

How to Apply Sum and Difference Formulas in Trigonometry
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
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
FAQs on Trigonometry Functions of Sum and Difference of Angles
1. What are the primary trigonometric formulas for the sum and difference of angles for sine and cosine?
The primary formulas, which are fundamental in trigonometry, express the trigonometric function of a sum or difference of two angles (A and B) in terms of the functions of the individual angles. The core identities for sine and cosine are:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(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)
These formulas are essential for deriving other identities and solving trigonometric equations as per the CBSE Class 11 syllabus.
2. How are the sum and difference formulas for the tangent function expressed?
The sum and difference identities for the tangent function are derived from the sine and cosine formulas by using the ratio identity tan(x) = sin(x)/cos(x). The resulting formulas are:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
These are crucial for problems involving the tangent of combined or subtracted angles.
3. How can you find the exact value of cos(15°) using a difference formula?
To find the exact value of cos(15°), you can express 15° as a difference of two standard angles whose trigonometric values are known, such as 45° and 30° or 60° and 45°. Using the formula for cos(A - B):
Let A = 45° and B = 30°.
The formula is cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Substitute the values: cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°).
cos(15°) = (1/√2) * (√3/2) + (1/√2) * (1/2).
Simplifying this gives the exact value: (√3 + 1) / 2√2.
4. Is sin(A + B) equal to sin(A) + sin(B)? Explain why or why not.
No, sin(A + B) is not equal to sin(A) + sin(B). This is a common misconception. Trigonometric functions do not distribute over addition or subtraction of angles. We can prove this with a simple counterexample:
Let A = 30° and B = 60°.
Then, sin(A + B) = sin(30° + 60°) = sin(90°) = 1.
However, sin(A) + sin(B) = sin(30°) + sin(60°) = (1/2) + (√3/2) = (1 + √3)/2.
Since 1 ≠ (1 + √3)/2, we can conclude that sin(A + B) ≠ sin(A) + sin(B).
5. Why are the sum and difference of angle formulas so important in trigonometry?
The sum and difference formulas are critically important for several reasons:
Calculating Exact Values: They allow us to calculate exact trigonometric values for angles that are not standard (e.g., 15°, 75°, 105°) by expressing them as sums or differences of standard angles (0°, 30°, 45°, 60°, 90°).
Deriving Other Identities: They are the foundation for deriving many other key identities, including the double-angle, half-angle, and product-to-sum formulas.
Simplifying Expressions: They are used to simplify complex trigonometric expressions into a more manageable form, which is essential for solving equations in calculus and physics.
Applications in Science: These formulas are applied in physics to analyse phenomena like wave interference and in engineering for signal processing and mechanics.
6. How are sum and difference formulas used to prove other trigonometric identities, like the co-function identity cos(π/2 - x) = sin(x)?
The sum and difference formulas provide a direct method for proving other identities. To prove that cos(π/2 - x) = sin(x), we use the cosine difference formula, cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Here, let A = π/2 and B = x.
Substitute these into the formula: cos(π/2 - x) = cos(π/2)cos(x) + sin(π/2)sin(x).
We know the standard values: cos(π/2) = 0 and sin(π/2) = 1.
The expression becomes: (0) * cos(x) + (1) * sin(x).
This simplifies to sin(x), thus proving the co-function identity.
7. What is the geometric reasoning behind the derivation of the cos(A - B) formula?
The formula for cos(A - B) can be derived geometrically using the unit circle. The key steps are:
Consider two points P and Q on the unit circle corresponding to angles A and B. Their coordinates will be P(cos A, sin A) and Q(cos B, sin B).
The angle between the position vectors OP and OQ is A - B.
Calculate the square of the distance between P and Q using the distance formula: (PQ)² = (cos A - cos B)² + (sin A - sin B)².
Alternatively, calculate the same distance using the Law of Cosines on triangle POQ: (PQ)² = OP² + OQ² - 2(OP)(OQ)cos(A - B). Since OP and OQ are radii of the unit circle, their length is 1.
By equating the two expressions for (PQ)² and simplifying, we arrive at the identity: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).

















