Question

How do you solve $\cos \theta =\sin \theta $ ?

Answer 1

Hint: In this question we will convert the trigonometric identity using trigonometric equivalences we will try to solve the range of values for the angle $\theta $ for which the value of $\sin \theta $ and $\cos \theta $ is same and then simplify the expression to get the required value of $\theta $ and write the solution in the general format which is the final answer.

Complete step-by-step answer:
 We have the given equation as:
$\Rightarrow \cos \theta =\sin \theta $
Now we know that the value of $\sin \theta $ and $\cos \theta $ toggles after every $\dfrac{\pi }{2}$ radians.
Therefore, we can write $\sin \theta $ in the form of $\cos \theta $ as:
$\Rightarrow \cos \theta =\cos \left( \pm \dfrac{\pi }{2}-\theta \right)$
Now since both the trigonometric functions are the same on both the sides of the equation, we can remove them. On removing it is to be remembered that the solution will toggle after even $2\pi $ radians.
Therefore, we can write the angles as:
$\Rightarrow \theta =\pm \dfrac{\pi }{2}-\theta +2k\pi $, where $k$is any integer.
Now on taking $\theta $ on the same side of the expression, we get:
$\Rightarrow \theta +\theta =\pm \dfrac{\pi }{2}+2k\pi $
On simplifying the left-hand side of the expression, we get:
$\Rightarrow 2\theta =\pm \dfrac{\pi }{2}+2k\pi $
Now on dividing both the sides of the expression by $2$, we get:
$\Rightarrow \theta =\pm \dfrac{\pi }{4}+k\pi $, which is the required solution.

Note: Now to check whether the solution, the above angle should be substituted in $\cos \theta $ and $\sin \theta $ and evaluated. If both the values are the same, the value of $\theta $ calculated is correct.
For simplicity purposes, we will consider $k=1$ and the solution in the first quadrant which implies that we are considering $\theta $ as positive.
Consider $\cos \left( \dfrac{\pi }{4}+1\times \pi \right)$
On simplifying, we get:
$\Rightarrow \cos \left( \dfrac{5\pi }{4} \right)$
Which has a value using calculator as: $-\dfrac{1}{\sqrt{2}}$
Now consider $\sin \left( \dfrac{\pi }{4}+1\times \pi \right)$
On simplifying, we get:
$\Rightarrow \sin \left( \dfrac{5\pi }{4} \right)$
Which has a value using calculator as: $-\dfrac{1}{\sqrt{2}}$
Therefore, we can conclude that $\cos \theta =\sin \theta $, therefore the solution is correct.
The trigonometric table should be remembered while doing these types of questions.
The principal values of $\sin $ and $\cos $ should be remembered.