- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Implicit differentiation and inverse trigonometric functions practice question 8
Find the exact value of .
Is an angle that arctan can return? If not, arctan and tan do not simply cancel.
arctan returns the one angle strictly between and with a given tangent. Work out first, then find that angle.
An exact value, or a decimal to three places.
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 arctan has that tangent. arctan undoes tan only for angles already in that range, and 3π/4 is not one of them.
Working
In the second quadrant the sine is positive and the cosine negative.
arctan gives the one angle in (−π/2, π/2) with that tangent.
Answer
The answer is negative pi over 4.
Check
A calculator gives arctan(tan(3π/4)) = arctan(−1) ≈ −0.785, which is −π/4; 3π/4 ≈ 2.356 could never come out, as arctan stays between −π/2 and π/2.
Where this goes wrong
Cancelling arctan against tan and answering 3π/4. arctan(tan θ) equals θ only when θ lies strictly between −π/2 and π/2; tangent repeats every π, so the answer is 3π/4 − π = −π/4.
Leans on
Inverse functions, from Functions, graphs and inverses
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