Trigonometric Functions of Any Angle MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011
Objectives In this lesson we will learn to: evaluate trigonometric functions of any angle, find reference angles, evaluate trigonometric functions of real numbers.
Arbitrary Angle x,y r Θ x
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
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 =
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 =
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 = 12 13 tan θ = 5 12 cot θ = 12 5 sec θ = 13 12 csc θ = 13 5
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
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 =
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 =
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 = 15 17 cos θ = 8 17 tan θ = 15 8 cot θ = 8 15 sec θ = 17 8 csc θ = 17 15
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.
Examples (1 of 4) Θ' Θ x θ = π θ (radians) θ = 180 θ (degrees)
Examples (2 of 4) Θ x Θ' θ = θ π (radians) θ = θ 180 (degrees)
Examples (3 of 4) Θ x Θ' θ = 2π θ (radians) θ = 360 θ (degrees)
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,
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 = 360 309 = 51 θ = 215, θ = 7π 6, θ = 11.6,
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 = 360 309 = 51 θ = 215, θ = 215 180 = 45 θ = 7π 6, θ = 7π 6 π = π 6 θ = 11.6, θ = 4π 11.6 0.9664
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.
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 300 405 7π 6 10π 3
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 300 60 405 7π 6 10π 3
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 300 60 3 2 405 7π 6 10π 3 1 2 3
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 300 60 3 2 405 45 7π 6 10π 3 1 2 3
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 300 60 3 2 405 45 2 7π 6 10π 3 2 1 2 2 1 2 3
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 300 60 3 2 405 45 2 7π 6 10π 3 π 6 2 1 2 2 1 2 3
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 300 60 3 1 2 2 3 405 45 2 2 2 2 1 π 6 1 2 3 3 2 3 7π 6 10π 3
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 300 60 3 1 2 2 3 405 45 2 2 2 2 1 7π π 6 1 2 3 3 2 3 6 10π 3 π 3
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 300 60 3 1 2 2 3 405 45 2 2 2 2 1 7π π 6 6 1 2 3 3 2 3 3 10π 3 π 3 3 2 1 2
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 =
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 9 + 1 = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ =
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 9 + 1 = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ = 10
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 = 10 10 cot 2 θ + 1 = csc 2 θ 9 + 1 = csc 2 θ cos θ = tan θ = cot θ = 3 sec θ = csc θ = 10
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 9 + 1 = csc 2 θ sin θ = 10 10 cos θ = 3 10 10 tan θ = 1 3 cot θ = 3 10 sec θ = 3 csc θ = 10
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 =
Using a Calculator Use a calculator to evaluate each of the following trigonometric functions. Round your answers to four decimal places. sec 220 = 1.3054 cos ( 170 = tan π ) = 9 csc 0.39 =
Using a Calculator Use a calculator to evaluate each of the following trigonometric functions. Round your answers to four decimal places. sec 220 = 1.3054 cos ( 170 = 0.9848 tan π ) = 0.3640 9 csc 0.39 = 2.6303
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.
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.
Homework Read Section 4.4. Exercises: 1, 5, 9, 13,..., 89, 93