ramp? Multiply the length of the hypotenuse by the scale factor. First, measure the hypotenuse. If the scale on the drawing is 1/2 : 1, the length of the ramp in the drawing would equal approximately 5.75 inches. Multiply the length of the drawn ramp by the scale factor to determine the length of the actual ramp = 138 inches = 11.5 feet Compare students answers and see the variation in measurement data. Almost all the answers should range from feet to feet. i. What is the sine of 10º? 2 sin 10º = Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt. U2-28

27 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Practice 2.1.1: Defining Trigonometric Ratios Use ABC to complete problems 1 4. Round your answers to the nearest thousandths. A 4 cm C 5.71 cm B 1. Set up and calculate the trigonometric ratios for the sine, cosine, and tangent of A. 2. Set up and calculate the trigonometric ratios for the cosecant, secant, and cotangent of A. 3. Set up and calculate the trigonometric ratios for the sine, cosine, and tangent of B. 4. Set up and calculate the trigonometric ratios for the cosecant, secant, and cotangent of B. continued U2-29

28 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Draw triangles using the given information to complete problems 5 and A right triangle has side lengths of a = 8 and b = 4. Find the angle measurements and the length of the hypotenuse. Then, find the sine, cosine, and tangent for angles A and B (across from sides a and b respectively). 6. A right triangle has a side length of a = 4 and a hypotenuse of 8. Find the angle measurements of A and B. Then, find the sine, cosine, and tangent for both acute angles. Use the given information to complete problems A carpenter needs to measure the length of the beams that will support a roof. The building is 50 feet wide. The roof will rise at an angle of 30º from the top of the walls. The peak of the roof is feet above the top of the walls. The side adjacent to the 30º angle is half the width of the building. How long is each supporting beam, b, to the nearest thousandth? Add 2 feet to the beam length so that the roof can extend 2 feet past the walls. What is the cosine of 30º? Beam (b) feet 30 Building width = 50 feet continued U2-30

29 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 8. A water park has a straight slide into a deep pool. The slide is 50 feet long. It rises from the pool at an angle of 45º. How tall do the vertical supports of the platform need to be to support the platform at the top of the slide? The bottom of the slide is feet from the vertical supports. What is the sine of 45º? Round to the nearest thousandth. Ladder Slide = 50 feet 45 O Pool continued U2-31

30 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 9. Students are having a contest to see who can find the tallest tree in a park. To win, a student must measure the height of the tree without climbing the tree. Martha locates a very tall oak tree. She measures that the tree s shadow is 45 feet long. Martha has a shadow that is 11.5 feet long. She is 5.75 feet tall. How tall is the oak tree? What is the tangent of A? Rays of sun A Oak tree shadow = 45 feet Rays of sun Height = 5.75 feet Shadow = 11.5 feet continued U2-32

31 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 10. The drawing below shows a ladder against a wall. How high on the wall does the ladder reach if the ladder is feet long? Use your answer to find the sine of B. A 20 b c C a = 5 feet B U2-33

32 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Lesson 2.1.2: Exploring Sine and Cosine As Complements Warm-Up The high school basketball team has won the division title, and a work crew is going to put a celebratory banner up on the gymnasium wall. The top of the banner will be 20 feet off the floor. The crew leader knows that the base of an extension ladder should be 7 feet away from the wall to form an angle of 76º with the floor. He is not sure if a 25-foot ladder will be long enough. 1. Create a drawing of this problem. 2. What side length needs to be found in order to determine if the ladder will be long enough to hang the banner so that the top of the banner is 20 feet off the floor? 3. Use the Pythagorean Theorem to find the height the ladder will reach on the wall. Will the ladder be long enough? 4. What is the measurement of A? 5. How does the sine of 75º compare to the cosine of 15º? U2-34

33 Lesson 1: Exploring Trigonometric Ratios Lesson 2.1.2: Exploring Sine and Cosine As Complements Common Core Georgia Performance Standard MCC9 12.G.SRT.7 Warm-Up Debrief The high school basketball team has won the division title. A work crew is going to put the banner up on the gymnasium wall. The location for the top of the banner will be 20 feet off the floor. The crew leader knows that the base of an extension ladder should be 7 feet away from the wall to form an angle of 75º with the floor. He is not sure if a 25-foot ladder will be long enough. 1. Create a drawing of this problem. A b c = 25 feet C a = 7 feet 75 B U2-35

