The Six Sum and Difference Formulas

These formulas allow us to express trigonometric functions of A + B or A - B in terms of trigonometric functions of A and B separately.

Sine Formulas

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

Cosine Formulas

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

Tangent Formulas

tan(A + B) = tanA + tanB/1 - tanA tanB

tan(A - B) = tanA - tanB/1 + tanA tanB

Finding Exact Values

These formulas are powerful for finding exact values of trigonometric functions at non-standard angles by decomposing them into standard angles (30°, 45°, 60°, 90°, etc.):

Common Decompositions

15°

= 45° - 30°

75°

= 45° + 30°

105°

= 60° + 45°

π/12

= π/3 - π/4

5π/12

= π/4 + π/6

7π/12

= π/3 + π/4

Applications

Sum and difference formulas are used to:

Find exact values of trig functions at non-standard angles
Simplify expressions by rewriting as a single trig function
Prove identities by expanding compound expressions
Derive double angle formulas by setting A = B
Solve equations that involve sums or differences of angles

Connection: The double angle formulas (sin 2A, cos 2A, tan 2A) are special cases of these formulas where B = A. We cover those in a separate lesson.

Key Vocabulary

Sum Formula

A formula that expresses trig(A + B) in terms of trig functions of A and B separately.

Difference Formula

A formula that expresses trig(A - B) in terms of trig functions of A and B separately.

Exact Value

A value expressed using surds and fractions rather than decimal approximations (e.g., √3/2 not 0.866).

Standard Angles

Angles whose trig values are known exactly: 0°, 30°, 45°, 60°, 90° (and their radian equivalents).

Worked Examples

1

Find the exact value of tan(75°).

Step 1: 75° = 45° + 30°

Step 2: tan(75°) = (tan45° + tan30°) / (1 - tan45° tan30°)

Step 3: = (1 + 1/√3) / (1 - 1 × 1/√3) = (√3 + 1)/(√3 - 1)

Step 4: Rationalise: multiply by (√3 + 1)/(√3 + 1) = (4 + 2√3)/2

Answer: tan(75°) = 2 + √3

2

Find the exact value of sin(π/12).

Step 1: π/12 = π/3 - π/4

Step 2: sin(π/12) = sin(π/3)cos(π/4) - cos(π/3)sin(π/4)

Step 3: = (√3/2)(√2/2) - (1/2)(√2/2)

Answer: sin(π/12) = (√6 - √2) / 4

3

Given sinA = 3/5 (A in Q1) and cosB = 5/13 (B in Q1), find sin(A + B).

Step 1: Find cosA: cosA = 4/5 (from 3-4-5 triangle). Find sinB: sinB = 12/13 (from 5-12-13 triangle).

Step 2: sin(A + B) = sinA cosB + cosA sinB

Step 3: = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65

Answer: sin(A + B) = 63/65

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is sin(A - B) equal to?

sinA cosB + cosA sinB cosA cosB + sinA sinB cosA cosB - sinA sinB sinA cosB - cosA sinB

Question 2

Which decomposition could be used to find cos(105°)?

cos(60° + 45°) cos(50° + 55°) cos(90° + 10° + 5°) cos(70° + 35°)

Question 3

What is the exact value of cos(15°)?

(√6 - √2) / 4 (√6 + √2) / 4 (√3 + 1) / 2 √2/2

Question 4

In the formula for tan(A + B), what is the denominator?

1 + tanA tanB tanA tanB - 1 1 - tanA tanB tanA + tanB

Question 5

If sinA = 4/5 and cosB = 12/13 (both angles in Q1), what is cos(A + B)?

63/65 56/65 16/65 33/65

Sum and Difference Formulas