Trigonometric Functions of Sum and Difference of Two Angles | JEE Trigonometry Sum and Difference of Angles

Table of Content

Is sin (A + B) = sin A + sin B?
How to derive sin (A + B) formula?
What are compound angles formulas?
How can we transform the product into sum or difference formulae?
How can we transform the sum or difference into product formulas?
Give some important identities related to compound angles
Highlight Trigonometric Ratios of multiple and submultiple angles
What are the applications for compound angle formulae?

Is sin (A + B) = sin A + sin B?

At the initial learning about trigonometric, every student has the basic doubt whether sin(A + B) is equals to sin A + sin B or not same as x(2 + 3) = 2x + 3x.

So let’s clear this doubt here by taking an example.

sin90° = sin(60° + 30°) = sin60° + sin30°

LHS: Sin90° = 1

Clearly, we can see that LHS ≠ RHS and thus sin (A + B) is not equals to sin A + sin B in general. Then let us find its actual simplification in the next section.

How to derive sin (A + B) formula?

For deriving sin (A + B) formula, let us now assume that A + B ≤ 90° thus we shall take two points P and N on the Cartesian plane such that ON makes an angle of A with the x – axis and OP makes an angle of (A + B). See the figure to understand the derivation below.

= cosA sinB + sinA cosB

Thus we can now say that sin (A + B) = sinA cosB + cosA sinB. ……. (1)

From here, we can also derive the other compound angles formula.

For sin(A – B), if we just replace B with –B in equation (1). We get

sin (A - B) = sinA cos(-B) + cosA sin(-B)

Thus on further simplification, we get

sin (A - B) = sinA cosB - cosA sinB ……………….(2)

For cos(A + B), if we replace A by π/2 + A in equation (1),

On simplification,

cos(A + B) = cosA cos B + sinA sinB ……… (3)

And for cos (A – B), if we just replace B with –B in equation (3). We get

cos(A - B) = cosA cos(-B) + sinA sin(-B)

cos(A - B) = cosA cosB - sinA sinB ………… (4)

Although, we have assumed at the beginning that A + B ≤ 90°, but still the above formulas are valid for other condition too.

What are compound angles formulas?

Compound angles formulas are the formulas in trigonometry involving sum or difference of two or more angles. We have also discussed and derived the compound angle formulas for sine and cosine. We will list all these and some addition formulas below:

sin(A + B) = sinA cosB + cosA sinB
sin(A - B) = sinA cosB - cosA sinB
cos(A + B) = cosA cos B - sinA sinB
cos(A - B) = cosA cosB + sinA sinB

Few other formulas:

Important Note: If we assume A = 45^o and B = A in above formulas,

We get,

How can we transform the product into sum or difference formulae?

We need the help of compound angles derived above to get the relation between the product and sum or difference formulae.

Since we know that:

sin(A + B) = sinA cosB + cosA sinB
sin(A - B) = sinA cosB - cosA sinB
cos(A + B) = cosA cos B + sinA sinB
cos(A - B) = cosA cosB - sinA sinB

Now, to calculate product formula for - 2sinA cosB, if we add sin (A + B) and sin (A – B) from the above formulas, we can easily get

2sinA cosB = sin (A + B) + sin (A – B) …………………………. (5)

Similarly using the above four formulas we can get other relations as well,

2cosA sinB = sin (A + B) - sin (A + B) …………………………… (6)

2cosA cosB = cos (A + B) + cos (A – B) ………………………… (7)

2sinA sinB = cos (A - B) - cos (A + B) …………………………… (8)

With this derived product formulas, we can now easily convert the product of two trigonometric ratio of different or same angle into sum or difference of the ratios.

How can we transform the sum or difference into product formulas?

For the conversion, from sum or difference into product formulae, let us assume:

A + B = C and A – B = D, which gives

Using these assumptions in equations 5, 6, 7 and 8. and simplification, we get

All these above formulae are very important in trigonometry.

Give some important identities related to compound angles

Below are few important identities, which is used to simplify the complex trigonometric equations.

Note:- All these above identities can be easily derived by using compound angles formulae.

Highlight Trigonometric Ratios of multiple and submultiple angles

So first question here may be that what do we mean by multiple and submultiple angle. So to understand it, multiple angles here mean if we have an angle A, then all the angles which are integer multiple of A, that is, nA, will be considered as the multiple angles of A. While all the angles of the form A/n, will be called as the Submultiple Angles. Here, n is any integer.

So,

sin2A = sin(A + A) = sinA cosA + cosA sinA

(using the formula for sin(A + B))

= 2 sinA cosA

Thus, sin2A = 2sinAcosA

Similarly, cos2A = cos(A + A) = cosA cosA – sinA sinA

(using the formula for cos(A + B))

= cos²A – sin²⁡A

= 2cos²A – 1 = 1 – 2 sin²⁡A

Or we may write,

Similarly, we may also derive it for other trigonometric ratios. Here we are presenting all the other important formulas for multiple and submultiple angles.

Formulae in terms of tan²A: