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]
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]
