Week 2 - From Acute Angles to Any Angle

Transcription

1 Week 2 - From Acute Angles to Any Angle The trigonometric functions in the previous lesson were defined for an acute angle only and based on a right triangle. This is a limiting situation for applications. The next step is to expand trigonometry to any angle. Overview Our goals in this weekly lesson with 4 parts are to: Explore angles in degree measure. Define and use radian measure. Define and use the trigonometric functions of any angle. Use reference angles. Geometry knowledge needed: Obtuse angle, supplementary angles, degree measure Circles, radius, circumference, arc, central angle Algebra knowledge needed: Origin, coordinate plane, Cartesian coordinates, π Cancelling, reducing, simplifying, negative fractions Degree Measure The following diagrams review briefly the types of angles we may use.

2 2 In the last lesson, you had to find a complementary angle. That is, if you have an angle of 32, then its complement is = 58. Now we may need to find the supplement of an angle. If I want the supplement of 32, then it is = 148. At this point it seems simple, and it is. We are, however, building up to using any angle. For that, we need more terms. Let s look at the terms for an angle s parts and place that angle in standard position on a coordinate plane with the x-axis, y-axis, and origin. Why is it important to put an angle in standard position? Because we want to draw upon our knowledge of algebra and geometry at the same time. This is called Analytic Geometry and is part of your algebra background. The next diagrams show us the ideas of negative angles and coterminal angles. Notice that a negative angle is just a different direction from the positive angle. Coterminal angles can happen with a positive angle and a negative angle where they share the same initial and terminal sides. But two positive angles can also share sides. Can you picture that? For that to happen, we would have to have the terminal side go more than one revolution, or more than 360. For example, if α = 50 and β = 410, which is , α and β are coterminal angles. Another angle that would be coterminal with α is 770, because it is So how many angles could be coterminal with α? Infinitely many.

3 3 You need to be able to picture angles in your mind as well as sketch them in standard position. To get used to where the terminal side of an angle ends up in the plane and to notice what we call quadrant angles (those whose sides reside on quadrant lines), please look at the following activity: á Now do the problem set for Degree Measure Problems for Degree Measure 1. For each of the following angles, find the complement and the supplement. a. 49 b. 73 c. 118 d Sketch each of the following in standard position, marking the angle as you have seen in the diagrams in this section. a. 100 b. 295 c. 210 d. 60 e Give an angle that is coterminal with each of the following. Sketch the given angle in standard position first. a. 135 b. 54 c Designate the following angles as acute, obtuse, quadrant, or none of these. Sketch the given angle in standard position first. a. 166 b. 270 c. 200

4 4 Answers to Problems for Degree Measure 1. a. b. c. d. complement = = = = 75 Not positive, so no complement supplement = = = = = = =260 Or = = =-460 There are many more possibilities. We will usually use positive angles, unless a negative angle has a particular meaning.

5 5 4. a. This is an obtuse angle since the measure is between 90 and 180. b. This is a quadrant angle over three quadrants. c. This is none of the above.

6 6 Radian Measure We now go from triangles to circles. We have seen that degree measure for a central angle is a way of stating the part of a revolution that the angle makes, since 1 is just 1/360 th of a revolution. However, radian measure of an angle looks at the ratio of the arc length in between the sides of the angle and the radius of a circle. We say the arc is subtended by the angle. First, we ll look at a circle centered at the origin with any radius r, the arc length denoted by s, and the angle denoted by θ (theta). The radian measure of an angle is found by where θ is always in radians, which is a real number. (I have not made this formula a red letter idea because we will not use it that much. You might need to know it for a test, however.) The unit of 1 radian is when s = r, as shown in the second diagram. You can see about how big 1 radian is in terms of what part of a revolution the angle makes. Let s look at one full revolution for θ. If we do that, then arc length s will be the same as the circumference of the circle. For one full revolution, radians. There are several observations we can make from the above formula for θ. Radian measure uses the properties of the circle, but degree measure is historically an arbitrary measure (to divide the central angle into 360 parts or degrees). The formula actually finds how any radii are in the arc length, therefore the name radian. No matter what the size of the circle, there are still only radians in one revolution. For a specific value of θ, the ratio s/r will be the same no matter how large of a circle you might have. One full revolution is 2π radians for the central angle, but not always the circumference of the circle. One full revolution is also 360. It is useful to note then that This relationship gives us the basis for converting from one unit of measure to another.

