Cos(a - b)

In trigonometry, cos(a - b) is one of the important trigonometric identities, that finds application in finding the value of the cosine trigonometric function for the difference of angles. The expansion of cos (a - b) helps in representing the cos of a compound angle in terms of trigonometric functions sine and cosine. Let us understand the cos(a-b) identity and its proof in detail in the following sections.

Cos (a - b) is the trigonometric identity for compound angles. We apply the cos (a-b) identity formula when the angle for which the value of the cosine function is to be calculated is given in the form of the difference of angles. The angle (a-b) represents the compound angle.

We refer to cos(a - b) formula as the subtraction formula in trigonometry. The cos(a - b) formula for the compound angle (a-b) can be given as,

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

The proof of expansion of cos(a-b) formula can be given using the geometrical construction method. Let us see the stepwise derivation of the formula for the cosine trigonometric function of the difference of two angles. In the geometrical proof of cos(a-b) formula, we initially assume that 'a' and 'b' are positive acute angles, such that angle a > angle b. This formula, in general, is true for any positive or negative value of a and b.

To prove: cos (a - b) = cos a cos b + sin a sin b

Construction: Draw a line OX in a plane and let us rotate it about O in the anti-clockwise direction to a point Z, making an acute ∠XOZ = a, from starting position to its final position. Again, rotate the line, this time in the backward direction, starting from the position OZ till it reaches a point Y, thus making out an acute angle given as, ∠ZOY = b. Therefore, ∠XOY = a - b.

Next, take a point P on OY, and draw PQ and PR perpendiculars to OX and OZ respectively. Draw perpendiculars RS and RT from point R upon OX and PQ respectively.

Now, from the right-angled triangle PQO we get,
cos (a - b) = OQ/OP
= (OS+SQ)/OP
= OS/OP + SQ/OP
= OS/OP + TR/OP
= OS/OR ∙ OR/OP + TR/PR ∙ PR/OP
= cos a cos b + sin ∠TPR sin b
= cos a cos b + sin a sin b, (since we know, ∠TPR = a)

Therefore, cos (a - b) = cos a cos b + sin a sin b.

The expansion of cos(a - b) can be used to find the value of the cosine trigonometric function for angles that can be represented as the difference of standard angles in trigonometry. We can follow the steps given below to learn to apply cos(a - b) identity. Let us evaluate cos(90º - 30º) to understand this better.

• Step 1: Compare the cos(a - b) expression with the given expression to identify the angles 'a' and 'b'. Here, a = 90º and b = 30º.
• Step 2: We know, cos (a - b) = cos a cos b + sin a sin b.
  ⇒ cos(90º - 30º) = cos 90ºcos 30º + sin 90ºsin 30º
  since, sin 90º = 1, sin 30º = 1/2, cos 90º = 0, cos 30º = √3/2
  ⇒ cos(90º - 30º) = (0)(√3/2) + (1)(1/2) = 0 + 1/2 = 1/2
  Also, we know that cos 60º = 1/2. Therefore the result is verified.

Let us have a look a few solved examples to understand cos(a-b) formula better.

What is Cos(a-b)?

Cos(a - b) is one of the important trigonometric identities, also called the cosine subtraction formula in trigonometry. Cos(a-b) can be given as, cos (a - b) = cos a cos b + sin a sin b, where 'a' and 'b' are angles.

What is the Formula of Cos(a-b)?

The cos(a-b) formula is used to express the cosine compound angle formula in terms of sine and cosine of individual angles. cos(a-b) trigonometry formula can be given as, cos (a - b) = cos a cos b + sin a sin b.

What is Expansion of Cos(a-b)

The expansion of cos(a-b) is given as, cos (a - b) = cos a cos b + sin a sin b. Here, a and b are the measures of angles.

How to Prove Cos(a-b) Formula?

The proof of cos(a-b) formula can be given using the geometrical construction method. We initially assume that 'a' and 'b' are positive acute angles, such that a > b. Click here to understand the stepwise method to derive cos(a-b) formula.

What are the Applications of Cos(a-b) Formula?

cos(a-b) can be used to find the value of cosine function for angles that can be represented as the difference of standard or simpler angles. Therefore, it makes the deduction easier. It can also be used in finding the expansion of other double and multiple angle formulas.

How to Find the Value of Cos 30º Using Cos (a - b) Identity.

The value of cos 30º using cos(a - b) identity can be calculated by first writing it as cos[(90º - 60º)] and then applying cos(a - b) identity.
⇒cos[(90º - 60º)] = cos 90ºcos 60º + sin 90ºsin 60º = (0)(1/2) + (1)(√3/2) = 0 + (√3/2) = √3/2.

How to Find Cos(a + b - c) using Cos(a - b)?

We can express cos(a+b-c) as cos((a+b)-c) and expand using cos(a-b), cos(a+b) and sin(a+b) formula as, cos(a+b-c) = cos(a+b).cos c + sin(a+b).sin c = cos c.(cos a cos b - sin a sin b) + sin c.(sin a cos b + cos a sin b) = cos a cos b cos c - sin a sin b cos c + sin a cos b sin c + cos a sin b sin c.