# How do you find the sin of theta if sin of theta is radical 5 times cos of theta and 0<theta<pi/2?

## How do you find the sinθ if sinθ=sqrt(5cosθ) and 0<θ<$\frac{\pi}{2}$?

May 6, 2018

No solution.

#### Explanation:

$\text{Given : } \sin \theta = \sqrt{5 \cos \theta} , \theta \in 0 < \frac{\pi}{2}$

sin^2 theta = 5 cos theta, color(crimson)(" squaring both sides"

1 - cos^2 theta = 5 cos theta, color(crimson)(" as " sin^2 theta + cos^2 theta = 1

${\cos}^{2} \theta + 5 \cos \theta - 1 = 0$

$L e t \cos \theta = x$

${x}^{2} + 5 x - 1 = 0$

$x = \frac{- 5 \pm \sqrt{{5}^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

$x = \frac{- 5 \pm \sqrt{21}}{2}$

$x = \frac{- 5 \pm 4.9}{2}$

$x = - 0.05 , - 4.95$

$\cos \theta = - 0.05 , \cancel{- 4.05} , \text{ as cos can have a value between -1 and 1}$

$\cos \theta = - 0.05$

$\text{Since " theta in 0 < pi/2, " where cos is always positive, we cannot have a solution for the given sum.}$