applications-of-de-moivres-theorem

Multiple angle formulae

de Moivre’s theorem can be applied to prove identities such as . This allows expressions involving multiple angles to be written as a polynomial function of a single trig function which makes it easier to solve equations involving different angles.

How do I write sinkθ or coskθ in terms of powers of sinθ or cosθ?

• STEP 1 Use de Moivre’s theorem to write

• STEP 2 Use the binomial expansion to expand <img alt=”open parentheses cos theta plus isin theta close parentheses to the power of k equals sum from r equals 0 to k of open parentheses table row k row r end table close parentheses straight i to the power of left parenthesis k minus r right parenthesis end exponent cos to the power of r theta sin to the power of left parenthesis k minus r right parenthesis end exponent theta” data-mathml='<math ><semantics><mrow><msup><mfenced><mrow><mi>cos</mi><mi>θ</mi><mo>+</mo><mi>isin</mi><mi>θ</mi></mrow></mfenced><mi>k</mi></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><mi>k</mi></munderover><mfenced><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable></mfenced><msup><mi mathvariant=”normal”>i</mi><mrow><mo>(</mo><mi>k</mi><mo>-</mo><mi>r</mi><mo>)</mo></mrow></msup><msup><mi>cos</mi><mi>r</mi></msup><mi>θ</mi><msup><mi>sin</mi><mrow><mo>(</mo><mi>k</mi><mo>-</mo><mi>r</mi><mo>)</mo></mrow></msup><mi>θ</mi></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”56″ role=”math” 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