### Home > PC > Chapter 5 > Lesson 5.2.1 > Problem5-51

5-51.

Yuka was trying to solve the following equation for $θ$, but was so confused by all the symbols; she did not know what to do.

$\frac { \operatorname { sin } ^ { 3 } \theta - 4 \operatorname { sin } \theta } { \operatorname { sin } ^ { 2 } \theta } = 3$

1. Her friend Carlo suggested that she make the substitution $u = \sin θ$. Make this substitution and write the equation in terms of $u$.

$\frac{u^3-4u}{u^2} = 3$

2. Solve the equation for $u$.

u$^{2} − 4u = 3u^{2}$
$u^{3} − 3u^{2} − 4u = 0$
$u\left(u^{2} − 3u − 4\right) = 0$
$u\left(u − 4\right)\left(u + 1\right) = 0$
$u = 0, u = 4, u = −1$
But $u ≠ 0$ because you would have division by zero.

3. Solve the equation for $θ$ if $0 ≤ θ ≤ 2π$.

$u = \sinθ$
$−1 = \sinθ$

$\theta=270^º\text{ or } \frac{3\pi}{2}$

$4 = \sinθ$
$u = \sinθ$