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C H. 4 – T RIGONOMETRIC F UNCTIONS 4.7 – Inverse Trig Functions
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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!
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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5. Mr. Weinmann, show the class how to solve this problems!
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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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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
![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.](http://images.slideplayer.com./36/10666195/slides/slide_5.jpg "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.")
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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
![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](http://images.slideplayer.com./36/10666195/slides/slide_6.jpg "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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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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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 !
![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.](http://images.slideplayer.com./36/10666195/slides/slide_9.jpg "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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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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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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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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F IND THE EXACT VALUE OF. 1. 2. 3. 4. 5.
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![Find the exact values:. Inverse Trig Functions Inverse: “the angle whose (trig function) is x” Arcsin x or [-90° to 90°] Arccos x or [0° to 180°] Arctan.](/19/5785510/big_thumb.jpg "Find the exact values:. Inverse Trig Functions Inverse: “the angle whose (trig function) is x” Arcsin x or [-90° to 90°] Arccos x or [0° to 180°] Arctan.>")
