Question 252 marksVery Short AnswerSection B

  1. (A) If tan⁡θ=247\tan \theta = \frac{24}{7}, then find the value of sin⁡θ+cos⁡θ\sin \theta + \cos \theta.

Introduction to Trigonometry
Trigonometric ratios from a given ratio

Official answer

sin⁡θ+cos⁡θ=3125=1.24\sin \theta + \cos \theta = \frac{31}{25} = 1.24. Since θ is acute, both ratios are positive and their sum is less than 2≈1.414\sqrt{2} \approx 1.414, which matches 1.241.24 — a quick sanity check that the answer is correct.

Answer keywords:tan θ24/7hypotenuse 25Pythagorassin θ 24/25cos θ 7/2531/251.24

Marking scheme

  1. 1.1 mark: correctly finding the hypotenuse=25\text{hypotenuse} = 25 using Pythagoras theorem (perpendicular 24, base 7).
  2. 2.1 mark: writing sin⁡θ=2425\sin \theta = \frac{24}{25}, cos⁡θ=725\cos \theta = \frac{7}{25} and obtaining sin⁡θ+cos⁡θ=3125\sin \theta + \cos \theta = \frac{31}{25} (accept 1.241.24).
  3. 3.Accept any equivalent method that reaches 3125\frac{31}{25}; award full marks even if written as 1.241.24.

Hint

tan⁡θ=perpendicularbase=247\tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{24}{7}; find the hypotenuse with Pythagoras (a 7-24-25 triple), then read off sin θ and cos θ.

Quick oral answer

Since tan⁡θ=247\tan \theta = \frac{24}{7}, I take perpendicular 24 and base 7, so hypotenuse=576+49=25\text{hypotenuse} = \sqrt{576+49} = 25; then sin⁡θ=2425\sin \theta = \frac{24}{25} and cos⁡θ=725\cos \theta = \frac{7}{25}, giving sin⁡θ+cos⁡θ=3125\sin \theta + \cos \theta = \frac{31}{25}, which is 1.241.24.

Common mistakes

  1. 1.Adding the numerators and the denominators separately (e.g. writing 2425+725=3150\frac{24}{25} + \frac{7}{25} = \frac{31}{50}) instead of keeping the common denominator 25.
  2. 2.Computing the hypotenuse incorrectly as 242−72\sqrt{24^2 - 7^2} or 24+7=3124 + 7 = 31 instead of 242+72=25\sqrt{24^2 + 7^2} = 25.
  3. 3.Swapping the roles of perpendicular and base, giving sin⁡θ=725\sin \theta = \frac{7}{25} and cos⁡θ=2425\cos \theta = \frac{24}{25} — the sum is the same here (3125\frac{31}{25}), but the individual ratios would be wrong and cost marks in follow-up parts.

Analysis and explanation

Given only tan θ, build a right triangle to read off sin θ and cos θ, then add them over a common denominator.


Concept

  • tan⁡θ=247\tan \theta = \frac{24}{7} fixes only the ratio of perpendicular to base, not actual lengths, so take perpendicular = 24, base = 7.
  • Pythagoras theorem gives the hypotenuse: 242+72=25\sqrt{24^2 + 7^2} = 25 (the well-known 7-24-25 Pythagorean triple).
  • sin⁡θ=2425\sin \theta = \frac{24}{25}, cos⁡θ=725\cos \theta = \frac{7}{25}, both read directly off the same triangle.

Common mistakes

  • Leaving the final answer as an unsimplified fraction instead of 3125\frac{31}{25} (or 1.241.24).
  • Forgetting that sin θ and cos θ share the same hypotenuse (25), so they add directly without a common-denominator step.
  • Not spotting the 7-24-25 triple and computing 625\sqrt{625} the long way.

Real-world

  • A surveyor who knows only the slope (tan) of a hill can use this exact method to recover the sine and cosine components of a distance along it.

Interesting facts

(7,24,25)(7, 24, 25) is one of the standard Pythagorean triples, discovered and used by ancient mathematicians long before trigonometry was formalised — it appears in problems attributed to the Pythagorean school around 500 BCE.

Because tan⁡θ=247≈3.43\tan \theta = \frac{24}{7} \approx 3.43, angle θ is about 73.7∘73.7^\circ, quite close to a right angle, which is why cos⁡θ\cos \theta (725=0.28\frac{7}{25} = 0.28) is small and sin⁡θ\sin \theta (2425=0.96\frac{24}{25} = 0.96) is large.

Question details

Q25 of 53 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Basic)
Year
2026
Section
Section B
Marks
2
Type
Very Short Answer
Chapter
Introduction to Trigonometry
Topic
Trigonometric ratios from a given ratio
Bloom's level
apply
Syllabus topic
Unit 5: Trigonometry — Introduction to Trigonometry: Finding trigonometric ratios from one given ratio (CBSE Class 10 Mathematics)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why can we take perpendicular = 24 and base = 7 directly?

Because tan⁡θ=247\tan \theta = \frac{24}{7} gives only the ratio of the two sides. The actual sides could be 24k24k and 7k7k for any k>0k > 0, but k cancels in every trigonometric ratio, so choosing k=1k = 1 (perpendicular 24, base 7) is valid and gives the correct sin θ and cos θ.

Do I need to rationalise or simplify 3125\frac{31}{25}?

No. 31 and 25 share no common factor, so 3125\frac{31}{25} is already in simplest form. You may optionally write it as the decimal 1.241.24; both are accepted for full marks.

Is θ acute or obtuse here?

In Class 10 all these ratios are defined for an acute angle in a right triangle, so θ is acute and both sin θ and cos θ are positive, making the sum positive.