Oct 3, 2024
# Overview of the Unit Circle and Trigonometry
## Quadrants in the Unit Circle
- **First Quadrant:** All functions are positive (All Students Take Calculus)
- **Second Quadrant:** Sine is positive
- **Third Quadrant:** Tangent is positive
- **Fourth Quadrant:** Cosine is positive
## Major Angles in Degrees and Radians
- **0° (0):** (1, 0)
- **30° (π/6):** (√3/2, 1/2)
- **45° (π/4):** (√2/2, √2/2)
- **60° (π/3):** (1/2, √3/2)
- **90° (π/2):** (0, 1)
- **180° (π):** (-1, 0)
- **270° (3π/2):** (0, -1)
- **360° (2π):** (1, 0)
## Reference Angles Values
- **Reference Angle for 30°:** 30°
- Second Quadrant (150°): (-√3/2, 1/2)
- Third Quadrant (210°): (-√3/2, -1/2)
- Fourth Quadrant (330°): (√3/2, -1/2)
- **Reference Angle for 45°:** 45°
- Second Quadrant (135°): (-√2/2, √2/2)
- Third Quadrant (225°): (-√2/2, -√2/2)
- Fourth Quadrant (315°): (√2/2, -√2/2)
- **Reference Angle for 60°:** 60°
- Second Quadrant (120°): (-1/2, √3/2)
- Third Quadrant (240°): (-1/2, -√3/2)
- Fourth Quadrant (300°): (1/2, -√3/2)
## Evaluating Trigonometric Functions Using the Unit Circle
- **Sine and Cosine:** Use Y and X values respectively from the unit circle
- Example:
- Sine 60° = √3/2
- Cosine 60° = 1/2
- Tangent 60° = (√3/2) / (1/2) = √3
## Special Triangles
- **30-60-90 Triangle:**
- Opposite side (30°): 1, Adjacent side (60°): √3, Hypotenuse: 2
- **45-45-90 Triangle:**
- Opposite side (45°): 1, Hypotenuse: √2
## Reference Angles for Other Quadrants
- **Second Quadrant:** Reference angle = 180° - angle
- **Third Quadrant:** Reference angle = angle - 180°
- **Fourth Quadrant:** Reference angle = 360° - angle
## Inverse Trigonometric Functions
- **Inverse Sine (sin⁻¹):** Range between -90° and 90° (First and Fourth quadrants)
- **Inverse Cosine (cos⁻¹):** Range between 0° and 180° (First and Second quadrants)
- **Inverse Tangent (tan⁻¹):** Range between -90° and 90° (First and Fourth quadrants)
## Evaluating Examples
- **Evaluate sin 150° (Inverse):**
- sin 150° = 1/2
- sin⁻¹(1/2) = 30°
- **Evaluate cos 330° (Inverse):**
- cos 330° = √3/2
- cos⁻¹(√3/2) = 30°
- **Evaluate tan 90°:** Undefined (because cos 90° = 0)
## Additional Examples
- **Finding Other Trigonometric Functions from Sine:**
- If sin θ = 3/5, use the Pythagorean theorem to find the adjacent side and evaluate cosine, tangent, secant, cosecant, and cotangent.
- **Finding Trigonometric Functions in Different Quadrants:**
- Use the Pythagorean theorem and special triangles to derive the missing sides and values.
- **Understanding Restrictions of Inverse Functions:**
- Sine and tangent are restricted to the first and fourth quadrants, while cosine is restricted to the first and second quadrants.