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

Step 1: Understanding the Concept:
Trigonometric ratios represent the ratios of sides in a right-angled triangle. We can use the given ratio to find the hypotenuse and then determine the other ratios.
Step 2: Key Formula or Approach:
1. \(\tan \theta = \frac{Opposite}{Adjacent}\)
2. Pythagoras' Theorem: \(Hypotenuse = \sqrt{Opp^2 + Adj^2}\)
3. \(\sin \theta = \frac{Opp}{Hyp}\), \(\cos \theta = \frac{Adj}{Hyp}\)
Step 3: Detailed Explanation:
1. Let Opposite = 24k and Adjacent = 7k. 2. Calculate Hypotenuse (\(h\)): \[ h = \sqrt{(24)^2 + (7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \] 3. Find ratios: \[ \sin \theta = \frac{24}{25}, \quad \cos \theta = \frac{7}{25} \] 4. Add the values: \[ \sin \theta + \cos \theta = \frac{24}{25} + \frac{7}{25} = \frac{31}{25} \]
Step 4: Final Answer:
The value is \(\frac{31}{25}\).