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Trigonometry Lecture Notes_part1-1

We can graph the inverse of the sine function with

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We can graph the inverse of the sine function with the above restricted domain by simple reversing the points. For example, (x, y) is switched to (y, x). Finding Exact Values of sin -1 x Let θ = sin -1 x .
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Rewrite step 1 as sin θ = x . Use the exact values from the unit circle to find the value of θ in [- π /2 , π /2] that satisfies sin θ = x . Example 35 Find the exact value of sin -1 (1/2) The Inverse Cosine Function The inverse cosine function , denoted by cos -1 , is the inverse of the restricted cosine function y = cos x , 0 < x < π . Thus, y = cos -1 x means cos y = x , where 0 < y < π and –1 < x < 1. Example 36 Find the exact value of cos -1 (- 3 /2) The Inverse Tangent Function The inverse tangent function , denoted by tan -1 , is the inverse of the restricted tangent function y = tan x , - π /2 < x < π /2 . Thus, y = tan -1 x means tan y = x , where - π /2 < y < π /2 and – < x < . Example 37 Find the exact value of 1 tan 3 -
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Inverse Properties The Sine Function and Its Inverse sin (sin -1 x ) = x for every x in the interval [-1, 1]. sin -1 (sin x ) = x for every x in the interval [- π /2 , π /2 ]. The Cosine Function and Its Inverse cos (cos -1 x ) = x for every x in the interval [-1, 1]. cos -1 (cos x ) = x for every x in the interval [0, π ]. The Tangent Function and Its Inverse tan (tan -1 x ) = x for every real number x tan -1 (tan x ) = x for every x in the interval (- π /2 , π /2 ). Example 38 Find the exact values if possible of: a. ( ) 1 cos cos 0.6 - b. 1 3 sin sin 2 π - c. ( ) 1 cos cos 2 π - Example 39 Find the exact value of 1 5 cos tan 12 - . Example 40 Find the exact value of 1 1 cot sin 3 - - . Example 41 Calculators and the Inverse Trigonometric Functions Use your calculator to find 1 1 sin 4 - and ( ) 1 tan 9.65 - - in radian mode.
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Graphs of the Three Basic Trigonometric Functions
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Arc tangent
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