Question

How do you solve for x in $\cos (x + 30) = \sin (5x + 12)$ ?

Answer 1

Hint: In this question, we are given an algebraic expression containing trigonometric functions of one unknown variable quantity. We know that to find the value of “n” unknown variables, we need “n” number of equations. In the given algebraic expression, we have 1 unknown quantity and exactly one equation to find the value of x. So we can easily find the value of x by using the knowledge of trigonometric ratios, we will convert either the sine function into the cosine function or the cosine function into sine function and then apply the given arithmetic operations.

Complete step by step answer:
We are given that $\cos (x + 30) = \sin (5x + 12)$
We know that –
$\cos \theta = \sin (90 - \theta )$ and $\cos \theta = \sin (90 + \theta )$
$ \Rightarrow \cos (x + 30) = \sin [90 - (x + 30)] = \sin (90 - x - 30) = \sin (60 - x)$ and $ \Rightarrow \cos (x + 30) = \sin (90 + x + 30) = \sin (120 + x)$
So, we get –
$\sin (60 - x) = \sin (5x + 12)$ and $\,\sin (120 + x) = \sin (5x + 12)$
We know that when $\sin A = \sin B \Rightarrow A = B$
So,
$60 - x = 5x + 12$ and $120 + x = 5x + 12$
$ \Rightarrow 60 - 12 = 5x + x$ and $120 - 12 = 5x - x$
$ \Rightarrow 6x = 48$ and $4x = 108$
$ \Rightarrow x = 8$ and $x = 27$

Hence, when $\cos (x + 30) = \sin (5x + 12)$ , we get $x = 8^\circ $ and $x = 27^\circ $ .

Note: The written algebraic expression contains the sine and the cosine functions that are trigonometric functions. Trigonometric ratios are the ratio of any two sides of a right-angled triangle. They tell the relation between any two sides of the right-angled triangle and one of its angles other than the right angle. All the trigonometric functions are related to each other by some identities, so one ratio can be converted into the other by using their knowledge. We know that trigonometric functions are periodic functions, that is, they give the same value as the answer after a fixed interval, so the given question can have infinite answers. The written answers are the base angles that satisfy the given equation.