Question

How do you solve sin A – cos A = 0?

Answer 1

We will first take the cosine part on the right hand side. Then, we will bring it in denominator and get one trigonometric ratio and then solve it.

Complete step-by-step answer:
We are given that we need to solve sin A – cos A = 0.
We will take cos A from subtraction in the left hand side to addition in right hand side so that we obtain the following equation:-
$\Rightarrow$ sin A = cos A + 0
Since we know that if we add 0 to anything, it gets that thing back. So, we will then obtain:-
$\Rightarrow$ sin A = cos A
Now, we can write this as:-
$\Rightarrow$ sin A = 1.cos A
Taking the cos A from multiplication in the right hand side to division in the left hand side. So, that we will then obtain the following equations:-
$\Rightarrow \tan A = 1$
Since tangent takes the value 1 at $n\pi + \dfrac{\pi }{4}$.

Therefore, $A = n\pi + \dfrac{\pi }{4}$ is the required answer.

Note:
We know that tangent takes value of 1 at 45 degrees and if we see that it is positive in the third quadrant as well. And we have value 1 of tangent again when we cover the whole triangle and come back to it.
If we put the values of n from 0 to any positive integer, we will always stay in either the first or third quadrant.
If we directly approach the problem, we will have a problem with going in the third quadrant because sin and cos both are together positive in only the first quadrant and in the third quadrant both are negative, therefore, they both can be equal to each other. Now, you can also see the same using the graph of sin, cosine or tangent as well because they will give you the same result but it is just extremely difficult to read this in a graph.