Question

If \( \tan \theta = \frac{3}{4} \) and \( \theta \) is in the first quadrant, find the value of \( \sin \theta + \cos \theta \).

• A. \( \frac{5}{4} \)
• B. \( \frac{7}{5} \)
• C. \( \frac{5}{7} \)
• D. \( \frac{4}{5} \)

Hint:

For trigonometric problems with \( \tan \theta \), construct a right triangle to find \( \sin \theta \) and \( \cos \theta \), using the hypotenuse \( \sqrt{\text{opposite}^2 + \text{adjacent}^2} \).

Answer 1

The Correct Option is B

Given \( \tan \theta = \frac{3}{4} \) in the first quadrant, where \( \sin \theta \) and \( \cos \theta \) are positive. Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), assume a right triangle with opposite side 3 and adjacent side 4. The hypotenuse is: \[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] Thus: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \] Calculate: \[ \sin \theta + \cos \theta = \frac{3}{5} + \frac{4}{5} = \frac{7}{5} \] Alternatively, use the identity: \[ \sin \theta + \cos \theta = \sqrt{2} \sqrt{\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta} = \sqrt{2} \sqrt{1 + 2 \cdot \frac{\tan \theta}{1 + \tan^2 \theta}} \] Since \( \tan \theta = \frac{3}{4} \), this is complex, so the triangle method is simpler. The value is: \[ {\frac{7}{5}} \]