Formula for cos (a – b) is,

Cos(a – b) = cos a cos b + sin a sin b

Cos (a – b) is one of the important trigonometric identities, cos (a – b) is also called the cosine subtraction formula in trigonometry. Cos(a-b) is given as, cos (a – b) = cos a cos b + sin a sin b. In this article, we will learn about, cos(A – B), Proof of this Identity, How to Apply cos(A – B) Formula, and Others in detail.

What is Cos(A – B)?

Cos(A – B) is a trigonometric identity for compound angles. We use this identity when the angle for which we want to calculate the cosine function is given as the difference of two angles, such as (90° – 30°) or (45° – 15°). Angle (A – B) represents the compound angle.

What is Cosine Function?

Cosine function is one of the fundamental functions in trigonometry, often abbreviated as “cos.” It relates the angles of a right triangle to the lengths of the sides of the triangle. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

Cos(A – B) Formula

Formula for cos(A – B) is given by,

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

This formula can express the cosine of a compound angle in terms of the sine and cosine functions of the individual angles.

Proof of Cos(A – B) Formula

Proof of cos(A – B) formula can be done using various methods, such as Geometrical Construction Method, Using Complex Numbers, etc.

Using Trigonometric Identities

The formula for the cosine of the difference of two angles can be derived using the angle addition and subtraction identities for sine and cosine.

As we know Cos a + b Formula is given as:

cos(a + b) = cos(a)cos(b) − sin(a)sin(b)

cos(a − b) = cos(a + (−b))

As we know, cos(−b) = cos(b) and sin(−b) = −sin(b)

⇒ cos(a − b) = cos(a) cos(−b) − sin(a)sin(−b)

⇒ cos(a − b) = cos(a)cos(b) − sin(a)(−sin(b))

⇒ cos(a − b) = cos(a)cos(b) + sin(a)sin(b)

Which is required identity.

Using Complex Numbers

Proof of cos(A – B) using complex numbers is discussed below,

Cos(A – B) formula can be derived using the complex numbers as,

e^ix = cos x + i.sin x

Let us assume x = (A – B)

e^i(A-B) = cos (A-B) + i.sin (A-B)

Now, applying exponent rule in e^i(A-B)

e^i(A-B) = e^i(A). e^i(-B)

cos (A-B) + i.sin (A-B) = {cos A + i.sin A}.{cos (-B) + i.sin (-B)}

cos (A-B) + i.sin (A-B) = cos A.cos B + sin A.sin B + i(sin A.cos B – cos A.sin B)

Comparing Real and Imaginary Parts,

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

Thus, cos (A-B) formula is derived.

How to Apply Cos(A – B)?

Cos(A – B) formula can be used to find the value of cosine function for angles that can be expressed as the difference of standard or simpler angles. For example, we can use this formula to find cos(75°) or cos(15°), which are not directly available in the trigonometric table.

To apply the cos(A – B) formula, we can follow these steps:

Step 1: Identify the angles A and B in the given expression, such that A – B is equal to the required angle.

Step 2: Substitute the values of sin and cos of A and B from trigonometric table or using other identities.

Step 3: Simplify expression to get the final answer.

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