Trigonometric Identities - The Essential Toolkit
Trigonometric identities are equations involving trigonometric functions that are true for every valid angle. Rather than giving you one answer, they hold for all values of the variable. Identities are powerful because they let you simplify expressions, solve equations, and prove other results.
The Pythagorean Identity
This is the most fundamental identity in trigonometry. It comes directly from the equation of the unit circle (x² + y² = 1) and the fact that x = cos(θ) and y = sin(θ):
sin²(θ) + cos²(θ) = 1
Two rearrangements that are equally important:
sin²(θ) = 1 − cos²(θ)
cos²(θ) = 1 − sin²(θ)
Related Pythagorean Identities
| Identity | Derived from |
|---|---|
| 1 + tan²(θ) = sec²(θ) | Divide sin²+cos²=1 by cos² |
| 1 + cot²(θ) = csc²(θ) | Divide sin²+cos²=1 by sin² |
Reciprocal Identities
Each of the three main ratios has a reciprocal function:
csc(θ) = 1 / sin(θ) (cosecant)
sec(θ) = 1 / cos(θ) (secant)
cot(θ) = 1 / tan(θ) (cotangent)
Quotient Identities
tan(θ) = sin(θ) / cos(θ)
cot(θ) = cos(θ) / sin(θ)
Even and Odd Identities
These describe what happens when the angle becomes negative:
sin(−θ) = −sin(θ) (sine is an odd function)
cos(−θ) = cos(θ) (cosine is an even function)
tan(−θ) = −tan(θ) (tangent is an odd function)
Double Angle Formulas
sin(2θ) = 2·sin(θ)·cos(θ)
cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ)−1 = 1−2sin²(θ)
tan(2θ) = 2·tan(θ) / (1 − tan²(θ))
Worked Examples
cos²(θ) = 1 − sin²(θ) = 1 − 9/25 = 16/25. cos(θ) = 4/5 (positive, Q1).
tan(θ) = (3/5) / (4/5) = 3/4.
sin² + cos² = 1. So the expression = 1 + tan²(θ) = sec²(θ).
sin(60°) = 2·sin(30°)·cos(30°) = 2 × (1/2) × (√3/2) = √3/2.
Key Takeaways
- The master identity: sin²(θ) + cos²(θ) = 1.
- Reciprocals: csc = 1/sin, sec = 1/cos, cot = 1/tan.
- tan = sin/cos, cot = cos/sin.
- Double angle: sin(2θ) = 2·sinθ·cosθ. cos(2θ) = cos²θ−sin²θ.
Practice Questions
- If cos(θ) = 5/13 and θ is in Q1, find sin(θ) and tan(θ).
- Simplify: (1 − sin²(θ)) / cos(θ).
- Show that tan²(θ) + 1 = sec²(θ) by starting from sin² + cos² = 1.
- Find the exact value of cos(90°) using the double angle formula with θ = 45°.
- Given tan(θ) = 2 and θ in Q3, find sin(θ) and cos(θ).