Unit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6)

Transcription

1 Unit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Understand the need for restricting the domains of trig functions in order to have an inverse function. Evaluate and graph inverse trig functions. Find compositions of trig functions. Inverse Trig / 1

2 Domain Restricted Sine Function The sine function is not One-to-One so it cannot have an inverse function. If the domain of y = sin x is restricted to the interval [ π, π ], however, the restricted function is One-to-One and has an inverse function. Inverse Trig 3 / 1 Inverse Sine Function Notation y = sin 1 x or y = arcsin x Domain [ 1, 1] Range [ π, π ] Inverse Trig 4 / 1

3 Inverse Sine and the Unit Circle Recall that sin t is the y-coordinate of the point on the unit circle corresponding to the angle (or arc length) t. Because the range of the inverse sine function is restricted to [ π, π ], the angle measures that output from the inverse sine are located on the right half of the unit circle. Inverse Trig 5 / 1 Example 1 ( ) a) arcsin b) sin 1 1 c) arcsin Inverse Trig 6 / 1

4 Domain Restricted Cosine Function The cosine function is not One-to-One so it cannot have an inverse function. If the domain of y = cos x is restricted to the interval [0, π], however, the restricted function is One-to-One and has an inverse function. Inverse Trig 7 / 1 Inverse Cosine Function Notation y = cos 1 x or y = arccos x Domain [ 1, 1] Range [0, π] Inverse Trig 8 / 1

5 Inverse Cosine and the Unit Circle Recall that cos t is the x-coordinate of the point on the unit circle corresponding to the angle (or arc length) t. Because the range of the inverse cosine function is restricted to [0, π], the angle measures that output from the inverse cosine are located on the upper half of the unit circle. Inverse Trig 9 / 1 Example a) cos 1 1 ( ) 3 b) arccos c) cos 1 ( ) Inverse Trig 10 / 1

6 Domain Restricted Tangent Function The tangent function is not One-to-One so it cannot have an inverse function. If the domain of y = tan x is restricted to the interval ( π, π ), however, the restricted function is One-to-One and has an inverse function. Inverse Trig 11 / 1 Inverse Tangent Function Notation y = tan 1 x or y = arctan x Domain (, ) Range ( π, π ) Inverse Trig 1 / 1

7 Inverse Tangent and the Unit Circle Recall that tant = y x = sin t cos t of the point on the unit circle corresponding to the angle (or arc length) t. Because the range of the inverse tangent function is restricted to ( π, π ), the angle measures that output from the inverse tangent are located on the right half of the unit circle. Inverse Trig 13 / 1 Example 3 a) tan b) arctan 1 c) arctan 3 Inverse Trig 14 / 1

8 Composition of Functions Recall that if x is in the domain of f (x) and f 1 (x) then f [ f 1 (x) ] = x and f 1 [f (x)] = x However... This is not quite always true for trigonometric functions because of the domain restrictions necessary to create the inverse functions. Inverse Trig 15 / 1 Example 4 Find an exact value for: sin ( [sin 1 1 )] 4 Inverse Trig 16 / 1

9 Example 5 Find an exact value for: arctan ( tan π ) Inverse Trig 17 / 1 Example 6 Find an exact value for: arcsin ( sin 7π ) 4 Inverse Trig 18 / 1

10 Example 7 Find an exact value for: cos ( [tan 1 3 )] 4 Inverse Trig 19 / 1 Example 8 Write as an algebraic expression of a that does not involve trigonometric functions. tan(arcsin a) Inverse Trig 0 / 1

11 What You Learned You can now: Understand the need for restricting the domains of trig functions in order to have an inverse function. Evaluate and graph inverse trig functions. Find compositions of trig functions. Do problems Chap 4.6 #1-14, 17, 1, 3, 9-35 odd, odd Inverse Trig 1 / 1