Question

What are the exact values of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians?

Answer 1

Hint: To find the exact values of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians, we will use the concept of reference angle. Using the concept of reference angle, we will write \[\cos \left( {\dfrac{{3\pi }}{4}} \right) = \cos \left( {\pi - \dfrac{\pi }{4}} \right)\] and \[\sin \left( {\dfrac{{3\pi }}{4}} \right) = \sin \left( {\pi - \dfrac{\pi }{4}} \right)\]. Then using the value of standard angles, we will find the value of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians.

Complete step by step answer:
There are six functions of an angle commonly used in trigonometry namely sine, cosine, tangent, cotangent, secant and cosecant. According to the question find the exact values of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians. As we know that the reference angle is the acute angle with the x-axis. Using the concept of reference angle, we will find the exact values of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians.

Let us consider the original angle is given by \[\theta \] and the auxiliary value is given by \[\alpha \].
For the first quadrant, we have \[\theta = \alpha \].
For the second quadrant, we have \[\theta = \pi - \alpha \].
For the third quadrant, we have \[\theta = \pi + \alpha \].
For the fourth quadrant, we have \[\theta = 2\pi - \alpha \].
Consider \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\]. \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] is in the second quadrant.
Therefore, \[\cos \left( {\dfrac{{3\pi }}{4}} \right) = \cos \left( {\pi - \dfrac{\pi }{4}} \right)\]

In the second quadrant, \[\cos \] is negative. So,
\[ \Rightarrow \cos \left( {\pi - \dfrac{\pi }{4}} \right) = - \cos \left( {\dfrac{\pi }{4}} \right)\]
\[\therefore \cos \left( {\dfrac{{3\pi }}{4}} \right) = - \dfrac{1}{{\sqrt 2 }}\]
Now, consider \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\]. \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] lies in the second quadrant.
Therefore, \[\sin \left( {\dfrac{{3\pi }}{4}} \right) = \sin \left( {\pi - \dfrac{\pi }{4}} \right)\].
In the second quadrant, \[\sin \] is positive.
So,
\[ \Rightarrow \sin \left( {\pi - \dfrac{\pi }{4}} \right) = \sin \left( {\dfrac{\pi }{4}} \right)\]
\[\therefore \sin \left( {\dfrac{{3\pi }}{4}} \right) = \dfrac{1}{{\sqrt 2 }}\]

Therefore, the exact values of \[\cos \left( {\dfrac{{3\pi }}{4}} \right)\] radians and \[\sin \left( {\dfrac{{3\pi }}{4}} \right)\] radians are \[ - \dfrac{1}{{\sqrt 2 }}\] and \[\dfrac{1}{{\sqrt 2 }}\] respectively.

Note: In the first quadrant, all trigonometric functions are positive. In the second quadrant, \[\sin \] and \[\cos ec\] are positive. In the third quadrant, \[\tan \] and \[\cot \] are positive. In the fourth quadrant, \[\cos \] and \[\sec \] are positive. Also, note that here we have used values of some standard angles i.e., \[\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}\] and \[\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}\].