Inverse trigonometric functions

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Precalculus › Trigonometric functions

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Inverse trigonometric functions

Warm-up

Three quick questions from earlier lessons.

Discover

Many angles have a sine of 12\dfrac{1}{2}: π6\dfrac{\pi}{6}, 5π6\dfrac{5\pi}{6}, 13π6\dfrac{13\pi}{6}… Which one should sin⁡−112\sin^{-1}\dfrac{1}{2} give?

  1. π6\dfrac{\pi}{6}, the one between −π2-\dfrac{\pi}{2} and π2\dfrac{\pi}{2}
  2. All of them
  3. 22, because 11/2=2\dfrac{1}{1/2} = 2

Key idea

Restricting sine, cosine and tangent to one part makes them invertible.

sin⁡−1x\sin^{-1}x is the angle in [−π2,π2]\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] whose sine is xx. cos⁡−1x\cos^{-1}x is the angle in [0,π][0, \pi] whose cosine is xx. tan⁡−1x\tan^{-1}x is the angle in (−π2,π2)\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) whose tangent is xx.

Worked example

Find cos⁡−1(−12)\cos^{-1}\left(-\dfrac{1}{2}\right).

We need θ∈[0,π]\theta \in [0, \pi] with cos⁡θ=−12\cos\theta = -\dfrac{1}{2}
cos⁻¹ answers between 0 and π.
θ=π−π3\theta = \pi - \dfrac{\pi}{3}
cos(π/3) = 1/2; a negative cosine means quadrant II.
cos⁡−1(−12)=2π3\cos^{-1}\left(-\dfrac{1}{2}\right) = \dfrac{2\pi}{3}
Subtract.

Your turn, guided

Your turn: sin⁡−1(−22)\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right).

We need θ\theta with sin⁡θ=−22\sin\theta = -\dfrac{\sqrt{2}}{2}
sin⁻¹ answers between −π/2 and π/2.
θ=−π4\theta = -\dfrac{\pi}{4}
sin(π/4) = √2/2; below the axis, the angle is negative.

Practice ★

  1. Find the exact value of sin⁡−1(12)\sin^{-1}\left(\frac{1}{2}\right), in radians.
  2. Find the exact value of sin⁡−1(−32)\sin^{-1}\left(- \frac{\sqrt{3}}{2}\right), in radians.
  3. Find the exact value of sin⁡−1(−22)\sin^{-1}\left(- \frac{\sqrt{2}}{2}\right), in radians.
  4. Find the exact value of sin⁡−1(32)\sin^{-1}\left(\frac{\sqrt{3}}{2}\right), in radians.

Watch out!

Practice ★★

  1. Find the exact value of cos⁡−1(−32)\cos^{-1}\left(- \frac{\sqrt{3}}{2}\right), in radians.
  2. Find the exact value of cos⁡−1(−12)\cos^{-1}\left(- \frac{1}{2}\right), in radians.
  3. Find the exact value of cos⁡−1(−22)\cos^{-1}\left(- \frac{\sqrt{2}}{2}\right), in radians.
  1. Find the exact value of cos⁡(sin⁡−135)\cos\left(\sin^{-1} \frac{3}{5}\right).
  2. Find the exact value of cos⁡(sin⁡−1513)\cos\left(\sin^{-1} \frac{5}{13}\right).

Exit ticket

Three questions to check you have got it.

  1. Find the exact value of cos⁡−1(−32)\cos^{-1}\left(- \frac{\sqrt{3}}{2}\right), in radians.
  2. Find the exact value of cos⁡−1(−12)\cos^{-1}\left(- \frac{1}{2}\right), in radians.
  3. Find the exact value of cos⁡−1(−22)\cos^{-1}\left(- \frac{\sqrt{2}}{2}\right), in radians.

Summary card

Inverse trig functions

  • sin⁡−1\sin^{-1}: angle in [−π2,π2]\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]. tan⁡−1\tan^{-1}: in (−π2,π2)\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right).
  • cos⁡−1\cos^{-1}: angle in [0,π][0, \pi].

Watch out: sin⁡−1x\sin^{-1}x is an angle, not 1sin⁡x\dfrac{1}{\sin x}.

End of the lesson

Back to the chapter

Answers: Inverse trigonometric functions

A

  1. π6\frac{\pi}{6}
  2. −π3- \frac{\pi}{3}
  3. −π4- \frac{\pi}{4}
  4. π3\frac{\pi}{3}

B

  1. 5π6\frac{5 \pi}{6}
  2. 2π3\frac{2 \pi}{3}
  3. 3π4\frac{3 \pi}{4}

C

  1. 45\frac{4}{5}
  2. 1213\frac{12}{13}

D

  1. 5π6\frac{5 \pi}{6}
  2. 2π3\frac{2 \pi}{3}
  3. 3π4\frac{3 \pi}{4}

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