Double Angle Formulae
Set B = A in the compound angle formulae to get sin 2A, cos 2A and tan 2A, and pick between the three forms of cos 2A to remove either a squared term or a doubled angle.
What a learner can do afterwards
- Derive sin 2A by writing it as sin(A + A)
- Choose the form of cos 2A that clears a sin²x from an equation
- Solve an equation containing both sin 2x and sin x
1 · Read
Set B to A in the addition rules. Sin of A plus A is sin A cos A plus cos A sin A, which is 2 sin A cos A. One substitution gives the famous formula. Every double angle rule is born this way.
Cosine offers three versions: cos squared minus sin squared, 2 cos squared minus 1, and 1 minus 2 sin squared. Pick the version whose squares match your equation. An equation in sin squared calls for 1 minus 2 sin squared.
Solve sin 2x equals sin x. Expand the left to 2 sin x cos x, then bring all across: sin x times 2 cos x minus 1 equals 0. Each factor gives angles: 0, 60, 180, and 300 degrees in a full turn.
Never divide by sin x to simplify, since that deletes the 0 and 180 degree solutions. Factorise and set each factor to zero. With sin A 0.6 and cos A 0.8, sin 2A is 0.96.
Derive from the sum rule, match the cosine form to the equation, and factorise instead of dividing.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.