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) y = Arcsin (x) Ex: Find the exact value for Ask yourself, where on the unit circle does sin = ? Remember, you must be between

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

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) y = Arctan (x) Ex: Find the exact value for Ask yourself, where on the unit circle does tan = 1? Remember, you must be between

More Examples:

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!**

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!!

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