Which expression is equivalent to [tex][tex]$\sin \frac{7 \pi}{6}$[/tex][/tex]?

A. [tex][tex]$\sin \frac{\pi}{6}$[/tex][/tex]

B. [tex][tex]$\sin \frac{5 \pi}{6}$[/tex][/tex]

C. [tex][tex]$\sin \frac{5 \pi}{3}$[/tex][/tex]

D. [tex][tex]$\sin \frac{11 \pi}{6}$[/tex][/tex]



Answer :

To determine which expression is equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex], let's explore each sine function provided:

1. [tex]\(\sin \frac{\pi}{6}\)[/tex]
2. [tex]\(\sin \frac{5 \pi}{6}\)[/tex]
3. [tex]\(\sin \frac{5 \pi}{3}\)[/tex]
4. [tex]\(\sin \frac{11 \pi}{6}\)[/tex]

Firstly, recall the value and properties of the sine function at key angles, specifically those related to [tex]\(\pi\)[/tex].

### Step-by-step Solution:

1. Evaluating [tex]\(\sin \frac{\pi}{6}\)[/tex]:

[tex]\[ \sin \frac{\pi}{6} = \frac{1}{2} \][/tex]

2. Evaluating [tex]\(\sin \frac{5 \pi}{6}\)[/tex]:

Knowing that sine is positive in the second quadrant:

[tex]\[ \sin \frac{5 \pi}{6} = \sin \left(\pi - \frac{\pi}{6}\right) = \sin \frac{\pi}{6} = \frac{1}{2} \][/tex]

3. Evaluating [tex]\(\sin \frac{5 \pi}{3}\)[/tex]:

To find the value of [tex]\(\sin \frac{5 \pi}{3}\)[/tex], note that:

[tex]\[ \frac{5 \pi}{3} = 2\pi - \frac{\pi}{3} \][/tex]

Hence, it lies in the fourth quadrant where sine is negative:

[tex]\[ \sin \frac{5 \pi}{3} = \sin \left(2\pi - \frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2} \][/tex]

4. Evaluating [tex]\(\sin \frac{11 \pi}{6}\)[/tex]:

To find the value of [tex]\(\sin \frac{11 \pi}{6}\)[/tex], note that:

[tex]\[ \frac{11 \pi}{6} = 2\pi - \frac{\pi}{6} \][/tex]

Hence, it also lies in the fourth quadrant:

[tex]\[ \sin \frac{11 \pi}{6} = \sin \left(2\pi - \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2} \][/tex]

Now, compare these values with [tex]\(\sin \frac{7 \pi}{6}\)[/tex].

Evaluating [tex]\(\sin \frac{7 \pi}{6}\)[/tex]:
[tex]\[ \frac{7 \pi}{6} = \pi + \frac{\pi}{6} \][/tex]

Since this angle lies in the third quadrant where sine is negative:

[tex]\[ \sin \frac{7 \pi}{6} = -\sin \frac{\pi}{6} = -\frac{1}{2} \][/tex]

From the above calculations, we see that:

- [tex]\(\sin \frac{7 \pi}{6} = -\frac{1}{2}\)[/tex]
- [tex]\(\sin \frac{11 \pi}{6} = -\frac{1}{2}\)[/tex]

Therefore, the expression equivalent to [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is:

[tex]\[ \boxed{\sin \frac{11 \pi}{6}} \][/tex]

So the answer to the question is the fourth option, [tex]\(\sin \frac{11 \pi}{6}\)[/tex].