Trigonometric Identities


Trigonometry is a branch of mathematics which is used to show the relationship among the angles and sides of a triangle. In other word if we want to find relation between sides and angles of triangle then we will use trigonometry to find relation. The equalities that contain trigonometric functions are called as trigonometric identities. Some functions and angles are also involved in trigonometric function. Let’s discuss about list of trigonometric identities. The list of trigonometric identity is as follows: let’s understand first even – odd Trigonometric Identities of trigonometric function.

Sin (-a) = - sin a,

Cos (-a) = cos a,

Tan (-a) = - tan a,

Cosec (-a) = - cosec a,

Sec (-a) = sec a,

Cot (-a) = - cot a, using these even – odd identities we can easily solve trigonometric functions. Now we will talk about sum – difference formula:

Sin (u + v) = sin u cos v + cos u sin v,

Cos (u + v) = cos u cos v + sin u sin v,

Tan (u + v) = (tan u + tan v) / (1 + tan u tan v).

These are all sum – difference formulas. Let’s discuss some fundamental trigonometric identities. Lets’ discuss first reciprocal identities of the trigonometric functions:

Sin a = 1 / Cosec a,

Cos a = 1 / Sec a,

Tan a = 1 / Cot a,

Cosec a = 1 / Sin a,

Sec a = 1 / Cos a,

Cot a = 1 / Tan a, Reciprocal identities are also important in trigonometric functions. We will now discuss some Pythagorean identities:

Sin2 a + Cos2 a = 1,

1 + tan2 a = sec2 a,

1 + cot2 a = cosec2 a; These are Pythagorean identities given for trigonometric functions. Using all above mention trigonometric identity we can easily solve the trigonometric functions.

Reciprocal identities

Equation in math are used to define the equality between two sides. An identity is also an equality which is true for values of a variable. Reciprocal identities are true for some values of a variable. We can represent reciprocal of variable 'x' as 1 /y.

Let us see how to represent the reciprocal identity:

x = 1 / y.

Here 'x' and 'y' are two variable...Read More

pythagorean identities

While studying a unit circle it should be noticed that a point on this circle can be represented by the coordinates (cos θ, sin θ). To define this point first Pythagorean Identity can be used which is-

sin 2 θ + cos 2 θ = 1 --------------------- (1)

This is one of the standard formula used in trigonometry.

Using this Pythagorean identity tw...Read More

Sum and Difference Formulas

Sum is addition of two or more numbers, matrices etc. Normally it is expressed by sum and difference formulas. If we want to find sum A and B then simple mathematical formula for adding them will be written as:

Sum = A + B. Symbol ‘+’ is normally used for indicating sum. Similarly, difference of two or more numbers or matrices can be performed ...Read More

Double angle formulas

Trigonometry defines the relationship between angles and sides of a triangle. Here we will talk about trig double angle formulas. Some important double angle formulas are given below:

⇒sin Ө / 2 = 2 sin Ө cos Ө,

⇒cos 2 Ө = cos2 Ө – sin2Ө;

Cos 2 Ө have three formulas which are as follows:

⇒cos 2 Ө = cos2 Ө – sin2Ө;
⇒cos 2 Ө = ...Read More

Tangent and Cotangent identities

Tangent function can be obtained by ratio of sine to cosine function and Cotangent function is inversely equal to tangent function. Thus, tangent and cotangent identities can be derived by using sine and cosine function identities.

Basic identity for tangent and cotangent functions for angle ‘a’ can be given as follows:

tan (a) = sin (...Read More

Half Angle Formulas

We can show the relationship among the angles and sides of a triangle with help of trigonometry. Here we will see some half angle formulas. Half angle formula is given below:

⇒cos ⊖/ 2 = + √ ½ (1 + cos ⊖), and

⇒sin ⊖/ 2 = + √ ½ (1 - cos ⊖),

Let’s see proof of half angle formula.
'α' is half of 2α, if we put value 2α in place of ⊖ then we get:

⇒ α...Read More

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