Trigonometry Formulas - Full list

Trigonometry

TL;DR

The complete list of trigonometry formulas covers seven categories: basic ratios (sin, cos, tan), reciprocal identities (csc, sec, cot), Pythagorean identities (sin² + cos² = 1), angle-sum and angle-difference formulas, double-angle formulas, half-angle formulas, and sum-to-product formulas.

What Are the Basic Trigonometry Formulas?

Primary Ratios (SOHCAHTOA)

For a right triangle with angle θ:

sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent

Reciprocal Identities

cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ

Quotient Identities

tanθ = sinθ/cosθ, cotθ = cosθ/sinθ

What Are the Pythagorean Identities?

Three identities derived directly from the Pythagorean theorem applied to the unit circle:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = csc²θ

The first is the most-used identity in all of trigonometry. The other two come from dividing the first by cos²θ and sin²θ respectively.

Worked example: If sinθ = 3/5 and θ is in Q1, find cosθ.

cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25, so cosθ = 4/5.

What Are the Sum and Difference Formulas?

Sine

sin(A+B) = sinA cosB + cosA sinB
sin(A−B) = sinA cosB − cosA sinB

Cosine

cos(A+B) = cosA cosB − sinA sinB
cos(A−B) = cosA cosB + sinA sinB

Tangent

an(A+B) = (tanA + tanB) / (1 − tanA tanB)
an(A−B) = (tanA − tanB) / (1 + tanA tanB)

Worked example: Compute sin(75°) = sin(45° + 30°).

What Are the Double-Angle Formulas?

1. sin(2θ) = 2sinθcosθ
2. cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

What Are the Half-Angle Formulas?

1. sin(θ/2) = ±√(1 − cosθ)/2
2. cos(θ/2) = ±√(1 + cosθ)/2
3. tan(θ/2) = (1 − cosθ)/sinθ = sinθ/(1 + cosθ)

What Are the Triple-Angle Formulas?

1. sin(3θ) = 3sinθ − 4sin³θ
2. cos(3θ) = 4cos³θ − 3cosθ

What Are the Even-Odd (Negative Angle) Identities?

Sine, tangent, cosecant, and cotangent are odd functions:

• sin(−θ) = −sinθ
• tan(−θ) = −tanθ
• csc(−θ) = −cscθ
• cot(−θ) = −cotθ

Cosine and secant are even functions:

• cos(−θ) = cosθ
• sec(−θ) = secθ

What Are the Cofunction Identities?

The cofunction identities express trig of complementary angles in terms of the co-function.

sin(90° − θ) = cosθ
cos(90° − θ) = sinθ

What Are the Sum-to-Product and Product-to-Sum Formulas?

Sum-to-Product

• sinA + sinB = 2sin((A+B)/2)cos((A−B)/2)
• sinA − sinB = 2cos((A+B)/2)sin((A−B)/2)
• cosA + cosB = 2cos((A+B)/2)cos((A−B)/2)
• cosA − cosB = −2sin((A+B)/2)sin((A−B)/2)

Product-to-Sum

• 2sinAcosB = sin(A+B) + sin(A−B)
• 2cosAcosB = cos(A+B) + cos(A−B)
• 2sinAsinB = cos(A−B) − cos(A+B)

What Are the Inverse Trigonometric Formulas?

The inverse functions answer the question _"what angle has this trig ratio?"

• arcsin(sinθ) = θ (θ ∈ [−π/2, π/2])
• arccos(cosθ) = θ (θ ∈ [0, π])
• arctan(tanθ) = θ (θ ∈ (−π/2, π/2))

What Is Euler's Identity? (The Most Beautiful Equation)

e^{iπ} + 1 = 0

This single equation connects five of the most important constants in mathematics: e, i, π, 1, and 0.

Why Are Trigonometry Formulas Important? (The Real-World GROUND)

Trig formulas are essential for modern technology:

• GPS satellites use sum/difference formulas for position triangulation.
• Audio compression relies on discrete cosine transforms.
• Image compression utilizes 2D discrete cosine transforms.
• Signal processing employs fast Fourier transform techniques.

A Worked Example

Prove that sin(2θ) = 2sinθcosθ.

The intuitive approach fails.
Correct method: Start from the sum formula with A = B = θ:

sin(θ + θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ

What Are the Most Common Mistakes With Trigonometry Formulas?

Mistake 1

Confusing sin(A + B) with sinA + sinB.
Correct way: sin(A + B) = sinAcosB + cosAsinB.

Mistake 2

Forgetting the ± in half-angle formulas.
Correct way: Determine the quadrant to choose the sign.

Mistake 3

Using the wrong form of the cosine double angle.
Correct way: Choose the form based on known values.

The Mathematicians Who Shaped Trigonometry Formulas

• Hipparchus of Nicaea (c. 190–c. 120 BCE, Greece) — Built the first systematic trig table.
• Leonhard Euler (1707–1783, Switzerland) — Standardized modern notation and unified trigonometry with complex numbers.
• Jean-Baptiste Joseph Fourier (1768–1830, France) — Demonstrated any periodic function can be represented as a sum of sines and cosines.

A Practical Next Step

Try these three before moving on to trigonometric equations.

1. Use the double-angle formula to compute cos60° from cos30° = √3/2.
2. Use sin(A+B) to compute sin105°.
3. If cosθ = 4/5 and θ is in Q1, find sin2θ.