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The unit circle is a powerful tool in trigonometry. It's a circle with a radius of one unit, centered at the origin of the coordinate plane. The unit circle helps to find the values of trigonometric functions for specific angles.

Imagine the circle overlapping the x and y axes. The angle in the circle is measured starting from the positive x-axis and moving counterclockwise. Each point on the circumference of the unit circle has coordinates \((x, y)\). The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle.

Key points include:

• 0°: \((1, 0)\)
• 90°: \((0, 1)\)
• 180°: \((-1, 0)\)
• 270°: \((0, -1)\)

When dealing with angles like 45°, the coordinates can be found using Pythagoras' theorem. The relationship in this scenario is \((\frac {\tiny \sqrt{2}}{2}, \frac {\tiny \sqrt{2}}{2})\). Therefore, the sine of 45° is simply \(\frac {\tiny \sqrt{2}}{2}\).

The sine function is one of the fundamental trigonometric functions. It's often written as \(\text{sin}\). When you input an angle into the sine function, it gives the y-coordinate of the corresponding point on the unit circle.

Here are some essential properties of the sine function:

• It's periodic with a period of 360° (or 2Ï€ radians).
• The range of the sine function is \([-1, 1]\).
• It's an odd function, meaning \(\text{sin}(-θ) = -\text{sin}(θ)\).

For example, for an angle of 45°, the y-coordinate on the unit circle is known as \(\text{sin} 45°\). Using the unit circle or trigonometric tables, we find that \(\text{sin} 45° = \frac {\tiny \sqrt{2}}{2}\). This is a vital value to remember as it often appears in trigonometry problems.

Special angles are specific angles used frequently in trigonometry because of their easily remembered sine, cosine, and tangent values. The most common ones are 0°, 30°, 45°, 60°, and 90°.

For each of these angles, both their sine and cosine values have unique properties:

• 0°: \(\text{sin} 0° = 0\) and \(\text{cos} 0° = 1\)
• 30°: \(\text{sin} 30° = \frac {1}{2}\) and \(\text{cos} 30° = \frac {\tiny \sqrt{3}}{2}\)
• 45°: \(\text{sin} 45° = \frac {\tiny \sqrt{2}}{2}\) and \(\text{cos} 45° = \frac {\tiny \sqrt{2}}{2}\)
• 60°: \(\text{sin} 60° = \frac {\tiny \sqrt{3}}{2}\) and \(\text{cos} 60° = \frac{1}{2}\)
• 90°: \(\text{sin} 90° = 1\) and \(\text{cos} 90° = 0\)

When solving trigonometric problems, quickly knowing these values is very useful. So, to find \(\text{sin} 45°\), you recognize it's one of these special angles, and its value is \(\frac {\tiny \sqrt{2}}{2}\).