http://www.ck12.org Chapter 6. Transcendental Functions
The range is based on limiting the domain of the original function so that it is a one-to-one function.
Example 1:
What is the exact value of sin−^1 (
√
3 / 2 )?
Solution:
This is equivalent to sinx=
√ 3
2. Thus sin−^1 (√ 3 / 2 ) =π/3. You can easily confirm this result by using yourscientific calculator.
Example 2:
Most calculators do not provide a way to calculate the inverse of the secant function, sec−^1 x.A practical trick
however is to use the identity
sec−^1 x=cos−^11 x(Recall that secθ=cos^1 θ.)
For practice, use your calculator to find sec−^1 ( 3. 24 ).
Solution:
Since
1
x=1
3. 24 =^0.^3086 ,
sec−^13. 24 =cos−^10. 3086 = 72 ◦.Here are two other identities that you may need to enter into your calculator:
csc−^1 x=sin−^11 x,
cot−^1 x=tan−^11 x.The Derivative Formulas of the Inverse Trigonometric Functions
Ifuis a differentiable function ofxthen the generalized derivative formulas for the inverse trigonometric functions
are (we introduce them here without a proof):