Evaluate sin(2sin^-14/5).

Evaluate $sin (2 sin^{- 1} \frac{4}{5})$ .

medium 2 marks Class 12 · Mathematics · Inverse Trigonometric Functions

Answer

\frac{24}{25}

The correct answer is $\frac{24}{25}$ — Since $2 θ$ is within $[- π, π]$ , the value is directly $\frac{24}{25}$ .

Let $θ = sin^{- 1} \frac{4}{5}$ , so $sin θ = \frac{4}{5}$ and $θ$ lies in $[- \frac{π}{2}, \frac{π}{2}]$ .
Construct a right triangle with opposite side $4$ and hypotenuse $5$ ; then adjacent side $5^{2} - 4^{2} = 9 = 3$ .
Thus $cos θ = \frac{3}{5}$ .
Use the double‑angle formula $sin 2 θ = 2 sin θ cos θ$ .
Substitute: $sin 2 θ = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25}$ .
Since $2 θ$ is within $[- π, π]$ , the value is directly $\frac{24}{25}$ .

Common mistake: Forgetting to find

cos θ

and using

sin 2 θ = 2 sin^{2} θ

which is incorrect

Concept tested: Using double‑angle identity with inverse sine