Survey

Survey

Transcript

CH. 4 – TRIGONOMETRIC FUNCTIONS 4.7 – Inverse Trig Functions INVERSE 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! 1 sin . 2 1 FIND THE EXACT VALUE OF 1. 3 Mr. Weinmann, show the class how to solve this problems! 6 2. 3. 5 6 4. 5. 3 4 0% 0% 0% 0% 0% arcsin 1 . FIND THE EXACT VALUE OF 1. 2 2. 3. 3 2 4. 5. 3 2 2 0% 0% 0% 0% 0% GRAPHING 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! X Y -1 -π/2 -½ -π/6 0 0 ½ π/6 1 π/2 by ½’s by π/6’s INVERSE 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 3 cos 2 1 FIND THE EXACT VALUE OF 1. 4 2 2. 3. 6 4. 5. . 3 6 0% 0% 0% 0% 0% 2 cos . 2 1 FIND THE EXACT VALUE OF 1. 4 3 4 2. 3. 4 4. 5. 5 4 5 4 0% 0% 0% 0% 0% INVERSE 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 ( 3 ) 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! tan 1 0 FIND THE EXACT VALUE OF 1. 0 4 2. 3. 2 4. 5. . 3 2 0% 0% 0% 0% 0% cos 0 FIND THE EXACT VALUE OF 1. 1 2. 0 3. 2 4. undefined 5. 4 0% 0% 0% . 0% 0% Ex: Find sin-1 (sin( 5 )). 6 Inverse sine and sine will cancel, but we have to make sure the angle is in the range of inverse sine. Since 5 is not in [-π/2, π/2], find an angle with that sine that is in the range…6 …so we get 6 3 2 Ex: Find tan(arccos( 3 )). 5 θ 2 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… 5 …to get 2 FIND THE EXACT VALUE OF 1. 4. 5. ) . 1 3 4 2. 3. 3 sin (sin 4 1 2 2 4 undefined 0% 0% 0% 0% 0% 3 tan(sin ) 4 1 FIND THE EXACT VALUE OF 1. 0 7 4 2. 3. 4. 5. . 3 7 7 3 4 undefined 0% 0% 0% 0% 0%