Question

If A lies in the third quadrant and 3 tan A - 4 = 0, then find the value of 25 sin 2A + 4sinA + 3cos A

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Answer 1

In third quadrant only tanA & cotA gives positive values. Other trigonometric function gives negative value.

Given 3 tan A - 4 =

tan A = $\frac{4}{3}$

sinx & cosx is negative in lll quadrant.

sinA = $-\frac{4}{5}$

cosA = $-\frac{3}{5}$

$\Rightarrow$ 25 sin 2A + 4sinA + 3cosA = 25 (2sinA $\times$ cosA) + 4sinA + 3 cosA

= 25 $\times$ 2 $\times$ $(-\frac{4}{5})$ $\times$ $\left(\frac{-3}{5}\right)$ + 4 $\times$ $\left(\frac{-4}{5}\right)$ + 3 $\times$ $\left(\frac{-3}{5}\right)$

= 24 - $\frac{16}{5}-\frac{9}{5}$

= 24 - $\frac{25}{5}$ = 24 - 5 = 19