Question

If sin (x+20°)=cos x. Find the value of x.

Answer 1

Sin(90-x)=cos(x), so 90-x=x+20. Solve for x:
90-x=x+20
70=2x
x=35 degrees.

Proof for the identity, sin(90-x)=cos(x):

Recall the following formulas:
$sin(x)= \frac{opposite}{hypotenuse} \\ cos(x)= \frac{adjacent}{hypotenuse}$

These sides are relative to the same reference angle, x. If you use the angle 90-x instead, then a few things change. The hypotenuse does not change, because the two triangles will share that side. Because the triangles are both right triangles, if they share the same hypotenuse, then they will form a rectangle. A rectangle has equal opposite sides. This is illustrated clearly in the attached image.

The side that was opposite of the angle x is the same length as the side that is adjacent to the angle 90-x, and the side that was adjacent to the angle x is the same length as the side that is opposite of the angle 90-x. So, if you have cos(x), it is the side adjacent to the angle x divided by the hypotenuse. The side adjacent of the angle x is equal to the side opposite of the angle 90-x. So cos(x) is also equal to the side opposite of the angle 90-x divided by the hypotenuse. Does this sound familiar? It is the trig function for sine. So sin(90-x)=cos(x).