Answer :
A = B is our answer.
We are given,
[tex]A = cos\pi \\B = sin\frac{3\pi }{2}[/tex]
we need to see how A and B are related.
What is the sign and domain [tex]\theta[/tex] for different trigonometric functions in different quadrants?
In the first quadrant, all trigonometric functions are positive and the domain is denoted by [tex]\frac{\pi}{2} -\theta[/tex].
In the second quadrant, only the sin and cosec functions are positive and the domain is denoted by [tex]\frac{\pi}{2}+\theta ~~or ~~\pi - \theta[/tex].
In the third quadrant, only the tan and cot functions are positive and the domain is denoted by [tex]\pi + \theta~~or~~\frac{3\pi}{2}-\theta[/tex].
In the fourth quadrant, only the cos and sec functions are positive and the domain is denoted by [tex]\frac{3\pi}{2} + \theta~~or~~2\pi-\theta[/tex].
We can write,
[tex]sin\frac{3\pi}{2} = sin(\frac{\pi}{2}+\pi)[/tex]
Now,
[tex]\frac{\pi}{2}+\theta[/tex] is in the second quadrant and in the second quadrant sin is positive.
We can change
[tex]sin(\frac{\pi}{2}+\pi)~~into~~cos\pi[/tex] in the second quadrant.
[tex]Thus,\\A =cos\pi\\B= sin\frac{3\pi}{2} = cos\pi\\A=B[/tex]
Learn more about the trigonometric functions comparison here:
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