Implicit differentiation and inverse trigonometric functions practice question 3
Find the exact value of .
Does always give back ? Think about which angles arcsin is allowed to return.
arcsin returns the one angle in with a given sine. Work out first, then find that angle.
Theory UBC Math 100 §4.7CLP-1 §2.12Key results
Worked solution
Strategy
Evaluate the inside first, then ask which angle in the range of arcsin has that sine. arcsin undoes sin only for angles already in that range, and 5π/6 is not one of them.
Working
The reference angle of 5π/6 is π/6, and sine is positive in the second quadrant.
arcsin gives the one angle in [−π/2, π/2] with that sine.
Answer
The answer is pi over 6.
Check
A calculator gives arcsin(sin(5π/6)) ≈ 0.5236, which is π/6; 5π/6 ≈ 2.618 could never come out, as arcsin never exceeds π/2.
Where this goes wrong
Cancelling arcsin against sin and answering 5π/6. arcsin(sin θ) equals θ only when θ lies in [−π/2, π/2]; 5π/6 does not, so the answer is the angle in that range with the same sine.
The idea behind it
The implicit differentiation and inverse trigonometric functions lesson, a figure to move, in a few steps
Leans on
Inverse functions, from Functions, graphs and inverses
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