Fundamental trigonometric identities, aka trigonometric identities or trigonometric identities are equations containing trigonometric functions that are valid for any value you put in their variables.
These IDs are essential tools if you want to solve trigonometric equations and perform complex calculations in mathematics, physics or engineering. Understanding all of the trigonometric identities can help you simplify seemingly complex problems, especially in geometry and mathematics.
Fundamentals of Trigonometry
Trigonometry is a branch of mathematics. At the heart of trigonometry are 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, and instructors often teach them using the mnemonic SOH-CAH-TOA on right-angled triangles.
From these basic trigonometric functions, we derive other important functions such as secant, cosecant and cotangent, which play a vital role in the further development of trigonometric theory.
You may hear people refer to sine, cosine, tangent, secant, cosecant, and cotangent as the six trigonometric ratios or trigonometric ratios.
Basic Trigonometric Identities
Trigonometric identities form the cornerstone of higher mathematics. They cover all trigonometric ratios and relationships in a framework that enhances the solution of equations and understanding of geometric and algebraic concepts.
Trigonometric identities cover a wide variety of formulas, but people often categorize them according to their specific applications and formats.
There are three main categories of eight basic trigonometric identities. These categories include reciprocal identities, Pythagorean identities, and partition identities.
Mutual Identities
These identities express the basic trigonometric functions in terms of their reciprocal functions:
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Sine and cosecant:csc(θ) = 1/sin(θ)
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Cosine and secant: second(θ) = 1/cos(θ)
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Tangent and cotangent: baby cot(θ) = 1/skin color(θ)
Pythagorean Identities
Pythagorean trigonometric identities originate from the Pythagorean theorem, also known as the Pythagorean theorem after the Greek scientist who coined the mathematical expression.
Trigonometric identities based on the Pythagorean theorem are fundamental for connecting the squares of primary trigonometric functions:
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Basic Pythagorean identity: sin2(θ) + because2(θ) = 1
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derived for tangent: 1 + skin color2(θ) = seconds2(θ)
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Derived for cotangent: baby bed2(θ) + 1 = csc2(θ)
Department IDs
These identities relate functions through division:
Of course, beyond 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
Double angle formulas are trigonometric identities expressing the trigonometric functions of double angles, that is, angles of the form 2.θ — in terms of trigonometric functions of odd angles (θ).
These formulas are very important in various mathematical calculations and transformations, especially in solving calculus, geometry and trigonometric equations.
Primary double angle formulas include sine, cosine, and tangent formulas.
Cosine Double Angle Formula
The cosine double angle formula is:
because(2θ) = because2(θ) – sin2(θ)
You can show this in two alternative ways using the Pythagorean identity. sin2(θ) + because2(θ) = 1:
because(2θ) = 2cos2(θ) – one
2cos2(θ) – 1 = 1 – 2sin2(θ)
Sine Double Angle Formula
The sine double angle formula is:
sin(2θ) = 2sin(θ)Because(θ)
This formula is derived from total identities and is useful in solving problems involving sine and cosine products.
Tangent Double Angle Formula
The tangent double angle formula is:
skin color(2θ) = (2(θ))/(1 – skin color2(θ))
This expression arises from dividing the sine double angle formula into the cosine double angle formula and simplifying it using the definition of tangent.
Three Angle Trigonometric Identities
Although triple angle formulas are less commonly used, they offer shortcuts in certain scenarios, such as definite integrals and polynomial equations. These are identities that allow calculating the sine, cosine, and tangent of three multiples of a given angle (3θ) using 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 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 certain mathematical and engineering contexts, such as simplifying complex trigonometric expressions or solving higher order 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 double angle identities and other basic trigonometric identities.
Half-angle identities for sine, cosine, and tangent use the following half-angle formulas:
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Sine half angle identity:sin(θ/2) = ±√((1 – becauseθ)/2)
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Cosine half angle identity:because(θ/2) = ±√((1 + becauseθ)/2)
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Tangent half angle identity:tan(θ/2) = sin(θ)/(1 + because(θ)) = 1 – (because(θ)/sin(θ))
In sine and cosine half angle formulas, the sign depends on the current quadrant. θ/2 resides. You can also express the tangent half angle formula directly in terms of sine and cosine.
These identities are obtained by processing double-angle identities. For example, cosine double angle identity because(2θ) = 2cos2(θ) can be rearranged to express Because2(θ) in terms of because(2θ)and then taking the square root (and adjusting the sign relative to the quarter of the angle) gives the half-angle formula for cosine.
Half-angle identities are very important for simplifying the integration of trigonometric functions, especially when their 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
Total identities in trigonometry are basic formulas that allow calculating 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 with AI technology, then had it fact-checked and edited by a HowStuffWorks editor.
Original article: Trigonometric Identities: A Crash Course in Complex Math Concepts
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