Step by Step
1
The sine sum and difference formulas
sin(A+B) = sin(A)cos(B) + cos(A)sin(B). sin(A−B) = sin(A)cos(B) − cos(A)sin(B). Notice the sign in the formula always matches the sign in the original expression (A+B uses +, A−B uses −).
2
The cosine sum and difference formulas — the sign flips!
cos(A+B) = cos(A)cos(B) − sin(A)sin(B). cos(A−B) = cos(A)cos(B) + sin(A)sin(B). Unlike sine, the sign in the cosine formula is the OPPOSITE of the sign in the original expression.
3
Using these to find exact values
These formulas let you find exact trig values for angles that aren't standard, by breaking them into a sum or difference of two known angles — for example, 75° = 45° + 30°, both of which have known exact values.
4
Connection to double angle formulas
The double angle formulas covered earlier are simply special cases of these formulas where A = B = θ — a good way to check your memory of both sets of formulas against each other.
Applied Walkthrough
1
Find the exact value of sin(75°) by rewriting it as sin(45° + 30°).
2
Apply the sine sum formula: sin(45°+30°) = sin(45°)cos(30°) + cos(45°)sin(30°).
3
Substitute known exact values: sin(45°)=√2/2, cos(30°)=√3/2, cos(45°)=√2/2, sin(30°)=1/2. This gives: (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4.
4
Combine into a single fraction: sin(75°) = (√6 + √2)/4 — the exact value, obtained without a calculator.
Exam Application
Exams test whether you remember that the cosine sum/difference formula flips the sign compared to sine's formula (which keeps the same sign), and whether you can use these formulas to find exact values for non-standard angles by breaking them into sums of known angles.
⚠ Common Trap
The most common trap is using the same sign pattern for both sine and cosine formulas — remember sine keeps the same sign as the original expression, but cosine always flips it.
✓ Quick Self-Check
1. What is the formula for sin(A+B)?
sin(A)cos(B) + cos(A)sin(B).
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2. What is the formula for cos(A+B)?
cos(A)cos(B) − sin(A)sin(B).
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3. How does the sign behavior differ between the sine and cosine sum/difference formulas?
Sine keeps the same sign as in the original expression; cosine always flips the sign.
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4. Find the exact value of sin(75°) using the angle sum formula.
(√6 + √2)/4.
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5. What are the double angle formulas special cases of?
The angle sum formulas, with A = B = θ.
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