Compound And Multiple Angles

Question

By repeatedly using the triple-angle identity, the exact value of $\sin 3	heta$ when $\sin	heta = \frac{1}{2}$ and $	heta$ is acute simplifies to which of the following numbers?

RankGuru · Accepted Answer

The controlling formula is the triple-angle identity $\sin 3	heta = 3\sin	heta - 4\sin^3	heta$, which expresses the sine of a tripled angle entirely in terms of the sine of the original angle. This identity is built by combining the angle-addition formula with the double-angle formulas, so it lets a tripled angle be evaluated algebraically without ever finding the angle itself. Substituting the given value $\sin	heta = \frac{1}{2}$ directly, the expression becomes $3 \cdot \frac{1}{2} - 4 \cdot \left(\frac{1}{2}ight)^3 = \frac{3}{2} - 4 \cdot \frac{1}{8} = \frac{3}{2} - \frac{1}{2} = 1$. The acute condition uniquely fixes $	heta = 30^\circ$, so the tripled angle is $90^\circ$ and $\sin 90^\circ = 1$, matching the algebraic result exactly. Option $\frac{\sqrt{3}}{2}$ confuses the answer with a cosine value. Option $\frac{1}{2}$ simply repeats the original sine without applying the tripling. Option $0$ misapplies the cubic term. This follows the standard triple-angle expansion pattern of JEE Advanced. As a final consistency check, the geometric route confirms that $	heta = 30^\circ$ gives $3	heta = 90^\circ$ and hence $\sin 3	heta = 1$, in full agreement with the identity-based computation.

Compound And Multiple Angles

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Compound And Multiple Angles

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Solution

If $cos θ = \frac{3}{5}$ with $θ$ in the first quadrant, then applying the half-angle iden...

Given that $tan^{- 1} x + tan^{- 1} y = \frac{π}{4}$ holds for positive reals with $x y < 1$ , the e...

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The fundamental period of the composite function $g (x) = sin^{4} x + cos^{4} x$ , after simplification ...

In a triangle with sides $a$ , $b$ , $c$ and semiperimeter $s$ , the area computed by Heron's formula f...

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Statement based: assertion that $sin^{- 1} (sin \frac{2 π}{3}) = \frac{2 π}{3}$ and reason that arc...

How many distinct values of $x$ in the open interval $(0, π)$ satisfy the equation $\sin x + \sqrt...

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