Trigonometric Functions of Any Angle

Transcription

1 Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle, find reference angles, evaluate trigonometric functions of real numbers.

3 Arbitrary Angle x,y r Θ x

4 Trigonometric Functions Definition Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and r = x 2 + y 2 0. sin θ = y r cos θ = x r tan θ = y x, x 0 cot θ = x y, y 0 sec θ = r x, x 0 csc θ = r y, y 0

5 Example Suppose the point with coordinates ( 12, 5) is on the terminal side of angle θ. Find the values of the six trigonometric functions of θ. sin θ = cos θ = tan θ = cot θ = sec θ = csc θ =

6 Example Suppose the point with coordinates ( 12, 5) is on the terminal side of angle θ. Find the values of the six trigonometric functions of θ. We see that r = ( 12) 2 + (5) 2 = 13. sin θ = cos θ = tan θ = cot θ = sec θ = csc θ =

7 Example Suppose the point with coordinates ( 12, 5) is on the terminal side of angle θ. Find the values of the six trigonometric functions of θ. We see that r = ( 12) 2 + (5) 2 = 13. sin θ = 5 13 cos θ = tan θ = 5 12 cot θ = 12 5 sec θ = csc θ = 13 5

8 Signs of the Trigonometric Functions Quadrant II Quadrant I Quadrant II Quadrant I Π 2 Θ Π x 0, y 0 0 Θ Π 2 x 0, y 0 sin Θ: cos Θ: tan Θ: sin Θ: cos Θ: tan Θ: x x x 0, y 0 Π Θ 3Π 2 x 0, y 0 3Π 2 Θ 2Π tan Θ: cos Θ: sin Θ: tan Θ: cos Θ: sin Θ: Quadrant III Quadrant IV Quadrant III Quadrant IV

9 Example Given that cos θ = 8/17 and tan θ < 0 find the values of the six trigonometric functions of θ. sin θ = cos θ = tan θ = cot θ = sec θ = csc θ =

10 Example Given that cos θ = 8/17 and tan θ < 0 find the values of the six trigonometric functions of θ. We see that θ must be an angle in Quadrant IV. sin θ = cos θ = tan θ = cot θ = sec θ = csc θ =

11 Example Given that cos θ = 8/17 and tan θ < 0 find the values of the six trigonometric functions of θ. We see that θ must be an angle in Quadrant IV. sin θ = cos θ = 8 17 tan θ = 15 8 cot θ = 8 15 sec θ = 17 8 csc θ = 17 15

12 Reference Angles Definition Let θ be an angle in standard position. Its reference angle is the acute angle θ formed by the terminal side of θ and the horizontal axis. Remark: we can find the values of the trigonometric functions of angles greater than 90 from the values at their corresponding reference angles.

13 Examples (1 of 4) Θ' Θ x θ = π θ (radians) θ = 180 θ (degrees)

14 Examples (2 of 4) Θ x Θ' θ = θ π (radians) θ = θ 180 (degrees)

15 Examples (3 of 4) Θ x Θ' θ = 2π θ (radians) θ = 360 θ (degrees)

16 Examples (4 of 4) Find the reference angles corresponding to each of the following angles. It may help if you sketch θ in standard position. θ = 309, θ = 215, θ = 7π 6, θ = 11.6,

17 Examples (4 of 4) Find the reference angles corresponding to each of the following angles. It may help if you sketch θ in standard position. θ = 309, θ = = 51 θ = 215, θ = 7π 6, θ = 11.6,

18 Examples (4 of 4) Find the reference angles corresponding to each of the following angles. It may help if you sketch θ in standard position. θ = 309, θ = = 51 θ = 215, θ = = 45 θ = 7π 6, θ = 7π 6 π = π 6 θ = 11.6, θ = 4π

19 Evaluating Trigonometric Functions To find the value of a trigonometric function of any angle θ: 1 Determine the function value for the associated reference angle θ. 2 Depending on the quadrant in which θ lies, affix the appropriate sign to the function value.

20 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π 3

21 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π 3

22 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π

23 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π

24 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π

25 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π 6 10π 3 π

26 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π π 6 10π 3

27 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π π π 3 π 3

28 Examples Evaluate the sine, cosine, and tangent of each of the following angles without using a calculator. θ θ sin θ cos θ tan θ π π π 3 π

29 Using Trigonometric Identities Using the trigonometric identities, find the values of the trigonometric functions for an angle θ whose terminal side is in Quadrant II and for which cot θ = 3. sin θ = cos θ = tan θ = cot θ = sec θ = csc θ =

30 Using Trigonometric Identities Using the trigonometric identities, find the values of the trigonometric functions for an angle θ whose terminal side is in Quadrant II and for which cot θ = 3. sin θ = cot 2 θ + 1 = csc 2 θ = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ =

31 Using Trigonometric Identities Using the trigonometric identities, find the values of the trigonometric functions for an angle θ whose terminal side is in Quadrant II and for which cot θ = 3. sin θ = cot 2 θ + 1 = csc 2 θ = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ = 10

32 Using Trigonometric Identities Using the trigonometric identities, find the values of the trigonometric functions for an angle θ whose terminal side is in Quadrant II and for which cot θ = 3. sin θ = cot 2 θ + 1 = csc 2 θ = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ = 10

33 Using Trigonometric Identities Using the trigonometric identities, find the values of the trigonometric functions for an angle θ whose terminal side is in Quadrant II and for which cot θ = 3. cot 2 θ + 1 = csc 2 θ = csc 2 θ sin θ = cos θ = tan θ = 1 3 cot θ = 3 10 sec θ = 3 csc θ = 10

34 Using a Calculator Use a calculator to evaluate each of the following trigonometric functions. Round your answers to four decimal places. sec 220 = cos ( 170 = tan π ) = 9 csc 0.39 =

35 Using a Calculator Use a calculator to evaluate each of the following trigonometric functions. Round your answers to four decimal places. sec 220 = cos ( 170 = tan π ) = 9 csc 0.39 =

36 Using a Calculator Use a calculator to evaluate each of the following trigonometric functions. Round your answers to four decimal places. sec 220 = cos ( 170 = tan π ) = csc 0.39 =

37 Solving Equations Find two angles which solve the equation sec θ = 2. Do not use a calculator and express the angles in radians and degrees.

38 Solving Equations Find two angles which solve the equation sec θ = 2. Do not use a calculator and express the angles in radians and degrees. The equation implies that cos θ = 1/2. From our knowledge of the special angles on the unit circle we know cos 2π 3 = 1 2 and cos 4π 3 = 1 2. Thus θ = 2π 3, 4π 3 (radians) and θ = 120, 240.

39 Homework Read Section 4.4. Exercises: 1, 5, 9, 13,..., 89, 93