Question

Find the cosine of the following angles.

1. Cos 7pi/4
2. Cos pi/6
3. Cos 90*
4. Cos 240*

A. 0
B. -1/2
C. V3/2
D. V2/2

Answer 1

To find the cosine of the specified angles, let's evaluate each one:

1. **Cosine of \(7\pi/4\)**:
- The angle \(7\pi/4\) is in the fourth quadrant. Its reference angle is \(2\pi/4 = \pi/2\).
- \(\cos(7\pi/4) = \cos(2\pi - \pi/4) = \cos(\pi/4) = \sqrt{2}/2\).
- So, \( \cos(7\pi/4) = \frac{\sqrt{2}}{2} \).

2. **Cosine of \(\pi/6\)**:
- The angle \(\pi/6\) is in the first quadrant.
- \(\cos(\pi/6) = \sqrt{3}/2\).
- So, \( \cos(\pi/6) = \frac{\sqrt{3}}{2} \).

3. **Cosine of \(90^\circ\)**:
- The angle \(90^\circ\) is at the top of the unit circle.
- \(\cos(90^\circ) = 0\).
- So, \( \cos(90^\circ) = 0 \).

4. **Cosine of \(240^\circ\)**:
- The angle \(240^\circ\) is in the third quadrant. Its reference angle is \(240^\circ - 180^\circ = 60^\circ\).
- \(\cos(240^\circ) = -\cos(60^\circ) = -\frac{1}{2}\).
- So, \( \cos(240^\circ) = -\frac{1}{2} \).

Now, from the options given:

1. **Cos \(7\pi/4\)** = \(\frac{\sqrt{2}}{2}\) (not listed)
2. **Cos \(\pi/6\)** = \(\frac{\sqrt{3}}{2}\) (not listed)
3. **Cos \(90^\circ\)** = \(0\) (Option A)
4. **Cos \(240^\circ\)** = \(-\frac{1}{2}\) (Option B)

The final answers for the specific angles are:
- **Cos \(7\pi/4\)**: \(\frac{\sqrt{2}}{2}\) (not listed)
- **Cos \(\pi/6\)**: \(\frac{\sqrt{3}}{2}\) (not listed)
- **Cos \(90^\circ\)**: **0** (Option A)
- **Cos \(240^\circ\)**: **-\frac{1}{2}** (Option B)

In conclusion:
- **Cos \(90^\circ\) = 0** (A)
- **Cos \(240^\circ\) = -\frac{1}{2}** (B)