7 7 You can memorize a formula in the book, but I will use the above relationship (which is easier to remember) and reasoning. Example 1: Convert 45 to radians. (If you find 1 first, then you can find 45.) Using the above relationship and solving for 1,. Then and. Since we want 45, we multiply both sides:, which simplifies to =. All of these steps can be shortened to and then reduced. With practice this is quick work, but make sure you understand the order of reasoning. Let s visualize the 45 and the π/4 radians. We will use a circle with radius 1 to make it easier. Notice that π is the length of the semicircle here, so that π/4 is ¼ of that semicircle, just as 45 is ¼ of 180. Notice we are still using standard position for the angle. Example 2: Convert to degrees. (If you can find 1 rad first, then you can find rad.) We start with the relationship and solve for 1 radian. We want, so Reducing and simplifying,

8 8 This can be done in one step with the reasoning here, then don t try to do it all at once. and reduced. If you don t see all of To visualize what 5π/6 rad looks like, we will use a circle of radius 1 again. See that 150 is of 180. or of the semicircle here, just as To help you with this visual, click on the following site, click on go to student worksheet, and ignore everything except the radian and degree measures as you change the arc length: Note: These problems illustrate radian measure where π is part of the number. Because it is an exact number, we will keep π unless it cancels in the simplifying or you are asked to round off an answer to so many places. Applications for I am focusing on main topics and skills here, so you can look in your textbook for applications. Linear and angular speed are common examples. Their formulas are in the book. The important thing to remember in those problems is that θ is in radians. So, if you are given degrees, then you must convert to radians in order to use the formula. á Now do the problem set for Radian Measure

9 9 Problems for Radian Measure 1. To the nearest two decimal places, how many degrees are in 1 radian? 2. A circle has a central angle of θ, a radius of 3.5, and the angle subtends an arc of 10. a. What is the measure of θ in radians to two decimal places? b. What is the circumference of this circle to two decimal places? c. Sketch a picture of the angle making your arc the appropriate length in comparison to the radius. 3. Convert each of the following angles to radian measure. Sketch in standard position. a. 240 b. 135 c. 330 d Convert each of the following angles to degree measure. Sketch in standard position. a. 2π/3 b. 7π/6 c. 7π/4 d. 10

10 10 Answers for Problems for Radian Measure 1. On a calculator, in degree mode, put in nd π enter. On a graphing calculator, π is usually a secondary position. If you do not have a 2 nd key, you may need to use inv key. 2. a. b. C = 2πr c. C = 2π 3.5 C = 7π or rad 3. a. b. c. d. 4. a. b. c.

11 11 (4. continued) a. b. c. d.

12 12 Trigonometric Functions of Any Angle Get used to thinking in terms of π and where the terminal side of the angle ends up. Look at the following diagram carefully. The points lie on the terminal side of the specified angle. Notice in the special angles in quadrant I that we found in the special right triangles. What do you notice about the other angles in quadrants II, III, and IV in relation to the special angles? Notice the geometric pattern that develops. Let θ be an angle in standard position with (x, y) a point on the terminal side of θ, r units from the origin such that. We now define the trigonometric functions for any angle. Observations. Notice the use of a right triangle and the Pythagorean Theorem to get r, if not given. The definitions for the trigonometric functions of an acute angle are preserved. We can still find the three functions in black by using reciprocal identities. Our definitions depend on the coordinates (x, y) and the point s distance from the origin. θ can be a negative angle or greater than 360 or 2π radians. The point on the terminal side of the angle is also on a circle of radius r, but we do not need to draw a circle.

