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

5. More Examples:

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!**

8. 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!!

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