Q10

1 markMCQSection A

If $\cos A = \frac{1}{2}$ , 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 is the fastest way to solve this?

Split $2\cos^2 A$ into $\cos^2 A + \cos^2 A$, group $\sin^2 A + \cos^2 A = 1$, leaving $1 + \cos^2 A$. With $\cos^2 A = \frac{1}{4}$, the answer is $\frac{5}{4}$.

Q: Can I just find the angle A instead?

Yes. $\cos A = \frac{1}{2}$ gives $A = 60°$, so $\sin A = \frac{\sqrt{3}}{2}$. Then $\sin^2 A + 2\cos^2 A = \frac{3}{4} + 2\left(\frac{1}{4}\right) = \frac{5}{4}$ — the same answer.

Introduction to Trigonometry

Evaluating trigonometric expressions

Options

(A)

\frac{3}{2}

(B)

\frac{5}{4}

(C)

-1

(D)

\frac{1}{2}

Official Answer

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

trigonometric identitysin²A + cos²A = 1cos A = 1/2A = 60°value 5/4evaluating expression

Marking Scheme

11 mark: correct option (b) $\frac{5}{4}$ .
2Internal justification: $1 + \cos^2 A = 1 + \frac{1}{4} = \frac{5}{4}$ (or direct $A = 60°$ substitution).

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 $\cos^2 A = \frac{1}{4}$ .

Quick Oral Answer

Group it as sin²A plus cos²A, which is 1, plus one more cos²A; since cos A is one-half, cos²A is one-fourth, giving 1 plus one-fourth, that is five-fourths.

Analysis & Explanation

Rewriting sin²A + 2cos²A as (sin²A + cos²A) + cos²A = 1 + cos²A instantly gives 1 + 1/4 = 5/4 when cos A = 1/2.

Concept

Identity sin²A + cos²A = 1 lets the expression be regrouped as 1 + cos²A.
Given cos A = 1/2 ⇒ cos²A = 1/4.

Key steps

$\sin^2 A + 2\cos^2 A = 1 + \cos^2 A = 1 + \frac{1}{4} = \frac{5}{4}$ → option (b).
Cross-check: $\cos A = \frac{1}{2} \Rightarrow A = 60° \Rightarrow \sin A = \frac{\sqrt{3}}{2}$ ; $\sin^2 A + 2\cos^2 A = \frac{3}{4} + 2\left(\frac{1}{4}\right) = \frac{5}{4}$ ✓.

Common mistakes

(a) 3/2: comes from wrongly writing 1 + 2cos²A instead of 1 + cos²A (double-counting one cos²A term).
(c) −1: impossible, since the expression is a sum of squares and must be positive — a sign-error trap.
(d) 1/2: results from dropping the sin²A term entirely.

Real-world

The grouping trick (splitting 2cos²A into cos²A + cos²A so one part combines with sin²A) is a standard time-saver in trigonometry exams.

Common Mistakes

1Getting $\frac{3}{2}$ by writing the expression as $1 + 2\cos^2 A$ instead of $1 + \cos^2 A$ (double-counting the cos² term).
2Choosing $-1$ despite the expression being a sum of squares that can never be negative.
3Forgetting to square: using $\cos A = \frac{1}{2}$ directly in place of $\cos^2 A = \frac{1}{4}$ .

Interesting Facts

cos 60° = 1/2 is one of the standard angle values every Class 10 student memorises; 60° is also the interior angle of an equilateral triangle.

Expressions like this are the algebra behind alternating-current and wave power calculations, where averaging sin² and cos² terms is routine in physics and engineering.

Spotted a mistake or something unclear?

Tell us — we fix reported answers fast.

Frequently Asked Questions

What is the fastest way to solve this?

Split $2\cos^2 A$ into $\cos^2 A + \cos^2 A$ , group $\sin^2 A + \cos^2 A = 1$ , leaving $1 + \cos^2 A$ . With $\cos^2 A = \frac{1}{4}$ , the answer is $\frac{5}{4}$ .

Can I just find the angle A instead?

Yes. $\cos A = \frac{1}{2}$ gives $A = 60°$ , so $\sin A = \frac{\sqrt{3}}{2}$ . Then $\sin^2 A + 2\cos^2 A = \frac{3}{4} + 2\left(\frac{1}{4}\right) = \frac{5}{4}$ — the same answer.

Why can the answer never be −1?

The expression is a sum of squared quantities multiplied by positive numbers, so it is always positive. Option (c) is a trap.