34 Lesson 1: Exploring Trigonometric Ratios 2. What side length needs to be found in order to determine if the ladder will be long enough to hang the banner so that the top of the banner is 20 feet off the floor? Length b needs to be found (the height on the wall). 3. Use the Pythagorean Theorem to find the height the ladder will reach on the wall. Will the ladder be long enough? a = 7 feet c = 25 feet b 2 = 25 2 b = 576 b = 24 feet The ladder will be long enough to reach the height of the banner on the wall. 4. What is the measurement of A? m A= 90 75= How does the sine of 75º compare with the cosine of 15º? Using your calculator, find sin 75º. sin 75º Using your calculator, find cos 15º. cos 15º The sine of 75º and the cosine of 15º have exactly the same values. Connection to the Lesson Students will explore further the relationship between the sine and cosine of complementary angles. Students will apply the Pythagorean Theorem and sine and cosine ratios. U2-36

35 Lesson 1: Exploring Trigonometric Ratios Prerequisite Skills This lesson requires the use of the following skills: knowing how to find the sine and cosine ratios in a right triangle solving proportions that have an unknown dividing with decimals Introduction In the previous lesson, we applied the properties of similar triangles to find unknown side lengths. We discovered that the side ratios of similar triangles are always the same. As a preparation to using trigonometry to solve problems, we will look more deeply into the relationship between sine and cosine in this lesson. Key Concepts Sine and cosine are side length ratios in right triangles. opposite The ratio for the sine of an angle is as follows: sinθ=. hypotenuse adjacent The ratio for the cosine of an angle is as follows: cosθ=. hypotenuse Examine ABC. A 3 5 C 4 B U2-37

36 Lesson 1: Exploring Trigonometric Ratios Determine the sine of A. 4 sin A 5 Determine the cosine of B. 4 cos B 5 This shows sin A = cos B. You can also see from the diagram that sinb 3 A 5 cos. Show that this relationship will work for any right triangle. A b c C a B sin A a cos B sinb b c c cosa In ABC, sin A = cos B, and sin B = cos A. This relationship between sine and cosine is known as an identity. An equation is an identity if it is true for every value that is used in the equation. Sine and cosine are called cofunctions because the value of one ratio for one angle is the same as the value of the other ratio for the other angle. The two acute angles in a right triangle have a sum of 90º. They are complementary angles. If one acute angle has a measure of x, the other angle has a measure of 90º x. U2-38

37 Lesson 1: Exploring Trigonometric Ratios For example, if one acute angle x has a measure of 70º, the other acute angle must measure 90 x. 90 x = 20 or 20º The sine-cosine cofunction can be written as: sin = cos (90º ) cos = sin (90º ) In other words, you can use the sine of one acute angle to find the cosine of its complementary angle. Also, you can use the cosine of one acute angle to find the sine of its complementary angle. This identity relationship makes sense because the same side lengths are being used in the ratios for the different angles. Cofunctions such as sine-cosine give you flexibility in solving problems, particularly if several ratios of trigonometry are used in the same problem. Postulate Sine and cosine are cofunction identities. sin = cos (90º ) cos = sin (90º ) Common Errors/Misconceptions losing track of which side length or angle is being solved for forgetting to take the complement of the angle when using the sine-cosine cofunction U2-39

38 Lesson 1: Exploring Trigonometric Ratios Guided Practice Example 1 Find sin 28º if cos 62º Set up the identity. sin = cos (90º ) 2. Substitute the values of the angles into the identity and simplify. sin 28º = cos (90º 28º) sin 28º = cos 62º 3. Verify the identity by calculating the sine of 28º and the cosine of 62º using a scientific calculator. sin 28º cos 62º Example 2 Complete the table below using the sine and cosine identities. Angle Sine Cosine 10º º 1. Determine the relationship between the two given angles. 10º and 80º are complementary angles. 10º + 80º = 90º U2-40

39 Lesson 1: Exploring Trigonometric Ratios 2. Apply the sine identity. sin = cos (90º ) sin 10º = cos 80º 3. Use the given value of sin 10º from the table to find sin 80º. The sine of 10º = 0.174; therefore, the cosine of 80º = Apply the cosine identity. cos = sin (90º ) cos 10º = sin 80º 5. Use the given value of cos 10º from the table to find sin 80º. The cosine of 10º = 0.985; therefore, the sine of 80º = Fill in the table. Angle Sine Cosine 10º º U2-41

