How to Apply Sum and Difference Identities in Trigonometry Problems
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.
sine
cosine
tangent
cotangent
secant
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
Relation Among sin and cos
Negative Angles Identities
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.
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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FAQs on Sum and Difference of Angles in Trigonometry: Key Formulas & Examples
1. What are the fundamental sum and difference formulas in trigonometry for sine and cosine?
The fundamental sum and difference formulas, also known as trigonometric identities, allow us to 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 primary formulas 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 are essential for solving various problems as per the CBSE Class 11 syllabus for 2025-26.
2. How can you use the sum of angles formula to find the exact value of sin(75°)?
To find the exact value of sin(75°), we can express 75° as the sum of two standard angles for which we know the trigonometric values, such as 45° and 30°. The process is as follows:
Recognise that 75° = 45° + 30°.
Apply the sum formula for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Substitute A = 45° and B = 30°: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).
Substitute the known values: sin(75°) = (1/√2)(√3/2) + (1/√2)(1/2).
Simplify the expression: sin(75°) = (√3 + 1) / 2√2.
3. What are the sum and difference formulas for the tangent function, and how are they derived?
The sum and difference formulas for the tangent function are derived from the sine and cosine formulas, using the identity tan(θ) = sin(θ)/cos(θ). The 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))
To derive tan(A + B), you start with sin(A + B) / cos(A + B) and divide both the numerator and the denominator by cos(A)cos(B).
4. Why are these sum and difference formulas important if we can use a calculator to find trigonometric values?
While calculators provide numerical approximations, the sum and difference formulas are crucial for conceptual understanding and analytical problem-solving. Their importance lies in:
Proving Identities: They are fundamental tools for proving more complex trigonometric identities.
Solving Equations: They help in solving trigonometric equations that involve variables within the angles.
Deriving Other Formulas: They are the basis for deriving other essential identities, such as the double-angle and half-angle formulas.
Calculus: These identities are indispensable in calculus for simplifying expressions before differentiation or integration.
5. What is a common mistake students make when applying the cosine sum and difference formulas?
A very common mistake is confusing the signs in the expansion. Students often forget that the sign in the cosine formula is inverted. For example:
For cos(A + B), the expansion uses a minus sign: cos(A)cos(B) - sin(A)sin(B).
For cos(A - B), the expansion uses a plus sign: cos(A)cos(B) + sin(A)sin(B).
Incorrectly matching the sign (e.g., using a plus for cos(A+B)) is a frequent error in exams.
6. How can you use the sum and difference formulas to simplify an expression like cos(80°)cos(20°) + sin(80°)sin(20°)?
This expression can be simplified by recognising its pattern. It matches the right-hand side of the difference formula for cosine: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Identify A = 80° and B = 20°.
Apply the formula in reverse: cos(80°)cos(20°) + sin(80°)sin(20°) = cos(80° - 20°).
Simplify the angle: cos(60°).
State the exact value: The expression simplifies to 1/2.
7. How do the sum and difference of angles formulas relate to the double angle formulas?
The double angle formulas are a direct application and special case of the sum of angles formulas. They are derived by setting the two angles to be equal (i.e., A = B). For example:
To find sin(2A), we use sin(A + B) and set B = A: sin(A + A) = sin(A)cos(A) + cos(A)sin(A) = 2sin(A)cos(A).
Similarly, for cos(2A), we use cos(A + B): cos(A + A) = cos(A)cos(A) - sin(A)sin(A) = cos²(A) - sin²(A).
This shows that the sum formulas are a more general set of identities from which other important results are built.
