Trigonometry Functions of Sum and Difference of Angles

Sum And Difference of Angles in Trigonometry Functions

How can you measure the height of the mountain? How will you calculate the distance between the Earth and the Sun? There are numerous impossible problems, we depend on the mathematical formulas to calculate the answers. The trigonometry identities which are commonly used in Mathematical proof are also used to calculate long distance.

In this section, we will learn the technique to solve complicated problems such as the one given above. The trigonometry functions of sum and difference of angles formulas which we will apply will simplify many trigonometric expressions and equations.

The following equations in trigonometry will be used in this article to establish the relation between sum and difference of angles in trigonometry functions

 cos(-x) = cos

Sin (-x) = sin

Trigonometry Functions

Trigonometry function is defined as the function of an angle of triangle i.e relationship between angles and triangles is derived by trigonometric functions. It is also known as circular functions. Some of the basic trigonometry functions are sine, cosine, cosecant, tangent, secant, and cotangent. These basic functions are also known as trigonometry ratios.

There are multiple trigonometry formulas and identities which represent the relation between the functions and enable them to find the unknown angle of a triangle.

The relation between sum and difference of angles in trigonometry functions

The sum and differences of angles in trigonometry functions are used to find out the functional values of any angles. However, the most practical use of this is to find out the exact values of an angle that can be written as sum or difference using the most familiar values of sine, cosine, and tangent of the 30°, 45°,60°,90°,180°, 270°, and 360°.

Trigonometry Functions of Sum and Difference of Angles Formulas

Trigonometric Functions of Sum and Differences of Angles

Examine the following figure:

A circle is formed with the center of a circle as the origin and radius 1 unit. A point P₁ is selected at an angle of x units from an x-axis. The coordinates of the circle are mentioned in the above figure. Another point P₂ is selected at an angle of y units form the line segment OP₁. P₃ is another point on the circle which lies at an angle of y units from the x-axis measured in the clockwise direction.

Now in the above-given figure, ▴ OP₁P₃ ≅ OP₂OP₄ through SAS congruence criteria.

 As we know the coordinates of all the 4 points given in the above figure, Through distance formula we can write:

[Cos x- cos (-y)] 2 + [sin x- sin (-y)] 2 =  [1- cos(x + y] 2 + sin2 (x + y)

After solving the above equation, we got the following identity.

Cos (x+y) = cosx cosy -sinx siny…. (1)

Substituting the y by –y in identity 1, we get,

Cos (x-y) = cosx cosy +sinx siny…….(2)

Also,

Cos (π/2 – x) = sin x……………(3)

Substituting x by π/2 and y by x in identity 2 we get,

Sin (π/2 – x) = cos x………..(4)

As sin (Cos (π/2 – x) = sin x

As, sin (π/2 – x) = cos (π/2 – (π/2-x)] (by using identity 3). We get,

Sin (π/2-x) = cos x

Now, we are aware of the expanded form of sum and difference of angle of cos. Now, we will use the above concept for finding the values of sum and difference of angle of sin.

Sin(x +y) can be written as cos [π/2 –(x + y)] which is equivalent to cos [(π/2- x)-y].

Now, with the help of identity (2), we can write

Cos [180/2 –x)-y)] = cos (π/2- x) cosy + sin [π/2 –x) siny

= sin x cos y + cos x sin y

Hence,

Sin (x+y) = sinx cosy + cosx siny………..(5)

Now if we substitute y by – y in the above formula, we get

Sin (x- y) = sinx cosy - cosx siny………..(6)

Now if we will substitute suitable values in the above identities (1), (2), (5) and (6), we will have the following equation:

• Cos (π/2 + x) = - sinx

• Sin (π/2 + x)= cos x

• Cos (π ±) x) = -cos x

• Sin (π - x) = sinx

• Sin (π + x) = - sinx

• Sin (2π - x) = - sinx

• Cos (2π - x)= cos x

After understanding about the expanded form the trigonometric functions of  sum and difference of angles of sin and cos, the expansion of tan and cot is derived by,

• Tan(α  + A) = (tan α  + tan A)/ (1-tan α  tan A)

• Tan(α - A) = (tan α  - tan A)/ (1+ tan α  tan A)

Similarly, we get the following

• Cot (α + A) = (cot α cot A -1 )/ (cot A + cot α)

• Cot α - A) = (cot α cot A+ 1 )/ (cot A - cot α)

 Solved Examples

1.   Prove cos (30 + Ѳ) = \[\sqrt{3}\] / 2 cos Ѳ - sin Ѳ /2

Using the formula,

Cos (Ѳ + A) = cos Ѳ cos A - sin Ѳ sin A , and with the help of above 30° - 60° angle, 

We will first solve the left hand side (LHS) of the equation,

LHS =  Cos(30° + Ѳ)

= Cos 30° cos Ѳ - sin 30° sin Ѳ

= \[\sqrt{3}\] /2 cos Ѳ - ½ sin Ѳ

LHS = RHS

Hence proved

2.  Show that cos(π /2 + Ѳ) = -sin Ѳ

 Solution: As we know, 

Cos ( Ѳ + A) = cos  Ѳ cos A - sin  Ѳ sin A

cos(π /2 + Ѳ) = cos π/2 cos Ѳ- sin π/2 sin Ѳ

= 0 * cos Ѳ -1 * sin Ѳ

= - sin Ѳ

LHS = RHS

Hence, proved

 Facts 

• Hipparchus compiled the first trigonometry table.

• The establishment of modern trigonometry was imposed by  “Aryabhatiya” and  Al- Biruni.

• Every trigonometry function of any angle can be constructed through a circle centered at 0 with a radius of 1.

 Quiz Time

1. Find the exact value of the expression (sin 5π/12) cos(π/4)- cos(5π/12)sin(π/4)

a.                   ½

b.                  -½

c.                   -(2)/2

d.                  3/2

 2.  Find the value of cos (75°) using sum or difference identities

a.                   ¼

b.                  6/4 -2/4

c.                   6 + 2/4

d.                  -6/4 -2/4

 3. The value of sin θ and cos (90°- θ)

a.                   Are same

b.                  Are different

c.                   No relation

d.                  Not adequate Information