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Ch 4.7: Inverse Trig Functions

Ch 4.7: Inverse Trig Functions. Inverse of Sine. Inverse : Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one Interval : , then y = sin(x) has an inverse Written : y = arcsin(x) or y = sin -1 (x) Remember : y = sin -1 (x) iff x = sin(y).

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Ch 4.7: Inverse Trig Functions

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  1. Ch 4.7: Inverse Trig Functions

  2. Inverse of Sine • Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one • Interval: , then y = sin(x) has an inverse • Written: y = arcsin(x) or y = sin-1(x) • Remember: y = sin-1(x) iff x = sin(y) y = Arcsin (x) Ex: Find the exact value for Ask yourself, where on the unit circle does sin = ? Remember, you must be between

  3. Inverse of Cosine • Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one • Interval: , then y = cos(x) has an inverse • Written: y = arccos(x) or y = cos-1(x) • Remember: y = cos-1(x) iff x = cos(y) Ex: Find the exact value for Ask yourself, where on the unit circle does cos = ? Remember, you must be between

  4. Inverse of Tangent • Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one • Interval: , then y = tan(x) has an inverse • Written: y = arctan(x) or y = tan-1(x) • Remember: y = tan-1(x) iff x = tan(y) Ex: Find the exact value for y = Arctan (x) Ask yourself, where on the unit circle does tan = 1? Remember, you must be between

  5. Approximating Values • By definition, inverses are supposed to be in radians • Check to see whether the number is in radians or degrees • To do inverse trig functions, hit “2nd” then the trig function • Round 4 places • Some will not work! Ex: Arcsin(0.2447) = Ex: sin-1(2) =

  6. Inverse Prop.: Recall f(f-1(x))=x & f-1(f(x))=x For -1  x  1 and sin(sin-1(x)) = x & sin-1(sin(y)) = y For -1  x  1 and cos(cos-1(x)) = x & cos-1(cos(y)) = y For x is a real number and tan(tan-1(x)) = x & tan-1(tan(y)) = y **Pay attention to make sure the values fall within the parameters of the inverse!**

  7. More complex problems • Determine the quadrant • Draw a triangle, label the parts • Using the triangle, answer the problem Thus, either Quadrant I or IV. Since -3/5, you are in IV!! Thus, either Quadrant I or IV. Since 3/2, you are in I!!

  8. Most complex problems • Follow the same rules from previous slide, but now you will have variables in your answer

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