Cos and Sin only

Geometry Level 3

If $\cos52°\cos24°+\cos38°\cos66° = \sin(x°)$

where $0°$ $\le$ $x°$ $\le$ $90°$

Then find the value of $x$ .

1 solution

Ram Mohith
Jun 26, 2018

The given expression is : $\cos 52 ^\circ \cdot \cos 24 ^\circ + cos 38 ^\circ \cdot \cos 66 ^\circ = \sin x ^\circ$

$\implies \cos (90 - 38) ^\circ \cdot \cos 24 ^\circ + \cos 38 ^\circ \cdot \cos (90 - 24) ^\circ + \sin x ^\circ$

$\implies \sin 38 ^\circ \cdot \cos 24 ^\circ + cos 38 ^\circ \cdot \sin 24 ^\circ = \sin x ^\circ$

Now the expression is in the form $\sin A \cdot \cos B + \sin B \cdot \cos A = \sin(A + B)$

So, $A = 38, B = 24$

Now, $\sin(A + B) = \sin x^\circ$

$\implies x = 38 + 24 = 62 ^\circ$