Sum and Difference Formulas

Key idea

Trig of a sum in terms of single angles.

In depth

sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B. They are proved with the unit circle or rotation. Note that functions do not distribute: sin(A + B) ≠ sin A + sin B.

Worked example

sin 75° = sin(45° + 30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Formula

Real-world uses

• Calculating exact values such as sin 75° in exams
• Combining two waves in physics

Common mistake

Getting the sign wrong in cosine formulas, where it flips.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
• Pythagorean Identity – The key identity from the unit circle.