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

Q: 24. (A) If $\tan \theta = \frac{24}{7}$, then find the value of $\sin \theta + \cos \theta$.

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

Introduction to Trigonometry

Trigonometric ratios from a given ratio

Official Answer

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

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

Marking Scheme

11 mark: correctly finding the $\text{hypotenuse} = 25$ using Pythagoras theorem (perpendicular 24, base 7).
21 mark: writing $\sin \theta = \frac{24}{25}$ , $\cos \theta = \frac{7}{25}$ and obtaining $\sin \theta + \cos \theta = \frac{31}{25}$ (accept $1.24$ ).
3Accept any equivalent method that reaches $\frac{31}{25}$ ; award full marks even if written as $1.24$ .

Hint

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

Quick Oral Answer

Since $\tan \theta = \frac{24}{7}$ , I take perpendicular 24 and base 7, so $\text{hypotenuse} = \sqrt{576+49} = 25$ ; then $\sin \theta = \frac{24}{25}$ and $\cos \theta = \frac{7}{25}$ , giving $\sin \theta + \cos \theta = \frac{31}{25}$ , which is $1.24$ .

Analysis & 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

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

Common mistakes

Leaving the final answer as an unsimplified fraction instead of $\frac{31}{25}$ (or $1.24$ ).
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 $\sqrt{625}$ 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.

Common Mistakes

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

Interesting Facts

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

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Frequently Asked Questions

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

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

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

No. 31 and 25 share no common factor, so $\frac{31}{25}$ is already in simplest form. You may optionally write it as the decimal $1.24$ ; 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.