Sum and Difference Identities
Formulas for sin or cos of A ± B.
About this topic
The sum and difference identities express trig functions of A ± B in terms of A and B: sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B. Note sin(A + B) is not sin A + sin B.
They give exact values for angles like 75° (45° + 30°) or 15° (45° − 30°) that are not on the standard chart. For differences, sin(A − B) = sin A cos B − cos A sin B and cos(A − B) = cos A cos B + sin A sin B; notice the sign flips in the cosine formula. For tangent, tan(A + B) = (tan A + tan B)/(1 − tan A tan B).
Example: sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4, about 0.966.
These formulas explain how two waves combine, which is vital in sound, radio and optics, and they underlie rotation matrices in graphics. Memory tip: sine formulas keep the sign, mixing sin and cos; cosine formulas flip the sign, pairing like with like. Never distribute sine over a sum as if it were multiplication.
Key ideas
- sin(A + B) = sin A cos B + cos A sin B
- cos(A + B) = cos A cos B − sin A sin B
- Derive exact values like sin 75°
Quick check
Use cases: where Sum and Difference Identities helps
- Homework: review the key idea behind Sum and Difference Identities before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Sum and Difference Identities without notes.
- Visual learning: play with the interactive lab to see how Sum and Difference Identities changes when you adjust its inputs.
- Teaching a friend: explain Sum and Difference Identities in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Sum and Difference Identities?
Formulas for sin or cos of A ± B.
What will I learn in Sum and Difference Identities?
sin(A + B) = sin A cos B + cos A sin B. cos(A + B) = cos A cos B − sin A sin B. Derive exact values like sin 75°.
Is Sum and Difference Identities part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Sum and Difference Identities lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Sum and Difference Identities?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.