Question

if 5 tan a = 4, show that , 5 sin a - 3 cos a/5 sin a + 2 cos a = 1/6

Answer 1

Task 1. If $5 \tan a = 4$, show that $\frac{5 \sin a - 3 \cos a}{5 \sin a + 2 \cos a} = \frac{1}{6}$.

Solution. Divide the numerator and denominator of the fraction by $\cos a$:

$$\frac{5 \sin a - 3 \cos a}{5 \sin a + 2 \cos a} = \frac{5 \frac{\sin a}{\cos a} - 3}{5 \frac{\sin a}{\cos a} + 2}.$$

But $\frac{\sin a}{\cos a} = \tan a$, therefore,

$$\frac{5 \sin a - 3 \cos a}{5 \sin a + 2 \cos a} = \frac{5 \tan a - 3}{5 \tan a + 2} = \frac{4 - 3}{4 + 2} = \frac{1}{6},$$

as desired.

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