# Unit 6 Trigonometric Identities, Equations, and Applications

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1 Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3

3 INTRODUCTION: Accelerated Mathematics III Unit 6 Trigonometric Identities Student Edition In previous units, students have defined trigonometric functions using the unit circle, and have also investigated the graphs of the six trigonometric functions. This unit builds on students understanding of the trigonometric functions by having them discover, derive, and work with some of the most important trigonometric identities. Before applying these identities in problem-solving contexts, it is important that students have a conceptual understanding of their origins. For this reason, most of the identities are derived in the tasks in which they are introduced. These identities are then applied to solve problems in which an identity gives an expression a more convenient form. In unit 7, these identities are revisited in the context of solving trigonometric equations. ENDURING UNDERSTANDINGS: An identity is a statement that is valid for all values of the variable for which the expressions in the equation are defined. Trigonometric identities are valuable in a wide variety of contexts because they allow for expressions to be represented in more convenient forms. KEY STANDARDS ADDRESSED: MA3A5. Students will establish the identities below and use them to simplify trigonometric expressions and verify equivalence statements. tan sin cos cot cos sin sec cos csc sin sin cos tan sec cot csc sin sin cos cos sin cos cos cos sin sin sin sin cos cos cos sin Unit 6: Page 3 of 3

4 MA3A6. Students will solve trigonometric equations both graphically and algebraically. a. Solve trigonometric equations over a variety of domains, using technology as appropriate. b. Use the coordinates of a point on the terminal side of an angle to express x as rcosθ and y as rsinθ. d. Apply the law of sines and the law of cosines. MA3A7. Students will verify and apply ½ absinc to find the area of a triangle. RELATED STANDARDS ADDRESSED: MA3A. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured in degrees and radians, including but not limited to 0, 30, 45, 60, 90, their multiples, and equivalences. b. Understand and apply the six trigonometric functions as functions of general angles in standard position. c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position. d. Understand and apply the six trigonometric functions as functions of arc length on the unit circle. e. Find values of trigonometric functions using the unit circle. MA3P. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. MA3P. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. MA3P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. MA3P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. Unit 6: Page 4 of 3

6 Vocabulary and formulas: Identity: an equation that is valid for all values of the variable for which the expressions in the equation are defined. Reciprocal Identities: and cot tan cos sec, sin csc, tan cot, sec cos, csc sin, Quotient Identities: tan sin cos, cot cos sin Pythagorean Identities: sin cos, tan sec, and cot csc Sum & Difference Identities: sin sin cos cos sin and cos cos cos sin sin Double Angle Identities: sin sin cos and cos cos sin Trigonometric Equation: An equation containing one of the basic six trigonometric functions. Trigonometric Function: A function containing one of the basic six trigonometric functions. Unit 6: Page 6 of 3

7 DISCOVERING THE PYTHAGOREAN IDENTITIES LEARNING TASK: An identity is an equation that is valid for all values of the variable for which the expressions in the equation are defined. You should already be familiar with some identities. For example, in Mathematics I, you learned that the equation x y x y x y is valid for all values of x and y.. You will complete the table below by first randomly choosing values for the x s and y s, then evaluating the expressions x y and x y x y. The first row is completed as an example. a. Since x y x y x y is an identity, what should be true about the relationship between the numbers in the last two columns of each row? b. Complete the table below. x y x y x y x y. An identity is a specific type of equation. Many equations are not identities, however, because an equation is not necessarily true for all values of the involved variables. Of the eight equations that follow, only four are identities. Label the equations that are identities as such and provide a counterexample for the equations that are not identities. a. x 5 x 5 x 5 e. 6 3 x x b. x 5 x 5 f. x y x y c. x x g. 3 3 a b a ab b a b d. 4 x x h. x y x y Unit 6: Page 7 of 3

8 3. In this unit you will investigate several trigonometric identities. This task looks at the Pythagorean Identities, which are three of the most commonly used trigonometric identities, so-named because they can be established directly from the Pythagorean Theorem. In the figure below, the point (x, y) is a point on a circle with radius c. By working with some of the relationships that exist between the quantities in this figure, you will arrive at the first of the Pythagorean Identities y (x, y) c b a x a. Use the Pythagorean Theorem to write an equation that relates a, b, and c. b. What ratio is equal to cos? c. What ratio is equal to sin? d. Using substitution and simplification, combine the three equations from parts a-c into a single equation that is only in terms of. This equation is the first of the three Pythagorean identities. 4. Since the equation from 3d is an identity, it should be true no matter what is. Complete the table below, picking a value for that is in the appropriate quadrant. Use your calculator to round values to the nearest hundredth if the angle you choose is not found on the unit circle. How can you use this data to verify that the identity is valid for the four values of that you chose? Unit 6: Page 8 of 3

