# Let tan x = 2.4, sin y = 0.6, and both x and y be between 0° and 90°. Then cos(x + y) equals?

May 12, 2018

$- \frac{16}{65}$

#### Explanation:  May 13, 2018

cos (x + y) = - 0.246

#### Explanation:

Use trig identity:
cos (x + y) = cos x.cos y - sin x.sin y (1)
First, find cos y knowing sin y = 0.6
${\cos}^{2} y = 1 - {\sin}^{2} y = 1 - 0.36 = 0.64$
$\cos y = 0.8$ (because y is in Quadrant 1)
Next, find sin x and cos x, knowing tan x = 2.4.
${\cos}^{2} x = \frac{1}{1 + {\tan}^{2} x} = \frac{1}{1 + 5.76} = \frac{1}{6.76}$
$\cos x = \frac{1}{2.6}$ (because x is in Q. 1)
${\sin}^{2} x = 1 - {\cos}^{2} x = 1 - \frac{1}{6.76} = \frac{5.76}{6.76}$
$\sin x = \frac{2.4}{2.6}$
Replace all numeric values into equation (1), we get:
$\cos \left(x + y\right) = \left(\frac{1}{2.6}\right) \left(0.8\right) - \left(\frac{2.4}{2.6}\right) \left(0.6\right) = \frac{0.8}{2.6} - \frac{1.44}{2.6}$
$\cos \left(x + y\right) = - \frac{0.64}{2.6} = - 0.246$
Check by calculator.
$\sin y = 0.6$ --> $y = {36}^{\circ} 87$
$\cos x = \frac{1}{2.6}$ --> $x = {67}^{\circ} 38$
$x + y = 36.87 + 67.38 = {104}^{\circ} 25$ --> cos 104.25 = - 0.246. OK