Evaluate sin(3sin^-11/3).\n\n(a) 23/27\n(b) 7/9\n(c) 1/2\n(d) 5/27

Evaluate $sin (3 sin^{- 1} \frac{1}{3})$ .\n\n(a) $\frac{23}{27}$ \n(b) $\frac{7}{9}$ \n(c) $\frac{1}{2}$ \n(d) $\frac{5}{27}$

medium MCQ 2 marks Class 12 · Mathematics · Inverse Trigonometric Functions

$\frac{23}{27}$
$\frac{7}{9}$
$\frac{1}{2}$
$\frac{5}{27}$

Answer

\frac{23}{27}

The correct answer is $\frac{23}{27}$ — Substitute $sin θ = \frac{1}{3}$ : $sin 3 θ = 3 (\frac{1}{3}) - 4 (\frac{1}{3})^{3} = 1 - \frac{4}{27} = \frac{23}{27}$ .

Step-by-step solution

Set $θ = sin^{- 1} \frac{1}{3}$ , so $sin θ = \frac{1}{3}$ and $θ \in [- π /2, π /2]$ .
Find $cos θ = 1 - sin^{2} θ = 1 - \frac{1}{9} = \frac{2 2}{3}$ .
Use the triple‑angle formula $sin 3 θ = 3 sin θ - 4 sin^{3} θ$ .
Substitute $sin θ = \frac{1}{3}$ : $sin 3 θ = 3 (\frac{1}{3}) - 4 (\frac{1}{3})^{3} = 1 - \frac{4}{27} = \frac{23}{27}$ .

Common mistake: Using

sin 3 θ = 3 sin θ - 4 cos^{3} θ

or mis‑computing

cos θ

Concept tested: triple‑angle identity with inverse sine

Published 6 September 2026 · How this solution was prepared and checked

← All Inverse Trigonometric Functions questions

Evaluate sin(3sin−131​).\n\n(a) 2723​\n(b) 97​\n(c) 21​\n(d) 275​

Step-by-step solution

Evaluate $sin (3 sin^{- 1} \frac{1}{3})$ .\n\n(a) $\frac{23}{27}$ \n(b) $\frac{7}{9}$ \n(c) $\frac{1}{2}$ \n(d) $\frac{5}{27}$