40 Lesson 1: Exploring Trigonometric Ratios Example 3 Find a value of for which sin = cos 15º is true. 1. Determine which identity to use. The cosine was given, so use the cosine identity. Since is used as the variable in the problem, use the variable phi ( ) for the identity. cos = sin (90º ) = 15º cos 15º = sin (90º 15º) The cosine of 15º is equal to the sine of its complement. 2. Find the complement of 15º. 90º 15º = 75º The complement of 15º is 75º. 3. Substitute the complement of 15º into the identity. cos 15º = sin 75º or sin 75º = cos 15º 4. Write the value of. = 75º U2-42

41 Lesson 1: Exploring Trigonometric Ratios Example 4 Complete the table below using the sine and cosine identities. Angle Sine Cosine 28º Use a graphing calculator to find the cosine of 28º. cos 28º = Analyze the two sets of values for the sine and cosine. The values are switched, suggesting that the second angle of interest is the complement of the first angle of interest, 28º. Find the complement of the first angle of interest. 90º 28º = 62º The complement of 28º is 62º. The second angle must be 62º. Therefore, sin 62º = cos 28º. 3. Verify that the second angle of interest is correct using a scientific calculator. sin 62º = cos 28º = The second angle of interest is correct. 4. Use your findings to fill in the table. Angle Sine Cosine 28º º U2-43

42 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.2: School Vegetable Garden A high school biology class is planning a vegetable garden. The garden will be on the west side of the school building. The students are concerned that the garden will not get enough direct sunlight in the morning if the building s shadow shades the garden too much. Starting May 1, the garden needs direct sunlight by 10 A.M. at the latest. The school building is 8 meters high. The garden is 6 meters from the building. The students find on the Internet that the angle of the sun s rays at 10 A.M. on May 1 will be 60º above the horizon. How long will the building s shadow be? Is the chosen location a good spot for the garden? What are the sine of 30º and the cosine of 60º? U2-44

43 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.2: School Vegetable Garden Coaching a. Draw a picture of the problem. b. What are the measures of the angles in the drawing s triangle? c. What is the ratio between the shortest side length and the hypotenuse in this triangle? d. What is the length of the hypotenuse if the shortest side length of this triangle is 1 inch? e. Find the length of the longer side by using the Pythagorean Theorem. f. What are the basic side lengths of such a triangle if the shortest side length is 1 unit? g. What is the scale factor of the triangle in the drawing? h. How can you use the scale factor to find the length of the building s shadow? i. Is the chosen location a good spot for the garden? Explain. j. What are the sine of 30º and the cosine of 60º? U2-45

44 Lesson 1: Exploring Trigonometric Ratios Problem-Based Task 2.1.2: School Vegetable Garden Coaching Sample Responses a. Draw a picture of the problem. School 8 m 6 m 60 Garden b. What are the measures of the angles in the drawing s triangle? The angles measure 30º, 60º, and 90º. c. What is the ratio between the shortest side length and the hypotenuse in this triangle? Draw a triangle that has angles of 30º, 60º, and 90º. Measure the shortest side length, a. Measure the hypotenuse, c. Divide the hypotenuse length by the shortest side length, a. The ratio is 2 : 1 (or 1 : 2). The hypotenuse is twice as long as the shortest leg. d. What is the length of the hypotenuse if the shortest side length of this triangle is 1 inch? Draw a 30º 60º 90º triangle so that the shortest side length = 1 inch. The length of the hypotenuse = 2 inches. e. Find the length of the longer side by using the Pythagorean Theorem. Substitute values for a and c b 2 = 2 2 b 2 = 3 b = 3 inches U2-46

45 Lesson 1: Exploring Trigonometric Ratios f. What are the basic side lengths of a 30º 60º 90º triangle if the shortest side length is 1 unit? The basic proportional lengths opposite the angles in a 30º 60º 90º triangle are: opposite side length of 30º angle (shortest side length): 1 opposite side length of 60º angle (longest side length): 3 opposite side length of 90º angle (hypotenuse): 2 g. What is the scale factor, k, of the 30º 60º 90º triangle in the drawing? The only known side length (the longer side opposite the 60º angle) is 8 meters. The basic length of the longer side in a 30º 60º 90º triangle is 3. 3 k 8meters 8 k 3 k meters h. How can you use the scale factor to find the length of the building s shadow? The shadow is opposite the 30º angle in the drawing. This makes it the shortest side length. The basic length of the shortest side in a 30º 60º 90º triangle is 1. Multiply the scale factor by the basic side length of the triangle that corresponds with the building s shadow. This is the short side of the triangle, and the basic length is 1. Therefore, the length of the building s shadow is approximately meters. i. Is the chosen location a good spot for the garden? Explain. Yes, because the garden will not have shade at 10 A.M. on May 1. Since the garden will be placed 6 meters from the building, and the building s shadow will be about meters long, the garden will be outside of the shade. j. What are the sine of 30º and the cosine of 60º? sin 30º = 0.5 cos 60º = 0.5 Recommended Closure Activity Select one or more of the essential questions for a class discussion or as a journal entry prompt. U2-47

