Question

PART A - 4 Points Problem 1) Identify the sine, cosine and tangent for the magic angles (30, 45, and 60 degrees) in the 1st Quadrant using exact values NOT decimal approximations. Throughout this course you must use the exact values, not the decimal approximations. Make sure you rationalize all denominators. Complete the table below. Angle Π 30° or 6 Π 45° or 4 60° or Π 3 cosine sine tangent

Answer 1

Here are the sine, cosine, and tangent values for the angles 30°, 45°, and 60° expressed in terms of \(\pi\), along with their exact value forms.

### Trigonometric Values Table

| Angle (Degrees) | Angle (Radians) | Cosine | Sine | Tangent |
|------------------|------------------|---------------------------|---------------------------|--------------------------------|
| 30° | \(\frac{\pi}{6}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\) |
| 45° | \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | 1 |
| 60° | \(\frac{\pi}{3}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\sqrt{3}\) |

### Explanation

1. **30° (or \(\frac{\pi}{6}\))**
- \(\cos(30°) = \frac{\sqrt{3}}{2}\)
- \(\sin(30°) = \frac{1}{2}\)
- \(\tan(30°) = \frac{\sin(30°)}{\cos(30°)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\) (rationalized)

2. **45° (or \(\frac{\pi}{4}\))**
- \(\cos(45°) = \frac{\sqrt{2}}{2}\)
- \(\sin(45°) = \frac{\sqrt{2}}{2}\)
- \(\tan(45°) = \frac{\sin(45°)}{\cos(45°)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\)

3. **60° (or \(\frac{\pi}{3}\))**
- \(\cos(60°) = \frac{1}{2}\)
- \(\sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\tan(60°) = \frac{\sin(60°)}{\cos(60°)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\)

These are the exact trigonometric values for the specified angles in the first quadrant.