C H. 4 – T RIGONOMETRIC F UNCTIONS 4.7 – Inverse Trig Functions

I NVERSE SINE FUNCTION Does y = sinx have an inverse? No, because it fails the horizontal line test! However, let’s restrict the domain of y = sinx to [-π/2, π/2] so that… The function is one-to-one y = sinx takes on its full range of values …then we have the inverse function, y = sin -1 x It’s also called y = arcsin x Domain: [-1, 1] Range: [-π/2, π/2] y = sin x if and only if sin -1 y = x Remember: y = sin -1 x outputs an angle!

F IND THE EXACT VALUE OF Mr. Weinmann, show the class how to solve this problems!

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G RAPHING Y = SIN -1 X Domain: [-1, 1], Range: [-π/2, π/2] Make a table recalling points from the unit circle Remember, it should look like a piece of an inverted sine function Y values are in radians! The graph should be one-to-one No arrows on the end of the graph this time! XY - π/2 -½- π/6 00 ½ π/6 1 π/2 by π/6’s by ½’s

I NVERSE COSINE FUNCTION Like the sine function, the cosine function must be restricted to have an inverse We restrict the domain of y = cosx to [0, π] so that… The function is one-to-one y = cosx takes on its full range of values …then we have the inverse function, y = cos -1 x It’s also called y = arccos x Domain: [-1, 1] Range: [0, π] y = cos x if and only if cosy = x

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I NVERSE TANGENT FUNCTION Since we don’t cover tangent graphs, just believe me when I say that for y = tan -1 x … Domain: [- ∞, ∞ ] or all real numbers Range: [-π/2, π/2] Ex: Find tan -1 ( ) in radians. This will be tough to figure out by hand, so just use your calculator! If we do this problem with our calculator in radians, the answer probably will be a decimal… …so do the problem in degrees and convert to radians! You should get 60 °, or π/3 !

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Ex: Find sin -1 (sin( )). Inverse sine and sine will cancel, but we have to make sure the angle is in the range of inverse sine. Since is not in [-π/2, π/2], find an angle with that sine that is in the range… …so we get Ex: Find tan(arccos( )). Draw a right triangle to help you! This question is asking, “what is the tangent if the cosine is 2/3?” Complete the right triangle… …to get 2 3 θ

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