Question

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

Hint:

For part (B), don't waste time finding \(\sin\) and \(\cos\) individually; always simplify the algebraic expression using trigonometric identities first!

Answer 1

Given:
\(\tan \theta = \frac{24}{7}\)

Step 1: Convert tanθ into sinθ and cosθ
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{24}{7} \]
Let:
\(\sin\theta = 24k\) and \(\cos\theta = 7k\)

Step 2: Use the identity
\[ \sin^2\theta + \cos^2\theta = 1 \]
Substitute the values:
\[ (24k)^2 + (7k)^2 = 1 \] \[ 576k^2 + 49k^2 = 1 \] \[ 625k^2 = 1 \] \[ k = \frac{1}{25} \]

Step 3: Find sinθ and cosθ
\[ \sin\theta = 24k = \frac{24}{25} \] \[ \cos\theta = 7k = \frac{7}{25} \]

Step 4: Add the two values
\[ \sin\theta + \cos\theta = \frac{24}{25} + \frac{7}{25} \] \[ = \frac{31}{25} \]

Final Answer:
\[ \boxed{\frac{31}{25}} \]