If x = sin 3A, then which of the following expressions correctly represents x in terms of sin A only?

Introduction / Context:
Triple angle formulas are an important part of trigonometry and are frequently tested in aptitude exams and entrance tests. They allow you to express trigonometric functions of a multiple angle, such as 3A, in terms of functions of the single angle A. In this question, x is defined as sin 3A, and you are asked to write x purely in terms of sin A. This requires remembering or deriving the standard triple angle identity for sine.

Given Data / Assumptions:

• x is defined as x = sin 3A.
• Angle A is such that all the involved trigonometric functions are defined.
• The target is an expression involving only sin A, not cos A.
• We rely on standard trigonometric identities for multiple angles.

Concept / Approach:
The known triple angle identity for sine states that sin 3A = 3 sin A - 4 sin^3 A. This formula can be derived from the sine of sum identity by writing 3A as 2A + A and then expressing sin 2A and cos 2A in terms of sin A. Once you know or derive this identity, the problem becomes a simple reading exercise where you choose the option that matches the standard form exactly. Some options are designed to look similar but have sign or coefficient errors, which you must spot carefully.

Step-by-Step Solution:
Step 1: Recall the triple angle identity for sine: sin 3A = 3 sin A - 4 sin^3 A. Step 2: Compare this identity with the options given in the problem. Step 3: Option a states 3 sin A - 4 sin^3 A, which exactly matches the triple angle formula. Step 4: Option e, sin A (3 - 4 sin^2 A), is just a factored form of 3 sin A - 4 sin^3 A, but the question normally expects the standard expanded form. Step 5: Therefore, we select 3 sin A - 4 sin^3 A as the correct representation of x.

Verification / Alternative check:
To verify, you can plug in a simple value for A, such as A = 30 degrees. Then sin 3A becomes sin 90 degrees, which equals 1. Now compute 3 sin A - 4 sin^3 A with sin 30 degrees = 1/2. This gives 3 * (1/2) - 4 * (1/2)^3 = 3/2 - 4 * 1/8 = 3/2 - 1/2 = 1. The match between both sides confirms the identity for this test value, which supports the correctness of the formula.

Why Other Options Are Wrong:

• Option b (4 sin A - 3 sin^3 A) swaps coefficients and does not match the triple angle identity.
• Option c (3 sin A + 4 sin^3 A) has the wrong sign in front of the cubic term, which changes the value significantly.
• Option d (4 sin^3 A - 3 sin A) reverses both the order and the signs compared to the correct formula.
• Option e, although algebraically equivalent to option a when expanded, is not in the standard expanded form that is usually expected, but option a already presents the identity clearly.

Common Pitfalls:
A common pitfall is confusing the triple angle formulas for sine and cosine or mixing them with double angle identities. Another frequent mistake is misremembering the coefficients or the sign of the cubic term. Some students also incorrectly factor or expand expressions, leading to inaccurate alternatives that look convincing. To avoid such issues, it helps to memorise the key multiple angle identities and to verify them with simple angle values when in doubt.

Final Answer:
The correct expression for x in terms of sin A is 3 sin A - 4 sin^3 A.