If 3θ = 4θ, find the value of θ + ²θ - 1; 0°< θ < 90°

MathsCBSE Class 10 · Basic · Sample paper, session 2026-27 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic sample paper 2026-27: question 22

From the official CBSE sample question paper for 2026-27, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerIntroduction to Trigonometry

If $3 sin θ = 4 cos θ$ , find the value of $sin θ + cos^{2} θ - 1$ ; $0^{\circ} < θ < 9 0^{\circ}$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

Open one hint at a time and try again before the next.

Hint 1

Divide both sides by $3 cos θ$ to get one trigonometric ratio.

Hint 2

Draw a right triangle with $tan θ = \frac{4}{3}$ and use Pythagoras to get $sin θ$ and $cos θ$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$3 sin θ = 4 cos θ \Rightarrow \frac{sin θ}{cos θ} = \frac{4}{3} \Rightarrow tan θ = \frac{4}{3}$
½ mark
Take a right triangle with perpendicular 4 and base 3. Then the hypotenuse is $3^{2} + 4^{2} = 5$ .
So $sin θ = \frac{4}{5}$ and $cos θ = \frac{3}{5}$ .
½ mark
$sin θ + cos^{2} θ - 1 = \frac{4}{5} + \frac{9}{25} - 1 = \frac{20 + 9 - 25}{25} = \frac{4}{25}$
1 mark

Answer: $\frac{4}{25}$

Where the marks are

Obtains $tan θ = \frac{4}{3}$ (½ mark)
Finds $sin θ = \frac{4}{5}$ and $cos θ = \frac{3}{5}$ (½ mark)
Substitutes and gets $\frac{4}{25}$ (1 mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Proving Simple Trigonometric Identities

Practise this chapterBoard-style questions from Introduction to Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2026-27: question 22

Hints

Step-by-step solution

Revise the chapter

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