9 sin * cos sin cos QI 4 QII QIII QIV * sin sin, cos cos, and so on. This is just a notational convention mathematicians use to avoid writing too many parentheses! Unit 6: Page 9 of 3

10 5. The other two Pythagorean identities can be derived directly from the first. In order to make these simplifications, you will need to recall the definitions of the other four trigonometric functions: tan sin cos cot cos sin sec cos csc sin a. Divide both sides of the first Pythagorean identity by result is the second Pythagorean identity. cos and simplify. The b. Divide both sides of the first Pythagorean identity by result is the third and final Pythagorean identity. sin and simplify. The 6. Since the equations from 5a and 5b are identities, they should be true no matter what is. Complete the table below, picking a value for that is in the appropriate quadrant. Use your calculator to round values to the nearest hundredth if the angle you choose is not found on the unit circle. How can you use this data to verify that identities found in 5a and 5b are both valid for the four values of that you chose? tan sec cot csc QI QII QIII QIV Unit 6: Page 0 of 3

11 THE SUM AND DIFFERENCE IDENTITIES LEARNING TASK: In this task, you will use the sum and difference identities to solve equations and find the exact values of angles that are not multiples of 6 and 4. Before you apply these identities to problems, you will first derive them. The first identity you will prove involves taking the sine of the sum of two angles. sin sin cos cos sin We can derive this identity by making deductions from the relationships between the quantities on the unit circle below. R Q O - P S. Complete the following congruence statements: a. ROP QOS b. RO QO PO SO c. By the? congruence theorem, ROP QOS d. RP QS. Write the coordinates of each of the four points on the unit circle, remembering that the cosine and sine functions produce x- and y- values on the unit circle. a. R = ( cos, sin ) Unit 6: Page of 3

12 b. Q = ( cos, sin ) c. P = (?,?) d. S = ( cos, sin ) 3. Use the coordinates found in problem and the distance formula to find the length of chord RP. 4. a. Use the coordinates found in problem and the distance formula to find the length of chord QS. b. Two useful identities that you may choose to explore later are cos cos and sin sin. Use these two identities to simplify your solution to 4a so that your expression has no negative angles. 5. From d, you know that RP QS. You can therefore write an equation by setting the expressions found in problems 3 and 4b equal to one another. Simplify this equation and solve for sin. Applying one of the Pythagorean Identities will be useful! When finished, you will have derived the angle sum identity for sine. The other three sum and difference identities can be derived from the identity found in problem 5. These four identities can be summarized with the following two statements. sin sin cos cos sin cos cos cos sin sin Recall that so far, you can only calculate the exact values of the sines and cosines of multiples of 6 and 4. These identities will allow you to calculate the exact value of the sine and cosine of many more angles. Unit 6: Page of 3

13 6. Evaluate sin 75 by applying the angle addition identity for sine and evaluating each trigonometric function: sin sin 30 cos45 cos30 sin Similarly, find the exact value of the following trigonometric expressions: a. cos 5 b. sin c. cos(345 ) d. sin 9 Unit 6: Page 3 of 3

14 RIDING ON THE FERRIS WHEEL LEARNING TASK:. Lucy is riding a Ferris wheel with a radius of 40 feet. The center of the wheel is 55 feet off of the ground, the wheel is turning counterclockwise, and Lucy is halfway up the Ferris wheel, on her way up. Draw a picture of this situation with Lucy s position and all measurements labeled.. If the wheel makes a complete turn every.5 minutes, through what angle, in degrees, does the wheel turn each second? 3. Draw a picture showing Lucy s position 0 seconds after passing her position in problem? What height is Lucy at in this picture? 4. Draw a picture showing Lucy s position 0 seconds after passing her position in problem? What height is Lucy at in this picture? 5. Draw a picture showing Lucy s position 40 seconds after passing her position in problem? What height is Lucy at in this picture? 6. Write an expression that gives Lucy s height t seconds after passing her position in problem, in terms of t. 7. In problems 4 and 5, the angle through which Lucy turned was twice that of the problem before it. Did her change in height double as well? Students commonly think that if an angle doubles, then the sine of the angle will double as well, but as you saw in the previous problems, this is not the case. The double angle identities for sine and cosine describe exactly what happens to these functions as the angle doubles. These identities can be derived directly from the angle sum identities, printed here for your convenience: cos cos cos sin sin sin sin cos cos sin 8. Derive the double angle identity for cosine by applying the angle sum identity. cos cos Unit 6: Page 4 of 3

