What is the exact value of sin (120 degrees)?

Question 32Select one:

a.

½

b.

.866

c.

√3/2

d.

-1/2

e.

None of these

To find the exact value of \(\sin(120^\circ)\), we can use the following steps:

### Step 1: Recognize the Reference Angle
The angle \(120^\circ\) is in the second quadrant. The reference angle for \(120^\circ\) is:

\[
180^\circ - 120^\circ = 60^\circ
\]

### Step 2: Determine the Sign of the Sine Function
In the second quadrant, the sine function is positive.

### Step 3: Use the Sine of the Reference Angle
The sine of \(60^\circ\) is known to be:

\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\]

### Step 4: Apply the Sign from Step 2
Since \(120^\circ\) is in the second quadrant and the sine is positive there:

\[
\sin(120^\circ) = \frac{\sqrt{3}}{2}
\]

### Conclusion
The exact value of \(\sin(120^\circ)\) is \(\frac{\sqrt{3}}{2}\).

So the correct answer is:

**Option C: \(\frac{\sqrt{3}}{2}\)**

Would you like more details or have any questions? Here are some related questions:

1. What is the sine of \(240^\circ\)?
2. How can you determine the sine of any angle using the unit circle?
3. What is the cosine of \(120^\circ\)?
4. What is the exact value of \(\sin(150^\circ)\)?
5. How can the sine of supplementary angles be related?

**Tip:** Remember that angles in the second quadrant have a positive sine and a negative cosine.

Discover the exact value of sin(120 degrees) using trigonometric principles. Learn about angle measurement, reference angles, and the sine function in the second quadrant. Suitable for high school students studying trigonometry.

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