Question

Evaluate the following:
$\dfrac{{\cos 37^\circ }}{{\sin 53^\circ }}$

Answer 1

Hint:
Here, we are asked to find the value of the trigonometric fraction $\dfrac{{\cos 37^\circ }}{{\sin 53^\circ }}$.
Now, use the property $\sin x = \cos \left( {90^\circ - x} \right)$ and find the value of \[\sin 53^\circ \] in the terms of cosine function.
Thus, to get the required answer, substitute the value of \[\sin 53^\circ \] in terms of cosine function in the given trigonometric equation.

Complete step by step solution:
Here, we are asked to find the value of the trigonometric fraction $\dfrac{{\cos 37^\circ }}{{\sin 53^\circ }}$ .
We know the property that, $\sin x$ can also be written as $\cos \left( {90^\circ - x} \right)$ i.e. $\sin x = \cos \left( {90^\circ - x} \right)$ .
So, using the above property, we can write $\sin 53^\circ $ as $\cos \left( {90^\circ - 53^\circ } \right)$
 $\therefore \sin 53^\circ = \cos \left( {90^\circ - 53^\circ } \right) = \cos 37^\circ $ .
Now, we will substitute the value of $\sin 53^\circ $ as $\cos 37^\circ $ in the given trigonometric fraction.
$\therefore \dfrac{{\cos 37^\circ }}{{\sin 53^\circ }} = \dfrac{{\cos 37^\circ }}{{\cos 37^\circ }} = 1$

Thus, we get the required value of the given trigonometric fraction $\dfrac{{\cos 37^\circ }}{{\sin 53^\circ }}$ as 1.

Note:
Alternatively, we can also write $\cos 37^\circ $ in the terms of the sine function by using the property $\cos y = \cos \left( {90^\circ - y} \right)$. Thus, by substituting the value of $\cos 37^\circ $ in terms of sine function in the given trigonometric fraction, we get the required answer.
Some angle properties of trigonometric functions:
1) $\sin x = \cos \left( {90^\circ - x} \right)$
2) $\cos x = \sin \left( {90^\circ - x} \right)$
3) $\tan x = \cot \left( {90^\circ - x} \right)$
4) $\sin x = \sin \left( {360^\circ + x} \right)$
5) $\cos x = \cos \left( {360^\circ + x} \right)$
6) $\tan x = \tan \left( {360^\circ + x} \right)$