Q10

1 markMCQSection A

If $\cos A = 1/2$ , then the value of $\sin^2 A + 2\cos^2 A$ is :

(a) $3/2$ (b) $5/4$ (c) $-1$ (d) 1/2

Q: What angle has $\cos A = 1/2$?

$A = 60^\circ$, since $\cos 60^\circ = 1/2$ is a standard trigonometric value.

Introduction to Trigonometry

Trigonometric Identities - Evaluating an Expression

Options

(A)

3/2

(B)

5/4

(C)

-1

(D)

1/2

Official Answer

(b) 5/4 — $\sin^2 A + 2\cos^2 A = 1 + \cos^2 A = 1 + 1/4 = 5/4$ .

trigonometric identitysin squared plus cos squaredcos A = 1/2evaluating trigonometric expression5/4

Marking Scheme

11 mark: correct option (b) 5/4; full credit reasoning: $\sin^2 A + 2\cos^2 A = 1 + \cos^2 A = 1 + 1/4 = 5/4$ .

Hint

Rewrite $\sin^2 A + 2\cos^2 A$ as $(\sin^2 A + \cos^2 A) + \cos^2 A = 1 + \cos^2 A$ , then substitute $\cos^2 A = 1/4$ .

Quick Oral Answer

Since $\cos A = 1/2$ , $\cos^2 A = 1/4$ and $\sin^2 A = 3/4$ by the identity $\sin^2 A + \cos^2 A = 1$ ; so $\sin^2 A + 2\cos^2 A = 3/4 + 1/2 = 5/4$ .

Analysis & Explanation

Tests quick simplification using the Pythagorean identity $\sin^2 A + \cos^2 A = 1$ .

Concept

Rewrite $\sin^2 A + 2\cos^2 A$ as $(\sin^2 A + \cos^2 A) + \cos^2 A = 1 + \cos^2 A$ — the fastest route to the answer.

Key points

Given $\cos A = 1/2$ , $\cos^2 A = 1/4$ , so the expression = $1 + 1/4 = 5/4$ — option (b).

Common mistakes

Forgetting the identity shortcut and making arithmetic slips while separately computing sin²A and 2cos²A.
Stopping at $\cos A = 1/2$ itself (option d) without squaring and combining terms.
Introducing an incorrect subtraction (e.g., $\sin^2 A - \cos^2 A$ ), leading to a negative distractor.

Common Mistakes

1Forgetting the identity $\sin^2 A + \cos^2 A = 1$ and trying to find angle $A = 60^\circ$ and compute sin A separately without squaring correctly.
2Arithmetic error while adding $3/4$ and $1/2$ (common denominator mistake).
3Selecting $\cos A = 1/2$ itself as the final answer without completing the computation.

Interesting Facts

The Pythagorean trigonometric identity $\sin^2\theta + \cos^2\theta = 1$ is a direct restatement of the Pythagorean theorem applied to the unit circle, linking algebra and geometry.

$\cos A = 1/2$ corresponds to the standard angle $A = 60^\circ$ , one of the most frequently used special angles $(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ)$ tested in CBSE trigonometry MCQs.

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Frequently Asked Questions

What is the basic trigonometric identity used here?

$\sin^2\theta + \cos^2\theta = 1$ for any angle θ, which lets you substitute one ratio in terms of the other.

What angle has $\cos A = 1/2$ ?

$A = 60^\circ$ , since $\cos 60^\circ = 1/2$ is a standard trigonometric value.