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.
More to read
9.3 Double-Angle, Half-Angle, and Reduction Formulas
Double-angle rules come from the addition rules with B = A: sin(A + A) = sin A cos A + cos A sin A = 2 sin A cos A.
Cosine offers three versions: cos squared − sin squared, 2cos squared − 1, and 1 − 2sin squared. Pick the version whose functions match your equation. With sin A = 0.6 and cos A = 0.8, sin 2A = 0.96.
7.3 Double-Angle, Half-Angle, and Reduction Formulas
Choose the cos 2A form that clears the unwanted function. An equation in sin squared calls for cos 2A = 1 − 2sin squared, since sin squared = (1 − cos 2x)/2 swaps it out. The (1 + cos) version belongs to cos squared instead.
Mixing the versions is the classic slip, so write which squares each form carries. Match form to target before substituting anything.
9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
Equations mixing sin 2x with sin x factorise after expansion: sin 2x = sin x becomes 2 sin x cos x − sin x = 0, so sin x (2cos x − 1) = 0. Each factor gives angles: 0°, 60°, 180°, 300° in a full turn.
Never divide by sin x to simplify, since that deletes the 0° and 180° solutions. Factorise, set each factor to zero, and gather the full set.
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2 · Watch
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Where it sits
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Optional connections
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Where this leads
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17 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.