5.8. Graphs of Inverse Trigonometric Functions http://www.ck12.org
Example C
Evaluate the following expression with and without a calculator using right triangles and your knowledge of inverse
trigonometric functions.
cot(csc−^1 (−^135 ))
Solution: When using a calculator it can be extremely confusing trying to tell the difference between sin−^1 xand
(sinx)−^1. In order to be able to effectively calculate this out it is best to write the expression explicitly only in terms
of functions that your calculator does have.
The hardest part of this question is seeing the csc as a function (which produces an angle) on a ratio of a hypotenuse
of 13 and an opposite side of -5. The sine of the inverse ratio must produce the same angle, so you can substitute it.
- csc−^1 (−^135 )=sin−^1 (− 135 )
- cot(θ) =tan^1 θ
cot(csc−^1 (−^135 ))=tan(sin−^11 (− 135 ))=−^125
Not using a calculator is usually significantly easier. Start with your knowledge that csc−^1 (−^135 )describes an
angle in the fourth or the second quadrant because those are the two quadrants where cosecant is negative. Since
csc−^1 θhas range−π 2 ,π 2 , it only produces angles in quadrant I or quadrant IV (see Guided Practice 2). This triangle
must then be in the fourth quadrant. All you need to do is draw the triangle and identify the cotangent ratio.
Cotangent is adjacent over opposite.