Question 252 marksVery Short AnswerSection B

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

Official answer

sin⁡θ+cos⁡θ=3125\sin\theta + \cos\theta = \frac{31}{25}. Taking tan⁡θ=247=OppositeAdjacent\tan\theta = \frac{24}{7} = \frac{\text{Opposite}}{\text{Adjacent}}, form a right triangle with legs 24k,7k24k, 7k; by Pythagoras the hypotenuse is 25k25k (7-24-25 triplet). So sin⁡θ=2425\sin\theta = \frac{24}{25} and cos⁡θ=725\cos\theta = \frac{7}{25}, giving sin⁡θ+cos⁡θ=2425+725=3125\sin\theta + \cos\theta = \frac{24}{25} + \frac{7}{25} = \frac{31}{25} (= 1.24).

Answer keywords:tan thetasin thetacos thetaPythagoras theorem7-24-25 triplethypotenusetrigonometric ratios

Marking scheme

  1. 1.1 mark: correctly finding hypotenuse=25\text{hypotenuse} = 25 (using 7-24-25 Pythagorean triplet) and writing sin⁡θ=2425,cos⁡θ=725\sin\theta = \frac{24}{25}, \cos\theta = \frac{7}{25}.
  2. 2.1 mark: correct final addition sin⁡θ+cos⁡θ=3125\sin\theta + \cos\theta = \frac{31}{25} (accept equivalent decimal 1.24).

Hint

Draw a right triangle with opposite=24,adjacent=7\text{opposite} = 24, \text{adjacent} = 7; find hypotenuse using Pythagoras theorem (7-24-25 triplet), then compute sin⁡θ\sin \theta and cos⁡θ\cos \theta.

Quick oral answer

Given tan⁡θ=247\tan \theta = \frac{24}{7}, I take the opposite side as 24 and adjacent as 7, find the hypotenuse as 25 using Pythagoras theorem, giving sin⁡θ=2425\sin\theta = \frac{24}{25} and cos⁡θ=725\cos\theta = \frac{7}{25}, so sin⁡θ+cos⁡θ=3125\sin\theta + \cos\theta = \frac{31}{25}.

Common mistakes

  1. 1.Assuming tan⁡θ=247\tan\theta = \frac{24}{7} means sin⁡θ=24\sin\theta = 24 and cos⁡θ=7\cos\theta = 7 directly, without dividing by the hypotenuse.
  2. 2.Computing the hypotenuse incorrectly (e.g., adding 24+724+7 instead of using Pythagoras theorem).
  3. 3.Forgetting to add sin⁡θ\sin\theta and cos⁡θ\cos\theta as separate fractions with the same denominator, leading to an arithmetic slip.

Analysis and explanation

Given tan θ = 247\frac{24}{7}, build a right triangle to get sin⁡θ\sin \theta and cos⁡θ\cos \theta, then add them.


Concept

  • tan⁡θ=OppositeAdjacent\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}, so take Opposite=24k,Adjacent=7k\text{Opposite} = 24k, \text{Adjacent} = 7k.
  • Use Pythagoras theorem to get the hypotenuse before finding sin⁡θ,cos⁡θ\sin \theta, \cos \theta.

Key Points

  • 7-24-25 is a Pythagorean triple, so hypotenuse=25k\text{hypotenuse} = 25k directly.
  • sin⁡θ=2425,cos⁡θ=725  ⟹  sin⁡θ+cos⁡θ=3125\sin \theta = \frac{24}{25}, \cos \theta = \frac{7}{25} \implies \sin \theta + \cos \theta = \frac{31}{25}.

Common Mistakes

  • Treating 247\frac{24}{7} directly as sin⁡θcos⁡θ\frac{\sin\theta}{\cos\theta} without constructing the triangle.
  • Mixing up which side is opposite and which is adjacent.

Real-world Application

  • The same ratio-to-triangle reasoning underlies height-and-distance problems using angles of elevation/depression.

Interesting facts

7-24-25 is a classic Pythagorean triple (72+242=49+576=625=2527^2 + 24^2 = 49 + 576 = 625 = 25^2), frequently used in CBSE papers because it gives clean whole-number answers.

The word 'trigonometry' comes from the Greek words 'trigonon' (triangle) and 'metron' (measure), reflecting its origin in measuring triangles for astronomy and navigation.

Ancient Indian mathematician Aryabhata (476 CE) is credited with introducing the sine function (called 'jya') which later influenced trigonometry as used today.

Question details

Q25 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section B
Marks
2
Type
Very Short Answer
Chapter
Introduction to Trigonometry
Topic
Trigonometric Ratios
Bloom's level
apply
Syllabus topic
Unit V: Trigonometry — Chapter: Introduction to Trigonometry (Trigonometric Ratios of an acute angle)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is the hypotenuse taken as 25 here?

Because tan⁡θ=oppositeadjacent=247\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}, and by Pythagoras theorem the hypotenuse=242+72=625=25\text{hypotenuse} = \sqrt{24^2+7^2} = \sqrt{625} = 25, which is a well-known Pythagorean triple.

Does the value of k (the common multiplier) affect the final answer?

No. Since sin⁡θ\sin\theta and cos⁡θ\cos\theta are ratios of sides, the multiplier k cancels out, so the final answer 3125\frac{31}{25} is independent of the scale of the triangle.