Question

How do you find the general solution of $\sin x = \cos 3x$?

Answer 1

In the above question, is based on the concept of trigonometry. The sine, cosine, tangent functions can be solved by using the trigonometric identities on these
functions or by using trigonometric functions whose sign depends on quadrants on the
cartesian plane.

Complete step by step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle.

The trigonometric functions having multiple angles are the multiple angle formula which can be written as $\cos \left( {nx} \right)$ where n is an integer.

The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant sine and cosecant are positive and in third only tangent and cotangent are positive. The fourth quadrant has only cosine and secant are positive.

First, we will consider the cosine function so here given function is $\cos 3x$.It can also be written as

$\sin \left( {90 - 3x} \right)$ because of the quadrant rules. It can also be written as

$\sin \left( {\dfrac{\pi }{2} - 3x} \right)$

So, equating this with $\sin x$we get,

$$\sin \left( {\dfrac{\pi }{2} - 3x} \right) = \sin x \\\ \\\$$

So, equating the angle since the sine function is the same on both sides.
$\Rightarrow \dfrac{\pi }{2} - 3x = x \\\ \Rightarrow \dfrac{\pi }{2} = 4x \\\ \Rightarrow x = \dfrac{\pi }{8} \\\ \Rightarrow 3x = \dfrac{{3\pi }}{8} \\\$
Therefore, we get the answer as $\dfrac{{3\pi }}{8}$

Note: An important thing to note is that $\sin \left( {{{90}^ \circ } - 3x} \right)$ changes to $\cos 3x$.

The reason is that $\sin x = \dfrac{{opposite}}{{hypotenuse}}$and $\cos x = \dfrac{{adjacent}}{{hypotenuse}}$.For example if 3x is 30 then 90-3x becomes 60. So, the side opposite to 60 is actually adjacent to the cos angle. Therefore, the sine function changes to cosine.