Sum and Difference Angles Formulas for Sin Cos Tan with Proof and Examples
Sum and Difference of Angles Identities
Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most feasible use of sum of angles trig identities is to identify the exact values of an angle that can be mathematically expressed as a sum or difference using the familiar values for the sine, cosine and tangent of the 30°, 45°, 60° and 90° angles as well their multiples.
The sum and difference formulas for sine and cosine are also helpful in dealing with the inverse trigonometric functions.
Sum and Difference Angle Formulas
The sum formula for cosines implies that the cosine of the sum of two angles is equivalent to the product of the cosines of the angles subtracting the product of the sines of the angles. On the other hand, the difference formula for cosines implies that the cosine of the difference of two angles is equivalent to the product of the cosines of the angles adding up the product of the sines of the angles.
Sum and Difference Angle Formulas
Sum Formula for Tangent
The sum formula for tangent trigonometry implies that the tangent of the sum of two angles is equivalent to the sum of the tangents of the angles further divided by 1 minus (-) the product of the tangents of the angles.
Difference Formula for Tangent
The Difference Formula for tangent implies that the tangent of the difference of two angles is equivalent to the difference of the tangents of the angles divided by 1 plus (+) the product of the tangents of the angles.
Use of Pythagorean Theorem With the Sum and Difference Formulas
The Pythagorean Theorem when used along with the sum and difference formulas helps to find multiple sums and differences of angles.
Solved Examples
Example 1:
Identify the exact value of sine 15° applying the difference formula.
Solution:
Seeing that, 15° is the Value of Difference between 45° and 30°. Thus, the Sine Difference formula can be applied
Now, using the sin difference formula i.e. sin(A - B) = sin A · cos B - cos A · sin B. We get,
Sin [45° - 30°] = sin 45° · cos 30° - cos 45° · sin 30°
= √2/2. √3/2 -√2/2. √1/2
= √6/4 - √2/4 =√6-√2/4
Hence, sin 15° =√6-√2/4
Example 2:
Identify the Exact value of cos 75° applying the sum formula.
Solution:
Seeing that, 75° is the sum of 45° and 30°. Thus, the cos sum formula can be applied
Now, using the cos sum formula i.e. cos (A + B) = cos A · cos B - sin A · sin B. We get,
cos (30° + 45°) = cos 30° · cos 30° · cos 45° - sin 30°· sin 45°
= √3/2. √2/2 -√1/2. √2/2
= √6/4 - √2/4 =√6-√2/4
Hence cos 75° =√6-√2/4
Key Takeaways
1. You can have more than one option. For Example If you want to Find the sine of 120°
Sin (90 + 30) applying the Sum Formula
or
sin (180 - 60) applying the Difference Formula
Both Formulas Will Produce Similar Results.
2. In order to identify a secant, cosecant, or cotangent function, you would need to change the expression to one of the 3 basic functions, and perform the requisite calculations.
3. Remember to identify in which quadrant the angle will lie.
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 express functions like sin, cos, and tan of (A ± B) in terms of A and B separately. The key identities are:
- sin(A ± B) = sinA cosB ± cosA 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 that only the sign in the middle changes:
- For addition, use +
- For subtraction, use −
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. Unlike sine, the sign changes opposite to the bracket sign:
- For addition, use −
- For subtraction, use +
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). These are derived from sine and cosine sum formulas and are valid where the denominator is not zero. They are useful in solving trigonometric equations and evaluating special angles.
5. How do you find the exact value of sin 75° using sum formulas?
The exact value of sin 75° is found using sin(45° + 30°). Apply the sine addition formula:
- sin(45° + 30°) = sin45° cos30° + cos45° sin30°
- = (√2/2)(√3/2) + (√2/2)(1/2)
- = (√6/4) + (√2/4)
6. How do you derive double angle formulas from sum formulas?
The double angle formulas are obtained by setting 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. Why does the cosine formula change signs for addition and subtraction?
The cosine sum and difference formulas change signs because cosine is an even function and its derivation from the unit circle geometry leads to opposite middle signs. Specifically:
- cos(A + B) = cosA cosB − sinA sinB
- cos(A − B) = cosA cosB + sinA sinB
8. What is the difference between sine and cosine sum formulas?
The main difference is the sign pattern used in their formulas. For sine:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
9. How are sum and difference formulas used to solve trigonometric equations?
Sum and difference formulas are used to rewrite complex expressions into simpler forms that can be solved easily. For example:
- Given sin(x + 30°) = 1/2
- Expand: sinx cos30° + cosx sin30°
- Solve the resulting equation using known values
10. What are common mistakes when using sum and difference angle formulas?
Common mistakes include using the wrong sign and confusing sine and cosine patterns. Key errors to avoid:
- Mixing up ± and ∓ signs
- Forgetting that cosine uses the opposite sign rule
- Ignoring denominator restrictions in tangent formulas
- Incorrect substitution of special angle values
