What are the compound-angle formulae?
They expand the sine, cosine or tangent of a sum or difference of two angles: sin(A±B)=sinA cosB ± cosA sinB, cos(A±B)=cosA cosB ∓ sinA sinB, and tan(A±B)=(tanA±tanB)/(1∓tanA tanB).
// maths › Trigonometric Identities
Compound-Angle Formulae
Expand and evaluate sin(A±B), cos(A±B) and tan(A±B) with the full compound-angle working shown step by step, in degrees or radians.
sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB
Angle unit
Frequently asked questions
Can you show a worked example?
sin 75° = sin(45°+30°) = sin45 cos30 + cos45 sin30 ≈ 0.9659. In radians the same angle is 75° = 5π/12 ≈ 1.3090 rad; switch the toggle and the working re-derives.
Why does cosine flip the sign?
Because cos(A+B) loses a sin·sin term with a minus, while cos(A−B) gains it with a plus. So cos(A+B) uses − between the terms and cos(A−B) uses +, the opposite of sine.
When is tan(A±B) undefined?
When the denominator 1∓tanA tanB is zero, or when any of A, B or A±B is an odd multiple of 90° (π/2 rad), where the tangent itself does not exist.
Where are they used?
They underpin every later identity and are used to combine waves of the same frequency, in AC circuit analysis, and in rotation formulae for graphics.