A crash course in complex mathematical concepts

Fundamental trigonometric identities, i.e trig identities or trigo identities are equations with trigonometric functions that hold for any value you substitute into their variables.

These identities are essential tools when 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 basics of trigonometry

Trigonometry is a branch of mathematics. At the heart of trigonometry are 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 triangles.

From these fundamental trig functions we derive other crucial functions such as secant, cosecant and cotangent, all of which play a crucial role in the further development of trigonometric theory.

You may hear people call sine, cosine, tangent, secant, cosecant, and cotangent the six trigonometric ratios or trig ratios.

Basic trigonometric identities

Trigonometric identities are a cornerstone of higher mathematics. They encapsulate all trigonometric ratios and relationships in a framework that improves equation solving and understanding of geometric and algebraic concepts.

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

There are three main categories that include eight basic trigonometric identities. These categories include mutual identities, Pythagorean identities, and quotient identities.

Mutual identities

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

  • Sinus and cosecant: csc(θ) = 1/sin(θ)

  • Cosine and secant: sec(θ) = 1/cos(θ)

  • Tangent and cotangent: cot(θ) = 1/brown(θ)

Pythagorean identities

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

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

  • Basic identity of Pythagoras: shame2(θ) + co2(θ) = 1

  • Derived for tangent: 1 + brown2(θ) = sec2(θ)

  • Derived for cotangent: cot2(θ) + 1 = csc2(θ)

Quotient identities

These identities connect the functions through divisions:

Of course, in addition to these core identities, there are many more trigonometric 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 formulas for double angles are trigonometric identities expressing trigonometric functions of double angles – that is, angles of the form 2θ — in terms of trigonometric functions of some angles (θ).

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

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

Cosine double angle formula

The cosine double angle formula is:

cos(2θ) = co2(θ) – shame2(θ)

You can also represent this in two alternative forms using the Pythagorean identity shame2(θ) + co2(θ) = 1:

cos(2θ) = 2cos2(θ) – 1

2cos2(θ) – 1 = 1 – 2sin2(θ)

Sine double angle formula

The formula for a double sine angle is:

sin (2θ) = 2sin(θ)because(θ)

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

Double angle tangent formula

The formula for the double angle tangent is:

brown(2θ) = (2tan(θ))/(1 – brown2(θ))

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

Trigonometric identities with three angles

Triple angle formulas, although less commonly used, provide shortcuts in specific scenarios, such as in certain integrals and polynomial equations. These are identities with which the sine, cosine and tangent of three times a certain angle (3θ) can be calculated using the trigonometric functions of the angle itself (θ).

For example, the formula for the sine-triple angle is:

sin (3θ) = 3sin(θ) – 4sin3(θ)

This formula is derived 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 a given angle.

Half-angle formulas are especially useful in solving trigonometric equations, integrating trigonometric functions, and simplifying expressions when the angle of interest is halved. Formulas for half angles are derived from the double angle identities and other basic 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: brown(θ/2) = sin(θ)/(1 + cos(θ)) = 1 – (cos(θ)/shame(θ))

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

These identities are derived by manipulating the double corner identities. For example, the cosine double angle identity cos(2θ) = 2cos2(θ) can be rearranged to express because2(θ) in terms of cos(2θ)and then taking the square root (and adjusting the sign based on the quadrant of the angle) gives the half-angle formula for cosine.

Half-angle identities are crucial for simplifying the integration of trigonometric functions, especially when integral limits include pi (π) or when integrating periodic functions. They also play a crucial 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 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 using AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.

Original article: Trig Identities: A Crash Course in Complex Mathematical Concepts

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