Sum and Difference of Angles in Trigonometry

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What are Trigonometry Functions? 

Trigonometry functions define the relationships among angles and sides of a right-angled triangle. The applications of such functions are wide-ranged and may be seen within the solutions of functional equations and differential equations. For instance, the sum and difference of trigonometric identities can be represented in any periodic process. 

There are 6 trigonometric functions and they are as follows.

  1. sine

  2. cosine

  3. tangent

  4. cotangent

  5. secant

  6. cosecant

All the functions mentioned above also have corresponding inverse trigonometric functions. 


Different Trigonometric Identities 

Before proceeding with the sum and difference of trigonometric identities, let us go through some of the important identities.


Relations Between tan, cot, sec and cosec with Respect to sin and cos

tan \[\theta\] = \[\frac{sin \theta}{cos \theta}\]         cot \[\theta\] = \[\frac{1}{tan \theta}\] = \[\frac{cos \theta}{sin \theta}\]  

sec \[\theta\] = \[\frac{1}{cos \theta}\]                        csc \[\theta\] = \[\frac{1}{sin \theta}\]


Relation Among sin and cos 

sin\[^{2}\]\[\theta\] + cos\[^{2}\]\[\theta\] = 1


Negative Angles Identities 

sin(-θ) = - sin θ

cos(-θ) = cos θ 

tan(-θ) = - tan θ


It can be seen from the identities that sin, tan, cot, and cosec amount to odd functions. On the other hand, sec and cos amount to even functions. 


Sum Difference Angles Trigonometry – What are the Angle Identities?

The angle difference identities and sum identities are used to determine the function values of any of the angles concerned. To that effect, finding an accurate value of an angle may be represented as difference or sum by using the precise values of cosine, sine, and tan of angles 30°, 45°, 60°, 90°, 180°, 270°, and 360° as well as their multiples and sub-multiples. 


The following table shows the sum and difference of trigonometric identities.


Sum of Angles Identities

Difference of Angles Identities

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

tan(A+B) = \[\frac{tanA+tanB}{1-tanA.tanB}\]

tan(A-B) = \[\frac{tanA-tanB}{1+tanA.tanB}\] 


Converting Product to Sum and Difference of Trigonometric Identities

For deriving the relationship between sum and difference with that of the product of trigonometric identities compound angles have to be utilized. Below are some of the important relations.

sin (A + B) = sin A cos B + cos A sin B …………………………………  (1)

sin (A - B) = sin A cos B – cos A sin B ………………………………….  (2)

cos (A + B) = cos A cos B + sin A sin B …………………………………  (3)

cos (A - B) = cos A cos B – sin A sin B ………………………………….  (4)

Therefore, for the calculation of the product formula, it may be derived -

  • 2sin A cos B = sin (A + B) + sin (A – B)

  • 2sin A sin B = cos (A - B) - cos (A + B)

  • 2cos A sin B = sin (A + B) - sin (A + B)

  • 2cosA cos B = cos (A + B) + cos (A – B)

In deriving the formulas of the products, the conversion to sum and difference of trigonometric identities can also be done. 


Few Solved Examples 

1. Value of sin 15° with Help of Difference Formula 

First step: sin (A - B) = (sin A X cos B) – (cos A X sin B)

Second step: sin (45 - 30) = (sin 45 X cos 30) – (cos 45 X sin 30)

By substituting the respective values, sin 15° comes to: \[\frac{\sqrt{6}-\sqrt{2}}{4}\]


2. Value of cos 75° with Help of Sum Formula 

First step: cos (A + B) = (cos A X cos B) – (sin A X sin B)

Second step: cos (30 + 45) = (cos 30 X cos 45) – (sin 30 X sin 45)

By substituting the respective values, cos 75° comes to: \[\frac{\sqrt{6}-\sqrt{2}}{4}\]

The following points should be noted while solving these sums –

  • There may be more than one option. 

  • Identify the exact quadrant in which the angle will be located. 

For further elaboration and clarification on the topic, you may avail of Vedantu’s online classes or download the free PDFs on sums from trigonometric identity from Vedantu.

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FAQ (Frequently Asked Questions)

1. How are the Sum and Difference of Trigonometric Identities Used?

Ans. The trigonometry sum and difference formulas are utilized for the determination of the function values. It would include any of the angles. Moreover, it is particularly suited for finding the accurate values of an angle. Such an angle may be noted as a difference or sum of values of cosine, sine, and tangent of the angles of 30°, 45°, 60°, and 90°.

2. What is the Process of Using Trig Sum Identities?

Ans. Here is an example of proving trigonometric identities using sum and difference formulas for finding sine of 75°. The steps would include –

(1) Determination of two such angles in which the resulting sum is 75. For instance, 30 and 45.

(2) The angle measurements have to be put into identity – sin (30° + 40°) = sin30°cos45° + cos30°sin45°.

(3) Substitution of angle functions with corresponding values. 

3. What are the Sum and Difference Trig Identities?

Ans. The sum and difference trig identities help us to calculate the values of trigonometric functions for any given angle measure easily. The sum and difference formula of trigonometry can be applied within inverse trigonometric functions. For instance, the formula of sum for tangent indicates that the summation of two angles would be equal to tangents of angles when the latter is divided by one, subtracted from the product of angles’ tangents.