Question

Evaluate: \[ \sin^3 10^\circ + \sin^3 50^\circ - \sin^3 70^\circ = ? \]

• A. \( -\dfrac{3}{8} \)
• B. \( \dfrac{3}{4} \)
• C. \( \dfrac{\sqrt{3}}{2} \)
• D. \( -\dfrac{1}{3} \)

Hint:

When exact algebraic identities are not applicable, plug in approximate trigonometric values and use calculator or known approximations for cube powers.

Answer 1

The Correct Option is A

Step 1: Use the identity for \( a^3 + b^3 - c^3 \), if applicable. 
But instead, compute the values numerically or via trigonometric symmetry. 
Use approximation or calculator: \[ \sin 10^\circ \approx 0.1736 \Rightarrow \sin^3 10^\circ \approx 0.1736^3 = 0.0052 \] \[ \sin 50^\circ \approx 0.7660 \Rightarrow \sin^3 50^\circ \approx 0.4493 \] \[ \sin 70^\circ \approx 0.9397 \Rightarrow \sin^3 70^\circ \approx 0.8306 \] \[ \sin^3 10^\circ + \sin^3 50^\circ - \sin^3 70^\circ \approx 0.0052 + 0.4493 - 0.8306 = -0.3761 \approx -\dfrac{3}{8} \]