Q25
2 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).

tan thetasin thetacos thetaPythagoras theorem7-24-25 triplethypotenusetrigonometric ratios

Marking Scheme

  • 11 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}.
  • 21 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}.

Analysis & 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.

Common Mistakes

  1. 1Assuming 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. 2Computing the hypotenuse incorrectly (e.g., adding 24+724+7 instead of using Pythagoras theorem).
  3. 3Forgetting to add sin⁡θ\sin\theta and cos⁡θ\cos\theta as separate fractions with the same denominator, leading to an arithmetic slip.

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.

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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.