Trigonometry
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.
Many of these identities are far older than the modern sine and cosine notation used to write them. The Greek astronomer Ptolemy, working in Alexandria around 150 CE, compiled a table of chords in his book the Almagest that is mathematically equivalent to a table of sine values, and he proved a geometric result – now called Ptolemy's theorem – that is essentially the angle-addition formula for sine written in the language of circles and chords rather than algebra. The double angle formulas taught on this page are a special case of those same angle-addition identities, rediscovered and re-expressed algebraically by Islamic Golden Age mathematicians such as Abu al-Wafa in the 10th century, before reaching the symbolic form used in textbooks today.
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)
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.