15 9. Derive the double angle identity for sine, also by applying the angle sum identity. sin sin Unit 6: Page 5 of 3

16 WHERE S THE IDENTITY? LEARNING TASK: Recall that a trigonometric identity is a trigonometric equation that is valid for all values of the variables for which the expression is defined. Sometimes it is very difficult to glance at a trigonometric equation and determine if it is an identity or not. For example, in each of the following sets of three equations, only one is an identity. There are a variety of strategies for determining the identity in each set. Your teacher will assign you to one of the following three strategies for identifying each identity: Graphical, Numerical, or Algebraic. You will then present your strategy to your classmates, listen to your classmates present their strategies, and finally you will practice using each strategy. Set : a. sin x tan x sin x b. cos x tan x sin x c. sin x cot x sin x Set : a. tan x sin x sin x cos x b. cot x csc x sin x cos x c. tan x sec x sin x cos x Set 3: a. sin x cos x sin x cos x sec x csc x b. sin x cos x sin x cos x sec x csc x c. cos x sin x sec x csc x Unit 6: Page 6 of 3

17 Graphical The key to spotting identities graphically is to think of each side of the equation as a function. Since the left side of the equation should always produce the same output as the right side, no matter what the input variable is, both sides should look the same graphically. If the two sides have different graphs, then the equation cannot be an identity.. Set : a. In each of the three sets of equations, the right side is the same. Sketch the function made from the right side of the equations in Set in the space provided. b. Now sketch each of the three functions made from the left sides of the equations in Set. Circle the graph that matches the right side, since that graph is from the identity.. Set : a. Sketch the function made from the right side of the equations in Set in the space provided. b. Now sketch each of the three functions made from the left sides of the equations in Set. Circle the graph that matches the right side, since that graph is from the identity. 3. Set 3: a. Sketch the function made from the right side of the equations in Set 3 in the space provided. b. Now sketch each of the three functions made from the left sides of the equations in Set 3. Circle the graph that matches the right side, since that graph is from the identity. Unit 6: Page 7 of 3

18 Numerical By making a table of values that compares the left side of each equation to the right side, we can rule out equations whose left and right sides do not match. Since we know one of the equations is an identity, the equation that we cannot rule out must be the identity. Complete the following tables, using values of your choosing for x, and use the data you collect to decide which equation in each set is an identity.. Set : a. Since the right side of the equation is the same for all three equations in set, first complete the following table of values for the right sides of the equations, so that you can compare these values to the left sides of each equation. x sin x b. Now complete each of the three tables of values for the left sides of the equations in Set, using the same x-values from part a. Circle the table that matches the right side (from part a), since that table is from the identity. x 0 sin x tan x x 0 cosxtanx x 0 sin x cot x Set : a. Since the right side of the equation is the same for all three equations in set, first complete the following table of values for the right sides of the equations, so that you can compare these values to the left sides of each equation. x sin x Unit 6: Page 8 of 3

19 b. Now complete each of the three tables of values for the left sides of the equations in Set. Circle the table that matches the right side (from part a), since that table is from the identity. x tan x sin x x cot x cscx x tanx secx 3. Set 3: a. Since the right side of the equation is the same for all three equations in set 3, first complete the following table of values for the right sides of the equations, so that you can compare these values to the left sides of each equation. x secx cscx b. Now complete each of the three tables of values for the left sides of the equations in Set 3. Circle the table that matches the right side (from part a), since that table is from the identity. x sin x cos x sin x cos x x sin x cos x sin x cos x x cos x sin x Unit 6: Page 9 of 3

20 Algebraic Recall that from the definitions of the trigonometric functions, we get the following fundamental identities: Quotient Identities sin x tan x cos x sin x Reciprocal Identities csc x csc x sin x cot x sin x cos x cos x sec x sec x cos x tan x cot x cot x tan x We can use these identities to rewrite trigonometric expressions in different forms. For example, in the following equation, we can rewrite the left side using two of the above identities, eventually making the left side identical to the right side, thus proving that the original equation is an identity. tan xcot x sin x cos x cos x sin x In each of the following three sets of equations, there is exactly one identity. Attempt to use the above identities to rewrite one side of each equation so that it matches the other side. Since there is only one identity, this will only be possible for one of the equations in each set. Circle that equation, since it is the identity. Set : sin x tan x sin x cos x tan x sin x sin x cot x sin x tan x sin x sin x cos x cot x Set : csc x sin x cos x tan x sec x sin x cos x sin x cos x sin x cos x sec x csc x Set 3: sin x cos x sec x sin x cos x csc x cos x sin x sec x csc x Unit 6: Page 0 of 3

