Evaluate the following expression:
[tex]\[
\sin \left(\frac{3 \pi}{4}\right)
\][/tex]

A. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]

B. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]

C. [tex]\(\frac{1}{2}\)[/tex]

D. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]



Answer :

To find the value of [tex]\( \sin \left(\frac{3 \pi}{4}\right) \)[/tex], let's take it step by step.

1. Understand the Angle:
- The angle [tex]\( \frac{3\pi}{4} \)[/tex] is in radians. To understand its position on the unit circle, convert it to degrees:
[tex]\[ \frac{3\pi}{4} \times \frac{180^\circ}{\pi} = 135^\circ \][/tex]
- [tex]\( 135^\circ \)[/tex] lies in the second quadrant.

2. Reference Angle:
- The reference angle for [tex]\( 135^\circ \)[/tex] is found by subtracting it from [tex]\( 180^\circ \)[/tex]:
[tex]\[ 180^\circ - 135^\circ = 45^\circ \][/tex]

3. Sine of Reference Angle:
- The sine of [tex]\( 45^\circ \)[/tex] (or [tex]\( \frac{\pi}{4} \)[/tex] radians) is well-known:
[tex]\[ \sin 45^\circ = \frac{\sqrt{2}}{2} \][/tex]

4. Sign in the Second Quadrant:
- In the second quadrant, the sine function is positive.

5. Combine Information:
- Therefore, [tex]\( \sin(135^\circ) \)[/tex] or [tex]\( \sin \left( \frac{3\pi}{4} \right) \)[/tex] is [tex]\( \frac{\sqrt{2}}{2} \)[/tex].

Comparing with the multiple-choice options:
[tex]\[ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]

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