Unit Circle in Trigonometry $$ x^2 + y^2 = 1 $$

Unit Circle in Trigonometry: Quadrants and Special Angles

In this lesson, we explore the unit circle in trigonometry, with a focus on quadrants and special angles. The unit circle helps us understand the values of sine, cosine, and tangent for different angles in a clear and visual way.

1. Theory of the Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. Every point on the circle satisfies the equation:

$$ x^2 + y^2 = 1 $$

Any point on the unit circle can be written as:

$$ P(\cos \theta, \sin \theta) $$

where θ is the angle measured from the positive x-axis.

2. Concept Explanation

Angles on the unit circle are measured counterclockwise for positive angles and clockwise for negative angles. The circle is divided into four quadrants, and the signs of sine, cosine, and tangent depend on the quadrant where the angle lies.

Important reference points on the unit circle include:

• 0° or 360° → (1, 0)
• 90° → (0, 1)
• 180° → (−1, 0)
• 270° → (0, −1)

3. Steps or Formula

To analyze angles on the unit circle, follow these steps:

1. Determine the quadrant of the angle.
2. Identify the signs of sine, cosine, and tangent in that quadrant.
3. Use special angle values if applicable.

Signs of trigonometric functions in each quadrant:

• Quadrant I: sin +, cos +, tan +
• Quadrant II: sin +, cos −, tan −
• Quadrant III: sin −, cos −, tan +
• Quadrant IV: sin −, cos +, tan −

4. Example Problem

Find the sine, cosine, and tangent values for special angles.

Key trigonometric values on the unit circle:

• 0°: sin = 0, cos = 1, tan = 0
• 30°: sin = \( \frac{1}{2} \), cos = \( \frac{\sqrt{3}}{2} \), tan = \( \frac{1}{\sqrt{3}} \)
• 45°: sin = \( \frac{\sqrt{2}}{2} \), cos = \( \frac{\sqrt{2}}{2} \), tan = 1
• 60°: sin = \( \frac{\sqrt{3}}{2} \), cos = \( \frac{1}{2} \), tan = \( \sqrt{3} \)
• 90°: sin = 1, cos = 0, tan = undefined

As a rotation example, the point (1, 0) rotated 90° counterclockwise becomes (0, 1).

5. Final Answer

The unit circle provides a simple and powerful way to understand trigonometric values. By mastering quadrants and special angles, students can easily determine sine, cosine, and tangent for any angle and build a strong foundation in trigonometry.