If 7 tan θ = 4 then \frac{\left(\right. 7 sinθ - 3 cosθ \left.\right)}{\left(\right. 7 sinθ + 3 cosθ \left.\right)} = ?

Foundation

Mathematics Foundation

Introduction to Trigonometry

Question

If 7 tan θ = 4 then $\frac{(7 sinθ - 3 cosθ)}{(7 sinθ + 3 cosθ)} = ?$

$\frac{3}{7}$

$\frac{5}{7}$

$\frac{5}{14}$

$\frac{1}{7}$

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Given 7 tan θ = 4, so tan θ = 4/7. We need to find (7 sin θ - 3 cos θ)/(7 sin θ + 3 cos θ).

Divide numerator and denominator by cos θ to express in terms of tan θ:

$\frac{(7 \frac{\sin θ}{\cos θ} - 3)}{(7 \frac{\sin θ}{\cos θ} + 3)} = \frac{7 \tan θ - 3}{7 \tan θ + 3}$

Substitute tan θ = 4/7:

$\frac{7 \times \frac{4}{7} - 3}{7 \times \frac{4}{7} + 3} = \frac{4 - 3}{4 + 3} = \frac{1}{7}$

Final Answer: $\frac{1}{7}$