Question

The value of $\sin 5^{\circ}\sin 10^{\circ}\sin 15^{\circ}\sin 20^{\circ}\dots\sin 240^{\circ}$ is equal to

• A. $\frac{1}{2^{24}}(\sin 15^{\circ}+\cos 15^{\circ})$
• B. $\frac{1}{2^{12}}(\sin 15^{\circ}+\cos 15^{\circ})$
• C. $\frac{1}{2^{22}}(\sin 15^{\circ}+\cos 15^{\circ})$
• D. $\frac{1}{2^{42}}(\sin 15^{\circ}+\cos 15^{\circ})$
• E. 0

Hint:

Math Tip: Always scan long, intimidating product or summation series for "trapdoor" terms that evaluate to zero! In trigonometry, watch out for $\sin(180^{\circ})$, $\cos(90^{\circ})$, and $\tan(180^{\circ})$ hiding in sequences.

Answer 1

Answer: (E)

Concept:
Trigonometry - Properties of the Sine Function and Zero Products.
Step 1: Analyze the sequence of angles.
The given expression is a product of sine functions: $$ \sin(5^{\circ}) \cdot \sin(10^{\circ}) \cdot \sin(15^{\circ}) \dots \sin(240^{\circ}) $$ The angles form an arithmetic progression starting from $5^{\circ}$ with a common difference of $5^{\circ}$.
Step 2: Identify critical values within the sequence.
List out some of the terms as the sequence progresses towards $240^{\circ}$: $$ \dots \cdot \sin(170^{\circ}) \cdot \sin(175^{\circ}) \cdot \sin(180^{\circ}) \cdot \sin(185^{\circ}) \dots $$
Step 3: Evaluate the specific critical term.
Notice that $\sin(180^{\circ})$ is explicitly one of the terms in this multiplication series. Recall the value of sine at $180^{\circ}$ (or $\pi$ radians): $$ \sin(180^{\circ}) = 0 $$
Step 4: Apply the Zero Product Property.
If any single term in a chain of multiplication is zero, the entire product becomes zero, regardless of the values of the other terms. $$ \text{Product} = \text{Something} \times 0 \times \text{Something Else} = 0 $$ Therefore, the value of the entire expression is $0$.