46 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Practice 2.1.2: Exploring Sine and Cosine As Complements Use what you have learned about the sine and cosine identities to complete the following problems. 1. sin 40º What is cos 50º? 2. cos 25º Find the sine of the complementary angle. 3. Find a value of for which sin = cos 22º is true. 4. Find a value of for which cos = sin 59º is true. 5. sin 5º Use this information to write an equation for the cosine. 6. cos 34º Use this information to write an equation for the sine. continued U2-48

47 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 7. sin 20º + sin 10º Find if cos + cos 80º Use the diagrams of ABC and DEF to fill in the empty boxes for the following equations. A b c sin A = a c = cos = f C a 70 B cos B = sin = c f F d E 20 e f D continued U2-49

48 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios 9. Lucia needs to buy a ladder to reach her roof. The edge of the roof is feet high. To use a ladder safely, the base of a ladder should be 1 foot away from a building for every 3.25 feet of building height. The ladder should extend 3 feet longer than the edge of the roof. How long does the ladder need to be? 10. Use the information you found in problem 9 to draw the triangle created by leaning the ladder against the house at the recommended distance. Label the angles so that A is where the ladder meets the building, B is where the ladder meets the ground, and C is where the building meets the ground. Then, calculate the sine and cosine values for angles A and B. U2-50

49 Common Core Georgia Performance Standard MCC9 12.G.SRT.8 Essential Questions 1. How could you use sine, cosine, tangent, and the Pythagorean Theorem to find missing angles and side lengths of a right triangle? 2. How could you use cosecant, secant, cotangent, and the Pythagorean Theorem to find missing angles and side lengths of a right triangle? 3. How are the angle of depression and angle of elevation used to solve real-world right triangle problems? 4. How do you determine which trigonometric function to use when solving real-world triangle problems? WORDS TO KNOW altitude angle of depression angle of elevation arccosine arcsine arctangent cosecant cotangent secant the perpendicular line from a vertex of a figure to its opposite side; height the angle created by a horizontal line and a downward line of sight to an object that is below the observer the angle created by a horizontal line and an upward line of sight to an object that is above the observer the inverse of the cosine function, written cos 1 or arccos the inverse of the sine function, written sin 1 or arcsin the inverse of the tangent function, written tan 1 or arctan 1 the reciprocal of the sine ratio; cscθ= sinθ 1 the reciprocal of the tangent ratio; cotθ= tanθ 1 the reciprocal of the cosine ratio; secθ = cosθ U2-57

50 Recommended Resources MathIsFun.com. Sine, Cosine, and Tangent. This website gives a brief overview of three trigonometric functions and shows some applications of the uses of the ratios. Toward the bottom of the page, a manipulative allows the user to drag the vertex of a triangle to change the acute angle and observe the relationships between the legs of the triangle and the trigonometric ratios when the hypotenuse is one unit. Purplemath. Angles of Elevation / Inclination and Angles of Depression / Declination. This website boasts a variety of resources regarding basic trigonometry. Specifically, the angles of depression and elevation links offer common misconceptions and additional tricks to mastering proper trigonometric equation setups. Wolfram MathWorld. Trigonometry. This online math encyclopedia contains definitions for terms as well as visuals to support continued terminology mastery. U2-58

51 NAME: RIGHT TRIANGLE TRIGONOMETRY Lesson 2.2.1: Calculating Sine, Cosine, and Tangent Warm-Up After seeing a great deal advertised online, your family bought a refurbished television. The television was advertised as having a 60" screen, but you doubt this after you unpack the actual TV. You measure the length of the television to be 48" and the height of the television to be 36", as shown in the diagram below. Given that television screen sizes are a measure of the diagonal distance across the screen, use the Pythagorean Theorem to check the validity of your family s purchase. Is the screen 60"? 48'' 36'' 1. Draw a sketch of the scenario, indicating the measurements as given. 2. Use the Pythagorean Theorem to solve for the diagonal distance. 3. Does the television have a 60" screen? Why or why not? U2-59

