Cos Double Angle Formula

Cos Double Angle Formula

The Cos Double Angle Formula is used to express the trigonometric ratio of the double angle (2θ) in terms of the trigonometric ratio of the single angle (θ). The Cos Double Angle Formula  is a special case of (and therefore derived from) the empirical formula for trigonometry, and some alternative formulas are derived using the Pythagorean identity. Let’s memorize  

sin(A + B) = sin A cos B + cos A sin B

cos (A + B) = cos A cos B – sin A sin B

tan (A + B) = (tan A + tan B) / (1 – tan A tan B)

What is the Cos Double Angle Formula? 

By substituting A = B in each summation formula above, we derive the double-angle formulas for sin, cos, and tan. Derivation of some alternative formulas based on the Pythagorean identity is also possible. Below are the double-angle formulas that follow the derivation of each formula. Refer to Cos Double Angle Formula on Extramarks.

Cos Double Angle Formula can be used for sin, cos, and tan.

The Cos Double Angle Formula can be written as:

sin 2 A = 2 sinA cosA (or) (2 tan A /(1 + tan2A)

cos 2A = cos2A – sin2A (or) 2cos2A – 1 (or) 1 – 2sin2A (or) (1 – tan2A) / (1 + tan2A)

tan 2A = (2 tan A)/(1 – tan 2A)

Derivation of the double angle formula 

Let’s derive the double-angle formulas one by one from sin, cos, and tan.

The sum formula for the sin

sin(A + B) = sin A cos B + cos A sin B

If A = B, then the above formula becomes

sin (A + A) = sin A cos A + cos A sin A

sin 2 A=2 sinAcos A

Let’s use the Pythagorean identity sec2A = 1 + tan2A to derive another expression for sin 2A with respect to tan.double angle formula for the sine function:

sin 2A = 2 sin A cos A (or) (2 tan A) / (1 + tan2A)

Cos Double Angle Formula:

The Cos Double Angle Formula is cos (A + B) = cos A cos B – sin A sin B

If A = B, then the above formula becomes

cos(A + A) = cos A cos A – sin A sin A

cos2A = cos2A – sin2A

Using this as a base equation, we derive two more equations from cos 2A using the Pythagorean identity sin2A + cos2A = 1.

(i) cos2A=cos2A − (1 − cos2A) = 2cos2A − 1

(ii)cos 2A =(1 – sin2A) – sin2A = 1 – 2sin2A

Introduction to Cos 2 Theta formula

start with the addition formula.

Cos(A + B) = Cos A cos B – Sin A sin B

Let’s equate B with A, i.e. A = B

And the first of these equations becomes: Cos(t + t) = Cos t cos t – Sin t sin t

Cos2t = Cos2t – Sin2t

This how one gets double angle formula, this is because of doubling the angle (like 2A). Learners are advised to learn Cos Double Angle Formula from Extramarks.

Deriving Double Angle Formulae for Cos 2t

Students can check the derivation on the website of Extramarks.

Practice Example for Cos 2:

Solve; cos 2a = sin a. – Π

a < Π

Solution: Use the double-angle formula cos 2a = 1 − 2 sin2 a.

1−2 sin2 a = sin a.

2 sin2a + sin a − 1=0,

Let’s factor this quadratic equation by the variable sin x

(2 sin a − 1)(sin a + 1) = 0

2 sin a − 1 = 0 or sin a + 1 = 0

sin a= 1/2 or 

sin a= −1

For more formulas and examples, one can visit the Extramarks website.