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Inverse trig functions

Define sin⁡−1x\sin^{-1}x, cos⁡−1x\cos^{-1}x and tan⁡−1x\tan^{-1}x with restricted domains and ranges; sketch the graphs of the inverse trig functions; evaluate exact values using knowledge of the unit circle and special triangles.

Worked examples

Evaluating inverse trig functions at exact values

Straightforward

Problem

Find the exact value of each of the following in radians: (a) sin⁡−1 ⁣(32)\sin^{-1}\!\left(\dfrac{\sqrt{3}}{2}\right), (b) cos⁡−1 ⁣(−22)\cos^{-1}\!\left(-\dfrac{\sqrt{2}}{2}\right), (c) tan⁡−1 ⁣(13)\tan^{-1}\!\left(\dfrac{1}{\sqrt{3}}\right). (Each inverse trig function has a restricted range: sin⁡−1(x)\sin^{-1}(x) maps to [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], cos⁡−1(x)\cos^{-1}(x) maps to [0,π][0, \pi], and tan⁡−1(x)\tan^{-1}(x) maps to (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right).)

Domain restrictions and the composition sin⁡−1(sin⁡(θ))\sin^{-1}(\sin(\theta))

Moderate

Problem

Evaluate sin⁡−1 ⁣(sin⁡ ⁣(7π6))\sin^{-1}\!\left(\sin\!\left(\dfrac{7\pi}{6}\right)\right).

Using inverse trig in a composition

Challenging

Problem

Find the exact value of tan⁡ ⁣(sin⁡−1 ⁣(−513))\tan\!\left(\sin^{-1}\!\left(-\dfrac{5}{13}\right)\right).

Practise

Q1·Straightforward
Find the exact value of tan⁡−1(1)\tan^{-1}(1) in radians. Give your answer as a decimal to 4 decimal places.
Q2·Straightforward
Find the exact value of sin⁡−1(0)\sin^{-1}(0) in radians.
Q3·Straightforward
Find the exact value of cos⁡−1(1)\cos^{-1}(1) in radians.
Q4·Straightforward
Find the exact value of sin⁡−1(1)\sin^{-1}(1) in radians. Give your answer as a decimal to 4 decimal places.
Q5·Straightforward
The function sin⁡−1(x)\sin^{-1}(x) is defined only for values of xx in what interval?
Q6·Moderate
Find the exact value of cos⁡−1 ⁣(−12)\cos^{-1}\!\left(-\dfrac{1}{2}\right) in radians. Give your answer as a decimal to 4 decimal places.
Q7·Moderate
Find the exact value of sin⁡−1 ⁣(−22)\sin^{-1}\!\left(-\dfrac{\sqrt{2}}{2}\right) in radians. Give your answer as a decimal to 4 decimal places.
Q8·Moderate
Find the exact value of tan⁡−1(−3)\tan^{-1}(-\sqrt{3}) in radians. Give your answer as a decimal to 4 decimal places.
Q9·Moderate
Solve sin⁡−1(x)=π3\sin^{-1}(x) = \dfrac{\pi}{3} for xx. Give your answer as a decimal to 4 decimal places.
Q10·Moderate
What is the range of tan⁡−1(x)\tan^{-1}(x)?
Q11·Challenging
Find the exact value of cos⁡ ⁣(sin⁡−1 ⁣(35))\cos\!\left(\sin^{-1}\!\left(\dfrac{3}{5}\right)\right).
Q12·Challenging
Find the exact value of sin⁡−1 ⁣(sin⁡ ⁣(5π6))\sin^{-1}\!\left(\sin\!\left(\dfrac{5\pi}{6}\right)\right) in radians. Give your answer as a decimal to 4 decimal places.
Q13·Challenging
Solve 2sin⁡−1(x)=π22\sin^{-1}(x) = \dfrac{\pi}{2} for xx. Give your answer as a decimal to 4 decimal places.