Trigonometry — Complete Guide to Formulas & Identities

Trigonometry — Complete Guide to Formulas & Identities

TL;DR

This trigonometry complete guide covers the six trig functions (sin, cos, tan, csc, sec, cot), the unit circle, the four families of identities (Pythagorean, reciprocal, sum-and-difference, double-angle), the six inverse trig functions and the standard derivatives.

A Discipline Older Than Greek Geometry

Babylonian astronomers tracking the moon's position around 1800 BCE were already using chord-length tables — the direct ancestor of sine.
By Hipparchus in 130 BCE, the tables were systematic. By Aryabhata in 499 CE, sine and cosine had distinct names. By Euler in 1748, the six trig functions were modern analytic objects with derivatives, series expansions, and a connection to complex numbers via eiθ=cos⁡θ+isin⁡θ. Trigonometry is the oldest continuously-developed branch of mathematics — 3,800 years and counting.

What Is Trigonometry?

Trigonometry is the branch of mathematics that studies relationships between angles and side-length ratios in triangles, extended through the unit circle to all real angles. Six functions — sine, cosine, tangent, cosecant, secant, cotangent — convert angles into ratios. A handful of identities convert between expressions involving those functions. From this scaffold, the entire mathematics of waves, oscillations, rotations, and complex analysis follows.

Three perspectives, one subject:

This guide moves through all three.

The Six Trigonometric Functions

Function Right-triangle Unit-circle
sin⁡θ \frac{\text{opp}}{\text{hyp}} y-coordinate of (cos⁡θ,sin⁡θ)
cos⁡θ \frac{\text{adj}}{\text{hyp}} x-coordinate of (cos⁡θ,sin⁡θ)
tan⁡θ \frac{\text{opp}}{\text{adj}} \frac{\sin\theta}{\cos\theta}
csc⁡θ \frac{\text{hyp}}{\text{opp}} \frac{1}{\sin\theta}
sec⁡θ \frac{\text{hyp}}{\text{adj}} \frac{1}{\cos\theta}
cot⁡θ \frac{\text{adj}}{\text{opp}} \frac{\cos\theta}{\sin\theta}

SOH-CAH-TOA gives the first three from the right triangle. The other three are reciprocals: csc⁡=1/sin⁡, sec⁡=1/cos⁡, cot⁡=1/tan⁡.

The Unit Circle and Standard Angles

The unit circle is centred at the origin with radius 1. A point at angle θ from the positive x-axis has coordinates (cos⁡θ,sin⁡θ).
The five "exact" angles every student must know cold:

θ 0 \frac{\pi}{6} \frac{\pi}{4} \frac{\pi}{3} \frac{\pi}{2}
sin⁡θ 0 \frac{1}{2} \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} 1
cos⁡θ 1 \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} \frac{1}{2} 0
tan⁡θ 0 \frac{1}{\sqrt{3}} 1 \sqrt{3} undefined

The memorisation trick. Sine's row is 0/2,1/2,2/2,3/2,4/2 — indexed by square roots of 0 through 4. Cosine runs the same sequence backwards. For angles in quadrants II, III, IV, use reference angles + ASTC.

The Four Identity Families

Trigonometric identities collapse complex expressions into simpler ones. The four major families:

Pythagorean identities

sin⁡2θ+cos⁡2θ=1
1+tan⁡2θ=sec⁡2θ
1+cot⁡2θ=csc⁡2θ

All three follow from the unit-circle equation x²+y²=1.

Sum-and-difference identities

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B
tan⁡(A±B)=\frac{tan⁡A±tan⁡B}{1∓tan⁡Atan⁡B}

These identities are the most-used in trigonometric problem-solving — they derive from them.

Double-angle identities

sin⁡(2θ)=2sin⁡θcos⁡θ
cos⁡(2θ)=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1
tan⁡(2θ)=\frac{2tan⁡θ}{1−tan²θ}

Sum-to-product (and product-to-sum) identities

sin⁡A+sin⁡B=2sin⁡(\frac{A+B}{2})cos⁡(\frac{A−B}{2})
cos⁡A−cos⁡B=−2sin⁡(\frac{A+B}{2})sin⁡(\frac{A−B}{2})

The Six Inverse Trigonometric Functions

Inverse function Domain Range (principal)
arcsin⁡x [-1,1] [-π/2,π/2]
arccos⁡x [-1,1] [0,π]
arctan⁡x (-∞,∞) (-π/2,π/2)
arccsc⁡x x
arcsec⁡x x
arccot⁡x (-∞,∞) (0,π)

Trigonometric Derivatives

Function Derivative
sin⁡x cos⁡x
cos⁡x −sin⁡x
tan⁡x sec²x
cot⁡x −csc²x
sec⁡x sec⁡xtan⁡x
csc⁡x −csc⁡xcot⁡x

The "co-" rule. Every "co-" function has a negative sign in its derivative. The three "non-co" functions don't.

Three Worked Examples — Quick, Standard, Stretch

Quick

Evaluate sin⁡(π/3)+cos⁡(π/6).
From the standard-angle table: sin⁡(π/3)=√3/2 and cos⁡(π/6)=√3/2.
sin⁡(π/3)+cos⁡(π/6)=√3/2+√3/2=√3.

The Mistake Worth Making Once — Standard Example

Prove that \frac{1 − cos⁡(2θ)}{sin⁡(2θ)} = tan⁡θ. The wrong path. A student writes... (show steps here)
So where's the wrong path?

Stretch

Find the exact value of sin⁡(15°).
15°=45°−30°, so... (show steps here)

Why Trigonometry Matters Beyond the Triangle

Trigonometry is the language of every periodic phenomenon in the physical world.

The Mathematicians Who Built Trigonometry

Trigonometry is the work of more than a hundred named contributors across 3,800 years. Four matter most.

Why it matters:

Where Solutions Go Off the Rails — Common Mistakes

Mistake 1: Working in degrees when the formula assumes radians

Mistake 2: Forgetting to use the reference angle

Mistake 3: Confusing sin⁡−1x with 1/sin⁡x

Key Takeaways