cos theta Formula

cos θ formula

cos θ is one of the most commonly used trigonometric functions along with sin θ and tan θ. The cos θ is calculated in a right angle triangle which is defined as the ratio of the base to the hypotenuse of the right angle triangle. 

The main functions in trigonometry are the right-angle triangle formulas. Cos θ is one of them. The main formula of cos θ is given by the ratio of base to the hypotenuse. 

There are several identities relating to cos θ and other trigonometric functions. These include: 

• sin² θ + cos² θ = 1 

• cos θ = 1 / sec θ

• cos θ = (1 – tan² θ /2) / (1 + tan² θ / 2)

• cos θ = sin θ * cot θ

• cos (- θ) = cos θ

• cos (90 – θ) = sin θ

• cos (180 + θ) = cos θ

• cos (180 – θ) = – cos θ

• cos 2 θ = cos² θ – sin² θ

• cos (x + y) = cos x cos y – sin x sin y

• cos (x – y) = cos x cos y + sin x sin y

Some solved examples

Q1. If sin θ = 6/ 13, find out the value of cos θ.

We know the formula that, 

cos² θ = 1 – sin² θ

cos² θ = 1 – (6/ 13)²

cos² θ = 1 – 36/ 169

cos² θ = 133/ 169

cos θ = (133)^1/2/ 13

Q2. If tan θ /2 = 5/ 8, find out the value of cos θ. 

We know the formula that, 

cos θ = (1 – tan² θ /2) / (1 + tan² θ /2)

cos θ = (1 – 25/ 64) / (1 + 25/ 64)

cos θ = (39/ 64) / (89/ 64)

cos θ = 39 / 89 

Q3. Find cos θ if the value of sin θ = 2 / 5?

We know the formula that, 

cos² θ = 1 – sin² θ

cos² θ = 1 – (2 / 5)²

cos² θ = 1 – 4 / 25

cos² θ = 16 / 25

cos θ = 4 / 5