13 Example 1: Let (-5,-12) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ. Let s sketch the angle in standard position with the given point. We need to find r first. 13 (r is always positive) Now we use the definitions. Notice we now have some negative values for the functions. Example 2: Given that and, find sin θ and sec θ. We have no point, so how do we determine where the point on the terminal side is found? We know that. Should we make the 4 negative or the 3 negative? Since we also know that cos θ > 0 (positive) and, then we have a positive x- value because r is always positive. So x = 3 and y = -4. Make a sketch. (You could use a reciprocal identity.) Example 3: Find the sine function for the four quadrant angles. Sketch the four quadrants and pick a point on the terminal side of each angle. We will use a radius of 1 for simplicity. Now we use the definition for sine for each angle. á Now do the problem set for Trigonometric Functions of Any Angle

14 14 Problems for Trigonometric Functions of Any Angle 1. We have noticed that the function values for any angle depend on (x,y). With that and the definitions in mind, complete the table as to whether each function is positive (+) or negative (-) in each quadrant. Quadrant Sin θ Cos θ Tan θ Csc θ Sec θ Cot θ I II III IV 2. Let (4,-3) be a point on the terminal side of θ. Find all six trigonometric functions for θ. Make a sketch of the angle in standard position. 3. Make a sketch of the quadrant angles as in the lesson. Find the cosine of 4. Given that and, find cos θ and sin θ. Make a sketch. (Hint: remember there is more than one way for the tangent fraction to be positive.)

15 15 1. Answers to Problems for Trigonometric Functions of Any Angle Quadrant Sin θ Cos θ Tan θ Csc θ Sec θ Cot θ I II III IV (4,-3) x = 4 and y = We will again use a radius of 1 for simplicity. We know that. 4. We know by definition that. So y =12 and x = 5, or y = -12 and x = -5. Since and (negative), then y must be negative. Therefore, y = -12 and x = -5.

16 16 Reference Angles So far, we have not worked with very many particular values for θ, except for some very special values. Since there are infinitely many angles (positive, negative, larger than 360 or 2π radians, etc.), it would really be very time consuming if we had to treat every angle as very different from the rest. Thus, we have what we call reference angles. It is too long to show here, but geometry theorems about angles and triangles along with the definitions of trigonometric functions prove that the functions of θ are the same values as functions of the reference angle except for some signs. If θ is an angle in standard position, then its reference angle α (alpha) is the acute angle formed by the terminal side of θ and the horizontal axis. Since all angles in quadrant I are acute, we have the following situations. Notice that θ is a positive angle (counterclockwise). So if you are given a negative angle, find the positive angle that is coterminal with it. It s easier that way. You could memorize the formula for each quadrant; but if you can picture the acute angle made with the x-axis and then decide what you need to do to get it, you won t get them mixed up. Example1: Find the reference angle for each part. a. 300 b. 2.3 rad c a. Sketch in standard position and then look at what you need to do to get α. Subtract 300 from 360. So α = = 60. The reference angle for 300 is 60.

17 17 b. We need to see what quadrant 2.3 rad is in. We know that π rad is 3.14 and rad. Since 2.3 is between 1.57 and 3.14, then θ = 2.3 is in quadrant II. α = π 2.3 =.84 rad The reference angle for 2.3 rad is.84 rad. c. We need to find an angle that is coterminal with Go counterclockwise one full revolution from the terminal side of -135 to get a coterminal positive angle. Notice that the reference angle is still the same. α = = 45 The reference angle for -135 is 45. Note: The signs of the trigonometric functions depend on what quadrant that the terminal side of θ lies in, so we say that θ lies in quadrant. In Example 1, 300 lies in quadrant IV, 2.3 rad lies in quadrant II, and -135 lies in quadrant III. Example 2: Find. In what quadrant does lie? Since lies in quadrant III. Make a sketch. Get used to thinking in terms of π. (The radian measure here is equivalent to 30.) Since is in quadrant II, then the cosine is negative. Just as you recognize 30, 45, and 60 as special angles, then you need to recognize their radian equivalents as. Also, certain multiples of these angles will have the special angles as reference angles. You will need to think, think, think, and practice! á Now do the problem set for Reference Angles.

