Trig Identities: A Crash Course in Complex Math Concepts

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Fundamental trigonometric identities, aka trig identities or trigo identities, are equations involving trigonometric functions that hold true for any value you substitute into their variables.

These identities are essential tools if you want to solve trigonometric equations and perform complex calculations in mathematics, physics or engineering. Understanding all the trigonometric identities can help you simplify seemingly complicated problems, especially in geometry and calculus.

The Foundation of Trigonometry

Trigonometry is a branch of mathematics. At the heart of trigonometry lie the trigonometric functions, which relate the angles of a triangle to the ratios of its sides.

The most basic trigonometric functions are sine, cosine and tangent, which instructors often teach using the mnemonic SOH-CAH-TOA in right-angled triangles.

From these basic trig functions, we derive other crucial functions, such as secant, cosecant and cotangent, all of which play vital roles in further developing trigonometric theory.

You might hear people refer to sine, cosine, tangent, secant, cosecant and cotangent as the six trigonometric ratios, or trig ratios.

Fundamental Trigonometric Identities

Trigonometric identities form a cornerstone of higher mathematics. They encapsulate all the trigonometric ratios and relationships in a framework that enhances the solving of equations and understanding of geometric and algebraic concepts.

Trigonometric identities encompass a wide range of formulas, but people generally group them into categories based on their specific applications and forms.

There are three main categories comprising eight fundamental trigonometric identities. These categories include reciprocal identities, Pythagorean identities and quotient identities.

Reciprocal Identities

These identities express the basic trigonometric functions in terms of their reciprocal functions:

Sine and cosecant: csc(θ) = 1/sin(θ)
Cosine and secant: sec(θ) = 1/cos(θ)
Tangent and cotangent: cot(θ) = 1/tan(θ)

Pythagorean Identities

The Pythagorean trigonometric identities stem from the Pythagorean theorem, also known as the Pythagoras theorem, after the Greek scholar who came up with the mathematical statement.

The trig identities based on the Pythagorean theorem are fundamental to connecting the squares of the primary trigonometric functions:

Basic Pythagorean identity: sin²(θ) + cos²(θ) = 1
Derived for tangent: 1 + tan²(θ) = sec²(θ)
Derived for cotangent: cot²(θ) + 1 = csc²(θ)

Quotient Identities

These identities relate the functions through division:

Tangent as a quotient: tan(θ) = sin(θ)/cos(θ)
Cotangent as a quotient: cot(θ) = cos(θ)/sin(θ)

Of course, there are many more trigonometric identities beyond just these core identities that have applications in specific scenarios, such as double angle, triple angle, half angle and sum and difference identities.

Double Angle Trigonometric Identities

The double angle formulas are trigonometric identities that express trigonometric functions of double angles — that is, angles of the form 2θ — in terms of trigonometric functions of single angles (θ).

These formulas are crucial in various mathematical computations and transformations, particularly in calculus, geometry and solving trigonometric equations.

The primary double angle formulas include those for sine, cosine and tangent.

Cosine Double Angle Formula

The cosine double angle formula is:

cos(2θ) = cos²(θ) – sin²(θ)

You can also represent this in two alternative forms using the Pythagorean identity sin²(θ) + cos²(θ) = 1:

cos(2θ) = 2cos²(θ) – 1

2cos²(θ) – 1 = 1 – 2sin²(θ)

Sine Double Angle Formula

The sine double angle formula is:

sin(2θ) = 2sin(θ)cos(θ)

This formula is derived from the sum identities and is useful for solving problems involving products of sine and cosine.

Tangent Double Angle Formula

The tangent double angle formula is:

tan(2θ) = (2tan(θ))/(1 – tan²(θ))

This expression arises from dividing the sine double angle formula by the cosine double angle formula and simplifying using the definition of tangent.

Triple Angle Trigonometric Identities

Triple angle formulas, while less commonly used, offer shortcuts in specific scenarios, such as in certain integrals and polynomial equations. These are identities that allow the calculation of the sine, cosine and tangent of three times a given angle (3θ) using the trigonometric functions of the angle itself (θ).

For example, the sine triple angle formula is:

sin(3θ) = 3sin(θ) – 4sin³(θ)

This formula is derived by using the sine double angle formula and the angle sum identity.

Triple angle formulas can be derived from double angle and sum identities and are useful in specific mathematical and engineering contexts, such as simplifying complex trigonometric expressions or solving higher-degree trigonometric equations.

Half Angle Identities

Half angle identities are trigonometric formulas that allow you to prove trigonometric identities for the sine, cosine and tangent of half of a given angle.

Half angle formulas are particularly useful in solving trigonometric equations, integrating trigonometric functions, and simplifying expressions when the angle involved is halved. Half angle formulas are derived from the double angle identities and other fundamental trigonometric identities.

The half angle identities for sine, cosine and tangent use the following half angle formulas:

Sine half angle identity: sin⁡(θ/2) = ±√((1 – cosθ)/2)
Cosine half angle identity: cos⁡(θ/2) = ±√((1 + cosθ)/2)
Tangent half angle identity: tan(θ/2) = sin(θ)/(1 + cos(θ)) = 1 – (cos(θ)/sin(θ))

In the case of the sine and cosine half angle formulas, the sign depends on the quadrant in which θ/2 resides. The tangent half angle formula you can also express in terms of sine and cosine directly.

These identities are derived by manipulating the double angle identities. For example, the cosine double angle identity cos(2θ) = 2cos²(θ) can be rearranged to express cos²(θ) in terms of cos(2θ), and then taking the square root (and adjusting for sign based on the angle's quadrant) gives the half angle formula for cosine.

Half angle identities are crucial for simplifying the integration of trigonometric functions, particularly when integral limits involve pi (π) or when integrating periodic functions. They also play a vital role in various fields of science and engineering where wave functions and oscillations are analyzed.

Sum and Difference Identities

Sum identities in trigonometry are essential formulas that allow for the calculation of the sine, cosine and tangent of the sum of two angles. Conversely, difference formulas allow you to calculate the sine, cosine and tangent of the difference between two angles.

These identities are incredibly useful for simplifying expressions, solving trigonometric equations and performing complex calculations.

We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.