If $\tan\theta = \frac{24}{7}$, then find the value of $\sin\theta + \cos\theta$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:24 · grounding rag
Model Answer
Given: $\tan\theta = \dfrac{24}{7}$
So, Opposite side = 24, Adjacent side = 7.
By Pythagoras theorem:
Hypotenuse $= \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25$
Therefore:
$$\sin\theta = \frac{24}{25}, \quad \cos\theta = \frac{7}{25}$$
$$\sin\theta + \cos\theta = \frac{24}{25} + \frac{7}{25} = \frac{31}{25}$$
Source: Chapter 8, Section 8.1
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Explanation
- Always draw a right triangle with the given ratio, then use Pythagoras to find the hypotenuse.
- Examiners award 1 mark for finding hypotenuse = 25 and 1 mark for the correct final value $\dfrac{31}{25}$.
- Write each step clearly; don't skip the Pythagoras step.