Transcript 4.7 PP
4.7 Inverse Trigonometric Functions Copyright © 2011 Pearson, Inc. What you’ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions … and why Inverse trig functions can be used to solve trigonometric equations. Copyright © 2011 Pearson, Inc. Slide 4.7 - 2 Inverse Sine Function –1 Copyright © 2011 Pearson, Inc. Slide 4.7 - 3 Inverse Sine Function (Arcsine Function) The unique angle y in the interval [ -p / 2, p / 2 ] such that sin y = x is the inverse sine (or arcsine) of x, denoted sin -1 x or arcsin x. The domain of y = sin -1 x is [ -1,1] and the range is [ -p / 2, p / 2 ]. Copyright © 2011 Pearson, Inc. Slide 4.7 - 4 Example Evaluate sin–1x Without a Calculator æ 1ö Find the exact value without a calculator: sin ç - ÷ è 2ø -1 Copyright © 2011 Pearson, Inc. Slide 4.7 - 5 Example Evaluate sin–1x Without a Calculator æ 1ö Find the exact value without a calculator: sin ç - ÷ è 2ø -1 Find the point on the right half of the unit circle whose y-coordinate is - 1 / 2 and draw a reference triangle. Recognize this as a special ratio, and the angle in the interval [-p / 2, p / 2] whose sine is - 1 / 2 is - p / 6. Copyright © 2011 Pearson, Inc. Slide 4.7 - 6 Example Evaluate sin-1x Without a Calculator æ æ p öö Find the exact value without a calculator: sin ç sin ç ÷ ÷ . è è 10 ø ø -1 Copyright © 2011 Pearson, Inc. Slide 4.7 - 7 Example Evaluate sin-1x Without a Calculator æ æ p öö Find the exact value without a calculator: sin ç sin ç ÷ ÷ . è è 10 ø ø -1 Draw an angle p / 10 in standard position and mark its y-coordinate on the y-axis. The angle in the interval [ - p /2,p /2] whose sine is this number is p /10. æ æ p öö p Therefore, sin ç sin ç ÷ ÷ = . è è 10 ø ø 10 -1 Copyright © 2011 Pearson, Inc. Slide 4.7 - 8 Inverse Cosine (Arccosine Function) c os Copyright © 2011 Pearson, Inc. cos –1 Slide 4.7 - 9 Inverse Cosine (Arccosine Function) The unique angle y in the interval [ 0, p ] such that cos y = x is the inverse cosine (or arccosine) of x, denoted cos –1 x or arccos x. The domain of y = cos -1 x is [ - 1,1] and the range is [ 0, p ] . Copyright © 2011 Pearson, Inc. Slide 4.7 - 10 Inverse Tangent Function (Arctangent Function) ta ta –1 Copyright © 2011 Pearson, Inc. Slide 4.7 - 11 Inverse Tangent Function (Arctangent Function) The unique angle y in the interval (-p / 2, p / 2) such that tan y = x is the inverse tangent (or arctangent) of x, –1 denoted tan x or arctan x. The domain of y = tan -1 x is (-¥,¥) and the range is (-p / 2, p / 2). Copyright © 2011 Pearson, Inc. Slide 4.7 - 12 End Behavior of the Tangent Function The graphs of (a) y = tan x (restricted) and (b) y = tan–1x. The vertical asymptotes of y = tan x are reflected to become the horizontal asymptotes of y = tan–1x. Copyright © 2011 Pearson, Inc. Slide 4.7 - 13 Composing Trigonometric and Inverse Trigonometric Functions The following equations are always true whenever they are defined: ( ) sin sin -1 ( x ) = x ( ) cos cos -1 ( x ) = x ( ) tan tan -1 ( x ) = x The following equations are only true for x values in the "restricted" domains of sin, cos, and tan: sin -1 ( sin ( x ) ) = x Copyright © 2011 Pearson, Inc. cos -1 ( cos ( x ) ) = x tan -1 ( tan ( x ) ) = x Slide 4.7 - 14 Example Composing Trig Functions with Arccosine Compose each of the six basic trig functions with cos –1 x and reduce the composite function to an algebraic expression involving no trig functions. Copyright © 2011 Pearson, Inc. Slide 4.7 - 15 Example Composing Trig Functions with Arccosine Compose each of the six basic trig functions with cos –1 x and reduce the composite function to an algebraic expression involving no trig functions. a triangle in which = cos –1 x. The side opposite (which is sin) is found by using the Pythagorean Theorem. Copyright © 2011 Pearson, Inc. Slide 4.7 - 16 Example Composing Trig Functions with Arccosine Use the triangle to find the required ratios: -1 sin(cos x)) = 1 - x 2 -1 cos(cos x)) = x 2 1 x -1 tan(cos x)) = x Copyright © 2011 Pearson, Inc. -1 csc(cos x)) = 1 1- x 2 1 sec(cos x)) = x -1 -1 cot(cos x)) = x 1 - x2 Slide 4.7 - 17 Quick Review State the sign (positive or negative) of the sine, cosine, and tangent in quadrant 1. I 2. III Find the exact value. 3. cos p 6 4p 4. tan 3 11p 5. sin 6 Copyright © 2011 Pearson, Inc. Slide 4.7 - 18 Quick Review Solutions State the sign (positive or negative) of the sine, cosine, and tangent in quadrant 1. I +,+,+ 2. III -,-,+ Find the exact value. 3. cos p 6 4p 4. tan 3 11p 5. sin 6 3/2 Copyright © 2011 Pearson, Inc. 3 1/ 2 Slide 4.7 - 19