18 18 Problems for Reference Angles 1. Find the reference angle for each of the following. Make a sketch of the angle in standard position. a. 150 b. 225 c Find the reference angle for each of the following. Make a sketch of the angle in standard position. a. 7π/4 b. 4π/3 c Find sin 210. Make a sketch of the angle in standard position. 4. Find cos. Make a sketch of the angle in standard position. 5. List an angle in each of quadrants II, III, and IV that has 30 as the reference angle. 6. List an angle in each of quadrants II, III, and IV that has as the reference angle. 7. Repeat the directions in #5 and #6 for 45 and. (Hint: Are 45 and the same angle?) 8. Repeat the directions in #5 and #6 for 60 and.

19 19 Answers to Problems for Reference Angles 1. a. b. c. α = α = α = α = 30 α = 45 α = a. b. c. In part c, to find in what quadrant 5.42 lies, you had to know approximate values for π/2, π, 3π/2, and 2π. You could do all calculations on your calculator with the π key, but we keep answers in terms of π to be exact, unless otherwise stated. In part c, 5.42 was not exact, so we approximated 2π. 3. α = α = 30 Since 210 is in quadrant III and sine depends on the y-value, sin 210 is negative. (Remember the sin 210 = right triangle.)

20 20 4. Since lies in quadrant IV, the cosine depends on x and is positive. (Remember the right triangle.) 5. The three situations are as follows. Θ = θ = θ = Θ = 150 θ = 210 θ = 330 (The sketches guide your thinking.) 6. The sketches are the same except α =. θ = π - θ = π + θ = 2π - θ = θ = θ = θ = θ = θ = These are equivalent to the degree measures in # = rad. The three situations are as follows. θ = , θ = π - θ = , θ = π + θ = , θ = 2π - θ = 135, θ = θ = 225, θ = θ = 315, θ = θ = θ = θ =

21 = rad. The three situations are as follows. II. θ = , θ = π - III. θ = , θ = π + IV. θ = , θ = 2π - θ = 120, θ = θ = 240, θ = θ = 300, θ = For more help with reference angles and to see how the functions change in value, go to

22 22 Putting It All Together Before going on to another topic, do the following to reinforce ideas and to build up familiarity with these trigonometric functions. 1. From last week s lesson, sketch the special triangles and Mark all angles and sides. Write the sine, cosine, and tangent values for 30, 60, and 45. Think while you are doing this, but move as quickly as you can. 2. From this week s lesson, sketch the coordinate axes and mark points on the axes for the quadrant angles 0 or 0 rad, 90 or rad, 180 or π rad, and 270 or. Then write the sine, cosine, and tangent for each of the quadrant angles 3. Check your answers to #1 and #2 on the next page. Then complete the chart on the page after answers to #1 and #2. One row is completed to show what is required. Think about these concepts as you go, but move as quickly as you can.

23 23 Answers to Putting It All Together 1. Sin 30 = sin 60 = sin 45 = Cos 30 = cos 60 = cos 45 = Tan 30 = tan 60 = tan 45 = 2. This is the circle with a radius of r = 1. Sin 0 = sin 0 = sin 90 = sin Cos 0 = cos 0 = cos 90 = cos Tan 0 = tan 0 = tan 90 = tan is undefined sin 180 = sin π = sin 270 = sin cos 180 = cos π = cos 270 = cos tan 180 = tan π = tan 270 = tan is undefined 3. Answers follow on next page after the chart.

24 TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES in Related Exact Values of in degrees in radians stand.pos. angle sin cos tan csc sec cot

25 TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES in Related Exact Values of in degrees in radians Stand.pos angle sin cos tan csc sec cot undefined 1 undefined undefined 1 undefined undefined -1 undefined undefined -1 undefined π undefined 1 undefined Use the and right triangles, as well as points on the axes (circle with radius 1) for the quadrant angles, to help.