by Basic trigonometric identities, aka trig identities or trigonometric identities, are equations related to 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 complex problems, especially in geometry and calculus.
The Basics of Triangulation
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 basic trigonometric functions, we derive other important functions, such as segment, cosecant and cotangent, all of which play vital roles in the further development of trigonometric theory.
You may hear people refer to sine, cosine, tangent, secant, secant and tangent as the six trigonometric ratios, or trig ratios.
Basic Triangular Identities
Trigonometric identities are a cornerstone of advanced mathematics. They include all trigonometric ratios and relationships in a framework that improves equation solving and understanding of geometric and algebraic concepts.
Trigonometric identities encompass a wide variety of formulas, but people generally group them into categories based on their specific applications and forms.
There are three main categories containing eight basic triangular 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:
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Line and cosecant:csc(θ) = 1/sin(θ)
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Cosine and segment:sec(θ) = 1/cos(θ)
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Touch and touch: coat(θ) = 1/ tan(θ)
Pythagorean Identities
Pythagorean trigonometric identities derive from the Pythagorean theorem, also known as the Pythagorean theorem, after the Greek scholar who came up with the mathematical statement.
The triple identities based on the Pythagorean theorem are fundamental to connecting the squares of the principal trigonometric functions:
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Fundamental identity of Pythagoras: sin2(θ) + leg2(θ) = 1
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Derived for tangents:1 + tan2(θ) = sec2(θ)
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Derived for tangent: coat2(θ) + 1 = csc2(θ)
Common identities
These identities relate to the functions by sharing:
Of course, there are many more trigonometric identities beyond those core identities that apply in specific situations, such as double angle, triple angle, half angle and sum and difference equalities.
Double Angle Trigonometric Identities
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 essential in various mathematical calculations and transformations, especially in calculus, geometry and solving trigonometric equations.
The double prime angle formulas include those for sine, cosine and tangent.
Double Cosine Angle Formula
The double cosine angle formula is:
leg(2θ) = leg2(θ) – sin2(θ)
You can express this in two alternative forms using the Pythagorean identity sin2(θ) + leg2(θ) = 1:
leg(2θ) = 2 feet2(θ) – 1
2 feet2(θ) – 1 = 1 – 2sin2(θ)
Double Sine Angle Formula
The formula for a double sine angle is:
sin (2θ) = 2 sin(θ)foot(θ)
This formula is derived from the sum identities and is useful for solving problems involving sine and cosine products.
Double Angle Tangent Formula
The double tangent angle formula is:
tan(2θ) = (2tan(θ))/(1 – tan2(θ))
This expression results from dividing the double sine angle formula by the double cosine angle formula and simplifying using the definition of tangent.
Triple Angle Triangular Identities
Triple angle formulas offer shortcuts, although less frequently used, in specific situations, such as in certain elements and polynomial equations. These are equivalents that allow the sine, cosine and tangent of three times a given angle (3θ) to be calculated using the trigonometric functions of the angle itself (θ).
For example, the triple sine angle formula is:
sin (3θ) = 3 sin(θ) – 4 sins3(θ)
This formula is derived using the double sine angle formula and the angle sum identity.
Three-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 advanced trigonometric equations.
Half Angle IDs
Half-angle identities are trigonometric formulas that allow you to create trigonometric identities for the sine, cosine, and tangent of a given half angle.
Half-angle formulas are very useful for solving trigonometric equations, integrating trigonometric functions, and simplifying expressions when the angle in question is extended. Half-angle formulas are derived from the double-angle identities and other basic trigonometry identities.
The common half-angle identities for sine, cosine and tangent use the following half-angle formulas:
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Sine half angle identity: that (θ/2) = ±√((1 – cosθ)/2)
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Half-angle cosine identity: foot(θ/2) = ±√((1 + cosθ)/2)
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Identity of half a tangent angle: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 it contains 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 angle identities. For example, the double angle cosine identity leg(2θ) = 2 feet2(θ) it can be rearranged to express foot2(θ) in terms leg(2θ)and then taking the square root (and adjusting for the sign based on the square of the angle) gives the half-angle formula for cosine.
Half-angle identities are essential for simplifying the integration of trigonometric functions, especially when fundamental 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 equalities are essential formulas in trigonometry that allow the sine, cosine and tangent of the sum of two angles to be calculated. Conversely, difference formulas enable you to calculate the sine, cosine and tangent of the difference between two angles.
These identities are extremely useful for simplifying expressions, solving trigonometric equations and performing complex calculations.
We created this article in collaboration with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.
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