21 Using All Three Approaches Set 4: Graphical a. 3 3sin x cos x b. sin x 3sin x cos x c. 3sin x 3sin x cos x a. In each of the three sets of equations, the right side is the same. Sketch the function made from the right side of the equations in Set 4 in the space provided, with the aid of a graphing utility. b. Now sketch each of the three functions made from the left sides of the equations in Set. Circle the graph that matches the right side, since that graph is from the identity. Unit 6: Page of 3

22 Set 5: Numerical a. sin x tan x cos x sin x b. tan x tan x cos x sin x c. cos xsin x tan x cos x sin x a. Since the right side of the equation is the same for all three equations in set, first complete the following table of values for the right sides of the equations, so that you can compare these values to the left sides of each equation. x tan x cos x sin x b. Now complete each of the three tables of values for the left sides of the equations in Set, using the same x-values from part a. Circle the table that matches the right side (from part a), since that table is from the identity. x sin x x tan x x cos xsin x Unit 6: Page of 3

23 Set 6: Algebraic Attempt to use the reciprocal and quotient identities to rewrite one side of each equation so that it matches the other side. Since there is only one identity, this will only be possible for one of the equations. Circle that equation, since it is the identity. tan x sec x cos x sin x cot x csc x csc x sin x tan x sec x sec x sin x Unit 6: Page 3 of 3

24 ESTABLISHING IDENTITIES LEARNING TASK: As you saw in Where s the Identity?, trigonometric identities can be difficult to recognize, but by thinking graphically, numerically, and algebraically, you can gain valuable insight as to whether equations are identities or not. Numerical and graphical information is not enough to verify an identity, however. Identities can be established algebraically by rewriting one side of the equation until it matches the expression on the other side of the equation. Rewriting is often done by applying a basic trigonometric identity, so the identities you have already established are listed here for reference. Quotient Identities Reciprocal Identities tan sin cos cos sec sec cos cot cos sin sin csc csc sin tan cot cot tan Pythagorean Identities sin cos tan sec cot csc sin cos tan sec csc cot cos sin sec tan cot csc Sum & Difference Identities sin sin cos cos sin Double Angle Identities sin sin cos cos cos cos sin sin cos cos sin Unit 6: Page 4 of 3

25 When establishing identities, it is important that each equation that you write is logically equivalent to the equation that precedes it. One way to ensure that all of your equations are equivalent is to work with each side of the equation independently. The following problem provides an example of how failing to work with each side of an equation independently can produce what appears to be a proof of a statement that isn t true. This example should serve as a reminder as to why you should work with each side of an equation independently when establishing identities.. As explained above, a helpful guideline when establishing identities is to change each side of the equation independently. Circle the two lines that were produced by failing to abide by this guideline, in the faulty proof below.. Explain why reasoning. sin cos sin cos is not an identity, using either graphical or numerical When establishing the following identities, keep the following two general rules of thumb in mind. They will not always lead to the most efficient solution, but they are usually beneficial when help is needed. Begin working on the most complex side, because it is usually easier to simplify an expression rather than make it more complex. When no other solution presents itself, rewrite both sides in terms of sines and cosines. Establish the following identities by rewriting the left, right, or both sides of the equation independently, until both sides are identical. Unit 6: Page 5 of 3

26 For problems 3-5, apply either the quotient or reciprocal identities. 3. sin xcot x cos x 4. csc tan cos 5. sec csc tan For problems 6-0, apply the Pythagorean Identities. 6. tan w 8 sec w cos 3sin 3 cos 8. sin tan cos sec 9. sec sin sec For problems 0-, apply the sum, difference, or double angle identities. 0. cos cot tan cos sin. sin cot tan sin cos. 3. cot xcot x cot x 4 4 cos sin cos Unit 6: Page 6 of 3

