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Question

Question: How do you solve \(\cos x = 0\)?...

How do you solve cosx=0\cos x = 0?

Explanation

Solution

Here we need to find the solution of the given trigonometric equation. For that, we will use the unit circle, and then we will use the basic trigonometric ratios inside that unit circle. Then we will use the given value of the cosine function there and at last, we will find the general solution of the given trigonometric equation.

Complete step by step solution:
Here we need to find the solution of the given trigonometric equation i.e. cosx=0\cos x = 0.
We know that the cosine function is defined as the ratio of the adjacent side to the hypotenuse.
Now, we will draw the unit circle with center OO. We know that the circumference of the unit circle is equal to 2π2\pi .

Now, we will consider the triangle OPMOPM and it is clear that;
cosx=OMOP\Rightarrow \cos x = \dfrac{{OM}}{{OP}}
It is given that cosx=0\cos x = 0, so we will substitute the value zero here.
0=OMOP OM=0 \Rightarrow 0 = \dfrac{{OM}}{{OP}} \\\ \Rightarrow OM = 0 \\\
We can clearly see that when OM=0OM = 0 then the final arm OPOP of the angle xx coincides with OYOY or OYOY'.
Similarly, the final arm OPOP coincides with OYOY or OYOY' when x=π2,3π2,5π2,7π2,......,π2,3π2,5π2,7π2x = \dfrac{\pi }{2},\dfrac{{3\pi }}{2},\dfrac{{5\pi }}{2},\dfrac{{7\pi }}{2},......, - \dfrac{\pi }{2}, - \dfrac{{3\pi }}{2}, - \dfrac{{5\pi }}{2}, - \dfrac{{7\pi }}{2} i.e. when angle xx is an odd multiple of π2\dfrac{\pi }{2} i.e. when x=(2n+1)π2x = \left( {2n + 1} \right)\dfrac{\pi }{2} when nZn \in Z is the solution of the given trigonometric equation.

Note:
Here we have obtained the solution of the given trigonometric equations. Trigonometric equations are defined as the equation which involves the trigonometric functions with angles as the variables. Also, we need to remember that trigonometric identities are defined as the equalities which include the trigonometric functions. They are always true for every value of the occurring variables where both sides of the equality are well defined. The cosine function is an even function, whereas the sine function is an odd function.