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Worked solution

Choose principal inverse-trigonometric values

Find the principal value of (a) sin⁡−1(1/2)\sin^{-1}(1/2); (b) cos⁡−1(1/2)\cos^{-1}(1/\sqrt2); (c) tan⁡−1(1/3)\tan^{-1}(1/\sqrt3); (d) sin⁡[cos⁡−1(−3/2)]\sin[\cos^{-1}(-\sqrt3/2)].

CJC · O Level revision · Example 17

(a) Inverse sine

Answer

π/6\pi/6.

Select the principal angle

sin⁡−1(12)=π6\sin^{-1}\left(\frac12\right)=\frac\pi6

The principal inverse-sine range is [−π/2,π/2][-\pi/2,\pi/2].

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(b) Inverse cosine

Answer

π/4\pi/4.

Select the principal angle

cos⁡−1(12)=π4\cos^{-1}\left(\frac1{\sqrt2}\right)=\frac\pi4

The principal inverse-cosine range is [0,π][0,\pi]. The denominator is √2, not 2.

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(c) Inverse tangent

Answer

π/6\pi/6.

Select the principal angle

tan⁡−1(13)=π6\tan^{-1}\left(\frac1{\sqrt3}\right)=\frac\pi6

The principal inverse-tangent range is (−π/2,π/2)(-\pi/2,\pi/2). The denominator is √3.

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(d) Evaluate the outer sine

Answer

1/21/2.

Select the principal angle

sin⁡[cos⁡−1(−32)]=sin⁡(5π6)=12\sin\left[\cos^{-1}\left(-\frac{\sqrt3}2\right)\right]=\sin\left(\frac{5\pi}6\right)=\frac12

The inner inverse cosine selects 5π/65\pi/6, in its principal range; then take sine.

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