Equations That Always Hold: Trig Identities
Trigonometric identities are equations involving trig functions that are true for all valid input values, not just specific angles. They allow you to simplify complex trig expressions, solve trig equations, and prove mathematical relationships. Mastering identities is essential for advanced trigonometry, calculus, and physics.
Components of Trig Identities
This section covers the most important identity families:
- Pythagorean Identities: sin²(θ) + cos²(θ) = 1, along with the derived forms 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ).
- Reciprocal Identities: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ).
- Quotient Identities: tan(θ) = sin(θ)/cos(θ) and cot(θ) = cos(θ)/sin(θ).
- Co-Function Identities: sin(90° - θ) = cos(θ), cos(90° - θ) = sin(θ), and tan(90° - θ) = cot(θ).
Examples of Trig Identities
Pythagorean Identity Examples
- If sin(θ) = 3/5, find cos(θ): Using sin²(θ) + cos²(θ) = 1, we get 9/25 + cos²(θ) = 1, so cos²(θ) = 16/25, and cos(θ) = 4/5 (in the first quadrant).
- Simplify sin²(θ) + cos²(θ) + tan²(θ): Replace sin² + cos² with 1 to get 1 + tan²(θ) = sec²(θ).
- Verify for θ = 30°: sin²(30°) + cos²(30°) = (1/2)² + (√3/2)² = 1/4 + 3/4 = 1 ✓.
Reciprocal Identity Examples
- If sin(θ) = 0.6, then csc(θ) = 1/0.6 ≈ 1.667.
- If cos(θ) = √2/2, then sec(θ) = 1/(√2/2) = 2/√2 = √2.
- If tan(θ) = 3/4, then cot(θ) = 4/3.
Quotient Identity Examples
- If sin(θ) = 5/13 and cos(θ) = 12/13, then tan(θ) = sin(θ)/cos(θ) = (5/13)/(12/13) = 5/12.
- Simplify sin(θ)/cos(θ) × cos(θ): This equals sin(θ) because the cos(θ) terms cancel.
- Verify for θ = 45°: tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1 ✓.
Co-Function Identity Examples
- sin(30°) = cos(60°) = 1/2. The sine of an angle equals the cosine of its complement.
- cos(25°) = sin(65°). Since 25° + 65° = 90°, these are co-functions.
- If tan(θ) = 2.5, then cot(90° - θ) = 2.5 as well, because tangent and cotangent are co-functions.