Question

If \[cos 42^{\circ} = \sin A \], then find the value of A.

Answer 1

Hint:
We can very easily and quickly solve the question by applying the formula which defines the relation between the values of cosine and sine, tangent and cotangent, or secant and cosecant, when we have to calculate or solve for the values of angles of such questions, compare the values or the angles in such questions or even convert one trigonometric function to another trigonometric function. The only catch is that we have to remember the values, formulae and relation between these trigonometric functions.

Formula Used:
We will use the formula for the relation between cosine and sine:
$\cos A = \sin (90-A)$

Complete step by step solution:
We have been given an expression in cosine and we have to find its equivalent in the sine.
The expression in cosine is given to be – \[\cos 42^{\circ} \].
To get a hold of what this is, let us assume that we have been given the value of \[\cos 40^{\circ} \] and we have to calculate or determine at what measure of the angle it should be so that the sine also gives the same answer. This thing of calculating the value of sine can be easily calculated by using the following formula:
$\cos A = \sin (90-A)$
and its converse is also true, that is:
$\sin A = \cos (90-A)$
Hence, the value of sine which is equal to the value of cos40\[^\circ \] is –
\[\cos 40^{\circ} \]= $\sin (90-40)$ = \[\sin 50^{\circ} \]
The variance of a given relation is also true for the value pairs of tangent-cotangent and secant-cosecant.
So, it can be said that:
$\tan A = \cot (90-A)$
or conversely,$\cot A = \tan (90-A)$
and finally,
$\sec A = \csc (90-A)$
or conversely, $\csc A = \sec (90-A)$
Using the above written formula, we get:
\[\cos 42^{\circ} \]= $\sin (90-42) = \sin 48$\[^\circ \]

Hence, the answer for the value of A is \[A=48^{\circ} \].

Note:
For solving such questions, the best thing is to first write down the known quantity, then write the unknown one. Then, think of the formula which defines the relation between the two. And then, plug in the values and you will have your answer