If \(sin(90^ - x) = cos(2x)\) , then what is x?

Question

If \(sin(90^ - x) = cos(2x)\) , then what is x?

1. A. 0° (Correct answer)
2. B. 30°
3. C. 25°
4. D. 15°

Correct Answer

Option A — 0°

Detailed Solution & Explanation

The correct answer is 0°.

Key Points

• Use the complementary-angle identity: sin(90° − x) = cos x.
• The equation becomes cos x = cos 2x.
• Cosine values are equal when the angles are equal, so x = 2x, giving x = 0°.
• Verify: sin(90° − 0) = sin 90° = 1, and cos(0) = 1 ✓.

Additional Information

• The complementary identities are worth knowing as a set: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.
• Testing the options directly is often faster than solving: at x = 30°, sin 60° = 0.866 while cos 60° = 0.5, so it fails immediately.
• Standard values to have ready: sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1, with cosine running in reverse.

प्रश्न (हिन्दी में)

यदि \(sin(90^ - x) = cos(2x)\) है, तो x क्या है?

1. A. 0°
2. B. 30°
3. C. 25°
4. D. 15°