The major trigonometry functions are sine, cosine, and tangent, while the other three functions are cot, sec, and cosec. All six trig functions are used to derive the trigonometric identities.
Trigonometric identities come in handy when trigonometric functions are used in an expression or equation. Trigonometric identities are true for every value of a variable that appears on both sides of an equation. Geometrically, these identities include one or more trigonometric functions (such as sine, cosine, and tangent).
What are Trigonometric Identities?
Trigonometric Identities are equalities that utilize trigonometry functions and hold for all variables in the equation.
There are several trigonometric identities relating to the angle and the length of the side of a triangle. Only the right-angle triangle has the trigonometric identities.
The six trigonometric ratios are the basics of all fundamental trigonometric identities. Sine, cosine, tangent, cosecant, secant, and cotangent are the arithmetic functions. All of these trigonometric ratios are defined using the sides of a right triangle, namely the adjacent, opposing, and hypotenuse sides.
Reciprocal Trigonometric Identities
We already know that cosecant, secant, and cotangent are the reciprocals of sin, cosine, and tangent.
As a result, the reciprocal identities are as follows:
- sin θ = 1/cosecθ ; cosec θ = 1/sinθ
- cos θ = 1/secθ ; sec θ = 1/cosθ
- tan θ = 1/cotθ ; cot θ = 1/tanθ
Pythagorean Trigonometric Identities
The Pythagorean trigonometric identities are derived from Pythagoras’ theorem in trigonometry. When we apply Pythagoras’ theorem to the right-angled triangle shown below, we get:
= +
is used to divide both sides.
/ + / = /
sin2θ + cos2θ = 1
Complementary and Supplementary Trigonometric Identities
A complementary angle is a pair of two angles whose total is equal to 90°. θ is (90 – θ) is the complement of an angle. The complementary angle trigonometric ratios are as follows:
- sin (90°- θ) = cos θ
- cos (90°- θ) = sin θ
- cosec (90°- θ) = sec θ
The supplementary angles are a pair of two angles whose total equals 180°. An angle’s supplement θ is (180 – θ). Supplementary angles are:
- sin (180°- θ) = sinθ
- cos (180°- θ) = -cos θ
- cosec (180°- θ) = cosec θ
Trigonometry Formulas
Trigonometry formulae are collections of formulas that employ trigonometric identities to solve issues involving the sides and angles of a right-angled triangle. For certain angles, these trigonometry formulas contain trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. In the following sections, we will go through the formulae relating Pythagorean identities, product identities, cofunction identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, and so on.
List of Trigonometry Formulas
Trigonometry formulae are categorize into many groups based on the trigonometry identities used. Let’s have a look at the following sets of trigonometric formulae.
- Formulas for Trig Ratios: These are trigonometry formulae for the basic trigonometric ratios sin, cos, tan, and so on.
- Reciprocal Identities: Trigonometry formulae that deal with the reciprocal connection between trig ratios are include in this category.
- Trigonometric Ratio Table: The trigonometry table depicts trigonometry values for standard angles.
- Periodic Identities: These are trigonometry formulae that aid in determining the values of trig functions for a shift in angles by 2.
- Trigonometry formulae for cofunction identities demonstrate the interrelationships between trigonometric functions.
- Sum and Difference Identities: These trigonometry formulae are use to calculate the value of the trigonometric function for the sum or difference of angles.
- Sum to Product Identities: These trigonometry formulae are use to express the sum or product of trigonometric functions.
- Formulas for Inverse Trigonometry: Inverse trigonometry formulae contain formulas for inverse trig functions such as sine inverse, cosine inverse, and so on.
The Three Pythagorean Identities
The three Pythagorean identities come from Pythagoras’ theorem. The first one is sin squared theta plus cos squared theta equals 1. The second one is 1 plus tan squared theta equals sec squared theta. The third one is 1 plus cot squared theta equals cosec squared theta. These three identities are used in many proofs. If you know one of them, you can often find the others by dividing by sin squared theta or by cos squared theta.
Sum and Difference Identities
These identities find the sine or cosine of the sum of two angles. The sine of A plus B is sin A cos B plus cos A sin B. The sine of A minus B is sin A cos B minus cos A sin B. The cosine of A plus B is cos A cos B minus sin A sin B. The cosine of A minus B is cos A cos B plus sin A sin B. These formulas let you work out the sine of 75 degrees as the sine of 45 plus 30 degrees, using known values.
Double Angle and Half Angle Identities
A double angle identity gives the value of a function at two times an angle. The sine of 2A is 2 sin A cos A. The cosine of 2A is cos squared A minus sin squared A. It can also be written as 2 cos squared A minus 1, or as 1 minus 2 sin squared A. These forms are useful in different problems, so learn all three forms of cos 2A. The tangent of 2A is 2 tan A divided by 1 minus tan squared A.
Standard Angles Table: Values to Memorise
| Angle | sin | cos | tan |
|---|---|---|---|
| 0 degrees | 0 | 1 | 0 |
| 30 degrees | 1/2 | square root of 3 over 2 | 1 over square root of 3 |
| 45 degrees | 1 over square root of 2 | 1 over square root of 2 | 1 |
| 60 degrees | square root of 3 over 2 | 1/2 | square root of 3 |
| 90 degrees | 1 | 0 | Not defined |
Learn these values by heart. They appear in almost every trigonometry test. Draw a right-angled triangle for 30, 60, and 90 degrees to check the values. For 45 degrees, use a triangle with two equal sides.
How to Prove a Trigonometric Identity in Five Steps
Step one is to start with the harder side. Step two is to change all functions into sine and cosine. Step three is to use the Pythagorean identities to simplify. Step four is to factor or to take a common factor where you can. Step five is to keep going until the two sides match. Do not start by assuming the identity is true. Work on one side only until it equals the other side. Practice with easy identities first, such as proving that tan theta times cos theta equals sin theta.
Where Trigonometry Identities Are Used
Engineers use these identities to study waves, sound, and light. Builders use them to find heights and distances that are hard to measure. In physics, the motion of a swing and the path of a projectile both use sine and cosine. In computer graphics, rotation of an image on screen uses the sum and difference identities. In navigation, sailors and pilots use trigonometry to find a course from one place to another.
Frequently Asked Questions
What is a trigonometric identity?
It is an equation with trigonometric functions that is true for every value of the variable. For example, sin squared theta plus cos squared theta equals 1.
What are the three Pythagorean identities?
They are: sin squared theta plus cos squared theta equals 1; 1 plus tan squared theta equals sec squared theta; and 1 plus cot squared theta equals cosec squared theta.
What is sin of A plus B?
It is sin A cos B plus cos A sin B. The cosine of A plus B is cos A cos B minus sin A sin B.
What is sin of 2A?
It is 2 sin A cos A. The cosine of 2A is cos squared A minus sin squared A.
How do I prove a trigonometric identity?
Start with the harder side. Change the functions into sine and cosine. Use the Pythagorean identities to simplify, until both sides match.
Disclaimer: This guide is for general information only. It is an aid for study and does not replace your class notes or textbook. Check the values and formulas against your course material.
Last Updated: 6 November 2026