27 Solving Equations Learning Task: As you have just seen with identities, two trigonometric expressions can always, sometimes, or never be equal. Identities occurred when the two expressions were always equal. Now, in this task, we consider the sometimes and never situations. We will continue to look at these situations algebraically, graphically, and numerically. Consider this situation: The rabbit population in a national park rises and falls each year, it reaches its minimum of 5000 rabbits in January. By July, as the weather warms up and food grows more abundant, the population triples in size. By the following January, the population again falls to 5000 rabbits, completing the annual cycle. Using the given data for the two-year cycle of rabbit population, write a formula for the number of rabbits, R, as a function of time in months, for 0 < t < 4. t R What is the midline for this function?. When is the rabbit population at the midline value? 3. How do you find all the values of t when R is the midline value? Algebraically? Graphically? Using a table? Why did you pick the method you picked? 4. When is the rabbit population,000? How did you find these values? 5. When is the rabbit population more than,000? How do you know? 6. When is the rabbit population 4000? How do you know? Unit 6: Page 7 of 3

28 The NON-Right Triangle Learning Task: The sine cosine and tangent functions are useful when solving within a right triangle situation. However, not all situations can be defined as right triangles. Therefore, we need other relationships between sides and angles of non-right triangles to solve any problem situation we may be presented with. In this task, you will do so by establishing that result in the Law of Sines and the Law of Cosines. The Law of Sines Given triangle ABC with altitude from A, consider the following relationship: A b h c C a B Sin C =? Sin B =? Solve each equation for h. If both equations now equal h, they are equal to each other, when you solve and re-write, you Sin B SinC should have:, you can solve a similar situation with side a and angle A which leads b c to the extended proportion known as the Law of Sines: Sin A Sin B SinC a b c The Law of Sines is useful when sides and opposite angles are given. Think back to the triangle congruence theorems you learned in Accelerated Mathematics I, which of those apply here? Unit 6: Page 8 of 3

29 Suppose an aerial tram starts at a point one half mile from the base of a mountain whose face has a 60 o angle of elevation. The tram ascends at an angle of 0 o. What is the length of the cable needed to span the distance from T to A? The Law of Cosines Given triangle ABC with altitude from A, consider the following relationship: A b h c C x a - x B a Since there is a right angle where the altitude intersects side BC, both of the triangles formed are right triangles. That means that the sides must satisfy the Pythagorean Theorem. Write the appropriate Pythagorean statement for the right triangle with sides c, h, and a x. Expand the squared binomial. But, we want a relationship between the sides and the angles of triangle ABC, this means we need to get rid of the x and h. What do we know about the relationship between x and h. Use that relationship to replace them in your equation. Unit 6: Page 9 of 3

30 We have almost gotten the equation in terms of the sides and angles of triangle ABC, but now we must find a replacement for x. What relationships can you find with the x, h, c, triangle that might help? How would you substitute that relationship into your equation? Thus, the Law of Cosines is a + b abcos C = c for the sides a, b, c and opposite angle C for any triangle. The Law of Cosines is useful when given two sides and the included angle or when given three sides of a triangle. Think back to the triangle congruence theorems you learned in Accelerated Mathematics I, which of those apply here? For background information that may be helpful with this task visit: To help predict eruptions from the volcano Mauna Loa on the island of Hawaii, scientists keep track of the volcano s movement by using a super triangle with vertices on the three volcanoes Mauna Loa, Mauna Kea, and Hualalai. For example, in recent year, Mauna Loa moved 6 inches as a result of increasing internal pressure. If the distance from Mauna Loa to Mauna Kea is.479 miles and the distance from Mauna Kea to Haulalai is 8.43 miles and the angle at Mauna Kea is o, how far is it from Mauna Loa to Haulalai? (GPS receivers are use to constantly monitor these line lengths) The Area of Any Triangle Given triangle ABC with altitude from A, consider the following relationship: A b h c C a B Using the same idea as we have for the Law of Sines and Cosines, we will now investigate the area of any triangle. Unit 6: Page 30 of 3

31 What is the area of the above triangle? Using what you know about trigonometry, how could you find a value to substitute for h? Now, make your substitution into the area formula. Can you generalize the formula in words? A real estate salesperson wants to find the area of a triangular lot. The surveyor takes measurements for her. He finds that the two sides are 74.3 m and 34.6 m, and the angle between them is 40. o. What is the area of the lot? The distance from a point to a line The final application of our trigonometry is an alternate method from the one you developed in Accelerated Mathematics I of finding the distance between a point and a line. Explore this relationship using this website: Unit 6: Page 3 of 3

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