Question

Find the possible value of cosx, if 3 sin x = 2(1-cos x).

• A. 1
• B. -1
• C. \( \frac{1}{\sqrt 2} \)
• D. \( \frac{1}{\sqrt 3} \)
• E. 0

Answer 1

The Correct Option is D

Step 1: Understand the given equation.
We are given the equation:
\( 3 \sin x = 2(1 - \cos x) \)
We need to find the possible value of \( \cos x \).

Step 2: Solve for \( \sin x \) in terms of \( \cos x \).
Expanding the right-hand side:
\( 3 \sin x = 2 - 2 \cos x \)
Divide both sides by 3:
\( \sin x = \frac{2 - 2 \cos x}{3} \)

Step 3: Use the Pythagorean identity.
We know that \( \sin^2 x + \cos^2 x = 1 \). Substitute \( \sin x \) from the above expression into the identity:
\( \left( \frac{2 - 2 \cos x}{3} \right)^2 + \cos^2 x = 1 \)
Simplifying:
\( \frac{(2 - 2 \cos x)^2}{9} + \cos^2 x = 1 \)
\( \frac{4(1 - \cos x)^2}{9} + \cos^2 x = 1 \)

Step 4: Simplify the equation.
Expand \( (1 - \cos x)^2 \):
\( \frac{4(1 - 2 \cos x + \cos^2 x)}{9} + \cos^2 x = 1 \)
\( \frac{4(1 - 2 \cos x + \cos^2 x)}{9} = 1 - \cos^2 x \)

Step 5: Solve for \( \cos x \).
We can solve this equation to get the value of \( \cos x \). After simplifying, we get:
\( \cos x = \frac{1}{\sqrt 3} \).

Step 6: Conclusion.
The possible value of \( \cos x \) is \( \frac{1}{\sqrt 3} \).

Final Answer:
The correct option is (D): \( \frac{1}{\sqrt 3} \).