Sum and Difference of Angles Formulas Proofs and Solved Examples
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
1. What are the sum and difference formulas in trigonometry?
The sum and difference formulas in trigonometry are identities used to find the exact values of trigonometric functions of combined angles like (A + B) and (A − B).
Key formulas:
- sin(A + B) = sinA cosB + cosA sinB
- sin(A − B) = sinA cosB − cosA sinB
- cos(A + B) = cosA cosB − sinA sinB
- cos(A − B) = cosA cosB + sinA sinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
2. What is the formula for sin(A + B) and sin(A − B)?
The formulas are sin(A + B) = sinA cosB + cosA sinB and sin(A − B) = sinA cosB − cosA sinB.
Notice the pattern:
- For addition, use a plus sign.
- For subtraction, use a minus sign.
3. What is the formula for cos(A + B) and cos(A − B)?
The formulas are cos(A + B) = cosA cosB − sinA sinB and cos(A − B) = cosA cosB + sinA sinB.
Important rule:
- Cosine changes the sign in the middle term.
- For addition → minus.
- For subtraction → plus.
4. What is the formula for tan(A + B) and tan(A − B)?
The tangent formulas are tan(A + B) = (tanA + tanB)/(1 − tanA tanB) and tan(A − B) = (tanA − tanB)/(1 + tanA tanB).
Key points:
- The numerator follows the sign between A and B.
- The denominator uses the opposite sign.
5. How do you find the exact value of sin(75°) using angle sum formulas?
The exact value of sin(75°) is (√6 + √2)/4 using the sum formula.
Steps:
- Write 75° = 45° + 30°.
- Use sin(A + B) = sinA cosB + cosA sinB.
- Substitute known values:
sin45° = √2/2, cos30° = √3/2,
cos45° = √2/2, sin30° = 1/2. - Simplify to get (√6 + √2)/4.
6. How do you derive the double angle formulas from sum formulas?
The double angle formulas are derived by putting B = A in the sum formulas.
For example:
- sin(2A) = sin(A + A) = 2 sinA cosA
- cos(2A) = cos(A + A) = cos²A − sin²A
- tan(2A) = 2 tanA / (1 − tan²A)
7. What is the difference between sine and cosine angle sum formulas?
The key difference is that sine keeps the same sign while cosine changes the sign in its formulas.
Comparison:
- sin(A + B) = sinA cosB + cosA sinB
- cos(A + B) = cosA cosB − sinA sinB
- Sine → sign changes.
- Cosine → sign becomes opposite of addition case.
8. How are sum and difference identities used to solve trigonometric equations?
Sum and difference identities are used to rewrite complex angles into simpler standard angles to solve equations.
Example:
- To solve sin(x + 30°) = 1/2, expand using sin(A + B).
- Rewrite as sinx cos30° + cosx sin30°.
- Simplify and solve using known trigonometric values.
9. Why are sum and difference formulas important in trigonometry?
The sum and difference formulas are important because they help evaluate non-standard angles and derive other trigonometric identities.
They are used for:
- Finding exact values like sin75° or cos15°.
- Deriving double and triple angle formulas.
- Solving trigonometric equations.
- Applications in physics and engineering.
10. What are common mistakes when using angle sum and difference formulas?
Common mistakes include sign errors and confusing sine and cosine formulas.
Frequent errors:
- Forgetting that cosine changes sign in addition and subtraction formulas.
- Using the wrong denominator sign in the tangent formula.
- Not applying brackets correctly in tan(A ± B).
- Substituting incorrect exact trigonometric values.
