Trig Identities

Ultimate Guide to Trig Identities: Mastering Trigonometric Relationships

The basic trigonometric identities are formed based on our understanding of the unit circle, reference triangles, and angles. In mathematics, a “Trig identities” is an equation that is always true, as nicely stated by There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot.

There are fundamentally six trigonometry ratios utilized for finding the components in Trigonometry. They are called trigonometric ratios or functions. The six trigonometric ratios are sine, cosine, and secant, and their reciprocals are cosecant, tangent, and cotangent, respectively.

Ever wonder how to navigate a world of sine, cosine, and radians? Trigonometry can be a breeze with the right tools! Unlock the secrets of trig identities with a cheat sheet or dive deeper with a worksheet. Explore the unit circle to visualize trigonometric ratios (SOH CAH TOA) and understand their relationship with degrees and radians. Don’t get lost in the conversion – a trig identities calculator can be your hero.

If we use the right angle triangle as a kind of perspective, then, the trigonometric ratios derived as:

sin θ = Opposite Side/Hypotenuse
cos θ = Adjacent Side/Hypotenuse
sec θ = Hypotenuse/Adjacent Side
tan θ = Opposite Side/Adjacent Side
cosec θ = Hypotenuse/Opposite Side
cot θ = Adjacent Side/Opposite Side


MyLab Math can be a helpful resource for studying trigonometry identities. It provides a platform to learn and practice these mathematical concepts, which are essential for understanding trigonometric relationships.  If you’re looking for further assistance with MyLab Math or trig identities, feel free to ask!

The Reciprocal Identities of the right angle triangle are given as:

cosec θ = 1/sin θ
cot θ = 1/tan θ
sec θ = 1/cos θ
cos θ = 1/sec θ
sin θ = 1/cosec θ
tan θ = 1/cot θ

Identities and equations compared: An identity is a statement that is always true, whereas an equation is only true under certain conditions. For example
3x + 2x = 5x
is an identity that is always true, no matter what the value of x, whereas
3x = 15
is an equation (or more precisely, a conditional equation) that is only true if x = 5.

Trig Identities

A Trigonometric identity is an identity that contains the trigonometric functions sin, cos, tan, cot, sec, or csc. Trigonometric identities can be used to:
Simplify trigonometric expressions.
Solve trigonometric equations.
Prove that one trigonometric expression is equivalent to another so that we can replace the first expression with the second expression. The second expression can give us new insights into some applications that the first one doesn’t show.
Basic and Pythagorean Identities

CSC(x)=1/ sin(x)


cot(x)=1/tan(x)​= cos(x)​// /sin(x)
tan(x)=1/cot(x)​= sin(x)/Cos(x)​
Notice how a “co-(something)” trig ratio is always the reciprocal of some “non-co” ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.
The following (particularly the first of the three below) are called “Pythagorean” identities.
sin2(t) + cos2(t) = 1

tan2(t) + 1 = sec2(t)

1 + cot2(t) = csc2(t)
Note that the three identities above all involve squaring and the number 1. You can see the Pythagorean-Thereom relationship clearly if you consider the unit circle, where the angle is t, the “opposite” side is sin(t) = y, the “adjacent” side is cos(t) = x, and the hypotenuse is 1.

Master Trig Identities with Quizlet and Euler’s Formula

Struggling with trig identities? Quizlet can help you learn all the important ones, from sine, cosine, and tangent to arctangent. Practice with problems involving derivatives, antiderivatives, and complex numbers. Master identities for addition, subtraction, double angles, half angles, and even convert between sums and products!  This will help you tackle trig equations with ease, along with applications like the Law of Sines and Cosines, all with Euler’s formula in your back pocket.

Certainly, I can do that. Trig identities serve as a cornerstone in mathematics, particularly trigonometry. They are equations that are valid for all angles and connect the trigonometric ratios (sine, cosine, tangent, and so on). These identities streamline intricate trigonometric expressions and assist in tackling mathematical problems that involve angles. In essence, trig identities function as a bridge between various domains of mathematics, enabling the graceful manipulation and examination of trigonometric functions.

(You may have noticed the radicals on the 1’s in the above. Yes, these simplify to just 1, so you can write things that way, too, and you certainly should do the simplification in your final hand-in answer. But notice how all the denominators are 2’s, and how the numerators go up or go down, 1, 2, 3. This can help remember the trig values.)
You might be given a complete unit circle, with the values for the angles in the other three quadrants, too. But you only need to know the values in the first quadrant. Once you know them, and because the values repeat (other than sign) in the other quadrants, you know everything you need to know about the unit circle.


Trig identities, or trigonometric identities, are equations that relate the different trigonometric functions (sine, cosine, tangent, etc.) to each other. These equations are true for all angles (within their domain) and are incredibly useful in simplifying trigonometric expressions and solving trigonometric equations.  Knowing trig identities allows you to rewrite expressions in terms of a single function, which can make them easier to work with.  In essence, they provide a toolbox for manipulating trigonometric expressions.

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Christophe Rude

Christophe Rude

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