Question

How do you find the exact value of $\sin \left( {\dfrac{{3\pi }}{4}} \right)?$

Answer 1

Hint: Here we can use the trigonometric identity $\sin \left( x \right) = \cos \left( {\dfrac{\pi }{2} - x} \right)$.

Complete step by step answer:
Let us find the exact value of $\sin \left( {\dfrac{{3\pi }}{4}} \right):$
$\sin \left( {\dfrac{{3\pi }}{4}} \right) = \cos \left( {\dfrac{\pi }{2} - \dfrac{{3\pi }}{4}} \right)$
$ = \cos \left( {\dfrac{{2\pi }}{4} - \dfrac{{3\pi }}{4}} \right)$ (
$ = \cos \left( {\dfrac{{2\pi - 3\pi }}{4}} \right)$
 $ = \cos \left( {\dfrac{{ - \pi }}{4}} \right) = \cos \left( { - \dfrac{\pi }{4}} \right) = \cos \left( {\dfrac{\pi }{4}} \right)$
 $ = \dfrac{{\sqrt 2 }}{2} = \dfrac{{1.41421}}{2}$
 $\therefore \sin \left( {\dfrac{{3\pi }}{4}} \right) = 0.70710$

Hence, the exact value is $0.70710$.

Note: The trigonometric identities are verified formulas that are used for evaluating the exact value of a trigonometric function. From one of these trigonometric identities, the sine of any variable is equal-
to the cosine of the difference of $\dfrac{\pi }{2}$and that variable.
$\sin \left( x \right) = \cos \left( {\dfrac{\pi }{2} - x} \right)$
In a right triangle, the cosine angle divides the length of the hypotenuse(H) by the length of the adjacent side(A). The word sine comes from the Latin word sinus and the prefix ‘co’ with sine makes the cosine word.
Since our angle is greater than and less than or equal to radians, it is located in quadrant two .In the second quadrant, the values for sin are positive only.
In this question we can apply the reference angle by finding the angle with equivalent trig values in the first quadrant.$\sin \left( {\dfrac{\pi }{4}} \right)$.
The exact value of $\sin \left( {\dfrac{\pi }{4}} \right)$ is $\dfrac{{\sqrt 2 }}{2}$.
The result can be shown in multiple forms.
Exact Form:
$\dfrac{{\sqrt 2 }}{2}$
Decimal Form:$0.70710678$.