Overview
This lecture covers quotient identities for tangent and cotangent in trigonometry, their relationships, calculation methods, and applications to key angles.
Quotient Identities
- Tangent θ = sin θ / cos θ
- Cotangent θ = cos θ / sin θ
- Tangent is also y / x; cotangent is x / y (using coordinates on the unit circle).
- Cotangent is the reciprocal of tangent: cotangent θ = 1 / tangent θ.
Evaluating Tangent and Cotangent with Fractions
- To find tangent θ when sin θ and cos θ are given, divide sin θ by cos θ, simplifying complex fractions as needed.
- Example: If sin θ = 4/5, cos θ = 3/5, then tangent θ = (4/5) / (3/5) = 4/3.
- Multiply numerator and denominator by the denominator's base to simplify complex fractions quickly.
- Cotangent can be found by flipping the tangent result: cotangent θ = 3/4 in the above example.
Additional Examples
- If sin θ = 5/13 and cos θ = 12/13, then tangent θ = (5/13) / (12/13) = 5/12; cotangent θ = 12/5.
- Alternatively, cotangent θ = cos θ / sin θ gives the same result.
Tangent and Cotangent on the Unit Circle
- Tangent π/4: coordinates (√2/2, √2/2) → tangent = y/x = 1.
- Tangent 2π/3: quadrant II, reference angle π/3, coordinates (−1/2, √3/2) → tangent = (√3/2) / (−1/2) = −√3.
- Cotangent 4π/3: quadrant III, reference π/3, both x and y negative, cotangent = x/y = (−1/2) / (−√3/2) = 1/√3, rationalized to √3/3.
Tangent and Cotangent for Special Angles
- Tangent of 0°: coordinates (1, 0) → y/x = 0/1 = 0.
- Tangent of 90°: coordinates (0, 1) → y/x = 1/0 = undefined.
- Cotangent of 180°: point (−1, 0) → x/y = (−1)/0 = undefined.
- Cotangent of 3π/2: point (0, −1) → x/y = 0/(−1) = 0.
- For 0°, 90°, 180°, and 270°, tangent and cotangent are always 0 or undefined.
Key Terms & Definitions
- Tangent θ — The ratio of sin θ to cos θ (tan θ = sin θ / cos θ), also y/x on the unit circle.
- Cotangent θ — The ratio of cos θ to sin θ (cot θ = cos θ / sin θ), also x/y on the unit circle.
- Quotient identities — Identities involving division of sine and cosine (or vice versa) for tangent and cotangent.
Action Items / Next Steps
- Practice evaluating tangent and cotangent for various angles using the unit circle.
- Memorize key quotient identities and their reciprocal relationships.
- Rationalize denominators when necessary in cotangent or tangent results.