52 Lesson 2.2.1: Calculating Sine, Cosine, and Tangent Common Core Georgia Performance Standard MCC9 12.G.SRT.8 Warm-Up Debrief After seeing a great deal advertised online, your family bought a refurbished television. The television was advertised as having a 60" screen, but you doubt this after you unpack the actual TV. You measure the length of the television to be 48" and the height of the television to be 36", as shown in the diagram below. Given that television screen sizes are a measure of the diagonal distance across the screen, use the Pythagorean Theorem to check the validity of your family s purchase. Is the screen 60"? 48'' 36'' 1. Draw a sketch of the scenario, indicating the measurements as given. Since the television is a rectangle, drawing either diagonal will yield two right triangles with legs that measure 36" and 48". The diagonal will be the same in both triangles and should be marked with a variable. 36'' x 48'' U2-60

53 2. Use the Pythagorean Theorem to solve for the diagonal distance = x = x = x x x = 60 The diagonal distance is 60 inches. 3. Does the television have a 60" screen? Why or why not? Since television sizes are a measure of the diagonal distance across a rectangular or square screen, you can use the Pythagorean Theorem on the subsequent right triangles created by the diagonal. In this case, the screen is 60" because the sum of the squares of the length and width of the television yields the square of the diagonal distance. Connection to the Lesson Students will extend their understanding of the Pythagorean Theorem. U2-61

54 Prerequisite Skills This lesson requires the use of the following skills: manipulating the Pythagorean Theorem given any two sides of a right triangle defining the three basic trigonometric ratios (sine, cosine, and tangent) given an acute angle of a right triangle Introduction In the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact Burj Khalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so. You can use measuring tools for direct measurements of distance, but you can use trigonometry to find indirect measurements. First, though, you must be able to calculate the three basic trigonometric functions that will be applied to finding those larger distances. Specifically, we are going study and practice calculating the sine, cosine, and tangent functions of right triangles as preparation for measuring indirect distances. Key Concepts The three basic trigonometric ratios are ratios of the side lengths of a right triangle with respect to one of its acute angles. As you learned previously: Given the angle, sinθ= opposite hypotenuse. Given the angle, cosθ= adjacent hypotenuse. Given the angle, tanθ= opposite adjacent. Notice that the trigonometric ratios contain three unknowns: the angle measure and two side lengths. Given an acute angle of a right triangle and the measure of one of its side lengths, use sine, cosine, or tangent to find another side. U2-62

55 If you know the value of a trigonometric function, you can use the inverse trigonometric function to find the measure of the angle. The inverse trigonometric functions are arcsine, arccosine, and arctangent. Arcsine is the inverse of the sine function, written sin 1 or arcsin Arccosine is the inverse of the cosine function, written cos 1 or arccos Arctangent is the inverse of the tangent function, written tan 1 or arctan There are two different types of notation for these functions: sin 1 = arcsin ; if sinθ= 2 3, then arcsin 2 3 =θ. cos 1 = arccos ; if cosθ= 2 3, then arccos 2 3 =θ. tan 1 = arctan ; if tanθ= 2 3, then arctan 2 3 =θ. Note that 1 is not an exponent; it is simply the notation for an inverse trigonometric function. Because this notation can lead to confusion, the arc- notation is frequently used instead. ( ) = sin 1 = arcsin but (sin ) 1 = sinθ 1 1 sinθ To calculate inverse trigonometric functions on your graphing calculator: On a TI-83/84: Step 1: Press [2ND][SIN]. Step 2: Type in the ratio. Step 3: Press [ENTER]. On a TI-Nspire: Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select sin 1. Step 2: Type in the ratio. Step 3: Press [enter]. Use the inverses of the trigonometric functions to find the acute angle measures given two sides of a right triangle. U2-63

56 Common Errors/Misconceptions not using the given angle as the guide in determining which side is opposite or adjacent forgetting to ensure the calculator is in degree mode before completing the operations using the trigonometric function instead of the inverse trigonometric function when calculating the acute angle measure confusing inverse notation (sin 1 ) with reciprocal notation (sin ) 1 U2-64

57 Guided Practice Example 1 Leo is building a concrete pathway 150 feet long across a rectangular courtyard, as shown below. What is the length of the courtyard, x, to the nearest thousandth? 150 ft 34 x 1. Determine which trigonometric function to use by identifying the given information. Given an angle of 34º, the length of the courtyard, x, is opposite the angle. The pathway is the hypotenuse since it is opposite the right angle of the triangle. hypotenuse 34 opposite of 34 Sine is the trigonometric function that uses opposite and hypotenuse, opposite sinθ=, so we will use it to calculate the length of the hypotenuse courtyard. U2-65

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