Question 101 markMCQSection A

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

(a) 3/23/2 (b) 5/45/4 (c) −1-1 (d) 1/2

Introduction to Trigonometry
Trigonometric Identities - Evaluating an Expression

Options

(A)3/23/2
(B)5/45/4
(C)−1-1
(D)1/21/2

Official answer

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

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

Marking scheme

  1. 1.1 mark: correct option (b) 5/4; full credit reasoning: sin⁡2A+2cos⁡2A=1+cos⁡2A=1+1/4=5/4\sin^2 A + 2\cos^2 A = 1 + \cos^2 A = 1 + 1/4 = 5/4.

Hint

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

Quick oral answer

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

Common mistakes

  1. 1.Forgetting the identity sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1 and trying to find angle A=60∘A = 60^\circ and compute sin A separately without squaring correctly.
  2. 2.Arithmetic error while adding 3/43/4 and 1/21/2 (common denominator mistake).
  3. 3.Selecting cos⁡A=1/2\cos A = 1/2 itself as the final answer without completing the computation.

Analysis and explanation

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


Concept

  • Rewrite sin⁡2A+2cos⁡2A\sin^2 A + 2\cos^2 A as (sin⁡2A+cos⁡2A)+cos⁡2A=1+cos⁡2A(\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 A = 1/2, cos⁡2A=1/4\cos^2 A = 1/4, so the expression = 1+1/4=5/41 + 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\cos A = 1/2 itself (option d) without squaring and combining terms.
  • Introducing an incorrect subtraction (e.g., sin⁡2A−cos⁡2A\sin^2 A - \cos^2 A), leading to a negative distractor.

Interesting facts

The Pythagorean trigonometric identity sin⁡2θ+cos⁡2θ=1\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\cos A = 1/2 corresponds to the standard angle A=60∘A = 60^\circ, one of the most frequently used special angles (0∘,30∘,45∘,60∘,90∘)(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ) tested in CBSE trigonometry MCQs.

Question details

Q10 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Introduction to Trigonometry
Topic
Trigonometric Identities - Evaluating an Expression
Bloom's level
apply
Syllabus topic
Unit 4: Trigonometry — Introduction to Trigonometry: Trigonometric Identities (sin²θ + cos²θ = 1)
Author
CBSE
Reviewed by
Padhantu Academic Team

Spotted a mistake or something unclear?

Tell us and we will fix it.

Frequently asked questions

What is the basic trigonometric identity used here?

sin⁡2θ+cos⁡2θ=1\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\cos A = 1/2?

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