Question

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

Hint:

Memorizing common Pythagorean triplets like (7, 24, 25) helps save time in competitive exams.
If \(\tan \theta = \frac{a}{b}\), then \(\sin \theta + \cos \theta = \frac{a+b}{\sqrt{a^2+b^2}}\).

Answer 1

Step 1: Understanding the Concept:
In a right-angled triangle, \(\tan \theta\) is the ratio of the opposite side to the adjacent side. We can find the hypotenuse using the Pythagorean theorem to determine \(\sin \theta\) and \(\cos \theta\).
Step 2: Key Formula or Approach:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Step 3: Detailed Explanation:
Given \(\tan \theta = \frac{24}{7}\).
Let Opposite side = \(24k\) and Adjacent side = \(7k\).
\[ \text{Hypotenuse} = \sqrt{(24k)^2 + (7k)^2} = \sqrt{576k^2 + 49k^2} = \sqrt{625k^2} = 25k \]
Now, calculate \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = \frac{24k}{25k} = \frac{24}{25} \]
\[ \cos \theta = \frac{7k}{25k} = \frac{7}{25} \]
Adding the values:
\[ \sin \theta + \cos \theta = \frac{24}{25} + \frac{7}{25} = \frac{31}{25} \]
Step 4: Final Answer:
The value of \(\sin \theta + \cos \theta\) is \(\frac{31}{25}\).