If sin θ + sin 5θ = sin 3θ for an acute angle θ with 0° < θ < 60°, then what is the value of θ in degrees?

Introduction / Context:
This trigonometry question tests the use of sum to product formulas and basic equation solving with sine functions. It also illustrates how adding a condition on the range of the angle ensures a unique answer, which is important for multiple choice questions in competitive exams.

Given Data / Assumptions:

sin θ + sin 5θ = sin 3θ.
0° < θ < 60°, so θ is an acute angle less than 60 degrees.
We work in degrees and standard trigonometric definitions.

Concept / Approach:
We use the identity sin C + sin D = 2 sin((C + D) / 2) cos((C − D) / 2). Applying this to sin θ + sin 5θ allows us to rewrite the left side in terms of sin 3θ and cos 2θ. Then we can equate both sides and solve for θ. The additional condition on θ eliminates extra solutions that might otherwise satisfy the equation but fall outside the specified range.

Step-by-Step Solution:
Apply the sum to product formula: sin θ + sin 5θ = 2 sin((θ + 5θ) / 2) cos((θ − 5θ) / 2).This simplifies to 2 sin(3θ) cos(−2θ).Since cos is an even function, cos(−2θ) = cos 2θ, so the left side becomes 2 sin 3θ cos 2θ.The given equation becomes 2 sin 3θ cos 2θ = sin 3θ.Rearrange: sin 3θ (2 cos 2θ − 1) = 0.So either sin 3θ = 0 or 2 cos 2θ − 1 = 0.From 2 cos 2θ − 1 = 0, we get cos 2θ = 1/2, so 2θ = 60° or 300°.For 0° < θ < 60°, only 2θ = 60° is valid, giving θ = 30°.

Verification / Alternative check:
Check θ = 30° in the original equation: sin 30° + sin 150° = 1/2 + 1/2 = 1. On the right side, sin 3θ = sin 90° = 1. Both sides are equal, so θ = 30° is indeed a solution. The other possibility sin 3θ = 0 would give 3θ = 180°, 360° and so on, which leads to θ = 60° or larger values. Because of the condition 0° < θ < 60°, these are excluded, ensuring a unique answer.

Why Other Options Are Wrong:
Angles 15°, 45°, 60°, and 75° do not satisfy sin θ + sin 5θ = sin 3θ within the specified interval. Substituting any of these values either breaks the equality or falls outside the allowed range. For example, θ = 45° gives sin 45° + sin 225° which does not equal sin 135°. Thus these options are incorrect.

Common Pitfalls:
Some students cancel sin 3θ from both sides without considering that it might be zero, which can hide valid solutions. Others forget to apply the range condition and report θ = 60° as a solution. Always use sum to product identities carefully, solve all resulting cases, and then apply the interval restriction to select the correct answer.

Final Answer:
The angle θ that satisfies the given condition in the range 0° < θ < 60° is 30°.