# How do I solve the equation 90 cos X - 14 sin X = 72?

## 90, 14 and 72 are parameters as well, but those are results from another equation.

May 3, 2018

Write $\sin x$ in terms of $\cos x$.

#### Explanation:

You can write the 14$\sin x$ as 14sqrt(1-cos^2 (x) and then rearrange the terms such that the equation is:

14sqrt(1-cos^2(x) = 90$\cos x$ - 72

Upon squaring on both sides and solving the quadratic in $\cos x$, you get the value of $\cos x$, from which you get the value of x.

May 3, 2018

$x = {28}^{\circ} 97 + k {360}^{\circ}$
$x = - {46}^{\circ} 61 + k {360}^{\circ}$

#### Explanation:

90cos x - 14sin x = 72
Divide both sides by 90
$\cos x - \left(\frac{14}{90}\right) \sin x = \frac{72}{90} = \frac{4}{5} = 0.8$ (1)
Call $\tan t = \sin \frac{t}{\cos t} = \frac{14}{90} = \frac{7}{45}$ . Calculator gives:-->
$t = {8}^{\circ} 84$, and cos t = 0.988
The equation (1) becomes:
$\cos x . \cos t - \sin t . \sin x = 0.8 \cos t = 0.79$
Reminder: cos a.cos b - sin a.sin b = cos (a + b)
Therefor,
$\cos \left(x + t\right) = 0.79 = \cos 37.81$-->
$\left(x + t\right) = \pm 37.81$
a. x + t = 37.81 --> $x = 37.81 - 8.84 = {28}^{\circ} 97$
b. x + 8.84 = - 37.81 --> $x = - 37.81 - 8.84 = - {46}^{\circ} 61$
For general answers, add $k {360}^{\circ}$
Check by calculator:
x = - 46.61 --> 90cos x = 61.83 --> 14sin x = - 10.17
90cos x - 14sin x = 61.83 + 10.17 = 72.00. Proved.