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The double angle

Trigonometry, grade 11 (5 units) Lesson 5 of 18Learning

The double angle

The double-angle formulas

Put β=α\displaystyle {\beta}={\alpha} in the sum formulas:

  • sin⁡(α+α)=sin⁡αcos⁡α+cos⁡αsin⁡α\displaystyle {\sin(\alpha+\alpha)}={\sin\alpha\cos\alpha+\cos\alpha\sin\alpha}:
sin⁡2α\displaystyle \sin2\alpha =\displaystyle = 2sin⁡α⋅cos⁡α\displaystyle 2\sin\alpha\cdot\cos\alpha
  • cos⁡(α+α)=cos⁡αcos⁡α−sin⁡αsin⁡α\displaystyle {\cos(\alpha+\alpha)}={\cos\alpha\cos\alpha-\sin\alpha\sin\alpha}. With sin⁡2α=1−cos⁡2α\displaystyle {\sin^2\alpha}={1-\cos^2\alpha} or cos⁡2α=1−sin⁡2α\displaystyle {\cos^2\alpha}={1-\sin^2\alpha} two more forms follow:
cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha
  • tan⁡(α+α)\displaystyle \tan(\alpha+\alpha):
tan⁡2α\displaystyle \tan2\alpha =\displaystyle = 2tan⁡α1−tan⁡2α\displaystyle \frac{2\tan\alpha}{1-\tan^2\alpha}

Choose the form of cos⁡2α\displaystyle \cos2\alpha by what is known, or by what should be left: only cos⁡α\displaystyle \cos\alpha, only sin⁡α\displaystyle \sin\alpha, or both.

From sin⁡α\displaystyle \sin\alpha to sin⁡2α\displaystyle \sin2\alpha

sin⁡α=35\displaystyle {\sin\alpha}={\frac{3}{5}}, and the angle α\displaystyle \alpha is in the second quadrant.

  • cos⁡α=−45\displaystyle {\cos\alpha}={-\frac{4}{5}}.
  • sin⁡2α=2⋅35⋅(−45)=−2425\displaystyle {\sin2\alpha}={2\cdot\frac{3}{5}\cdot\left(-\frac{4}{5}\right)}={-\frac{24}{25}}.
  • cos⁡2α=1−2sin⁡2α=1−2⋅925=725\displaystyle {\cos2\alpha}={1-2\sin^2\alpha}={1-2\cdot\frac{9}{25}}={\frac{7}{25}} – here cos⁡α\displaystyle \cos\alpha is not needed.
Lowering a power

Isolate cos⁡2α\displaystyle \cos^2\alpha and sin⁡2α\displaystyle \sin^2\alpha from cos⁡2α=2cos⁡2α−1=1−2sin⁡2α\displaystyle {\cos2\alpha}={2\cos^2\alpha-1}={1-2\sin^2\alpha}:

cos⁡2α\displaystyle \cos^2\alpha =\displaystyle = 1+cos⁡2α2\displaystyle \frac{1+\cos2\alpha}{2}
sin⁡2α\displaystyle \sin^2\alpha =\displaystyle = 1−cos⁡2α2\displaystyle \frac{1-\cos2\alpha}{2}

A square of a sine or a cosine turns into an expression with no square, with the double angle. This also gives values of a half angle:

cos⁡2π8=1+cos⁡π42=2+24\displaystyle {\cos^2\frac{\pi}{8}}={\frac{1+\cos\frac{\pi}{4}}{2}}={\frac{2+\sqrt{2}}{4}}, so cos⁡π8=2+22\displaystyle {\cos\frac{\pi}{8}}={\frac{\sqrt{2+\sqrt{2}}}{2}} (the angle is acute, and the cosine positive).

Exercise 1

Work it out without a calculator.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Exercise 2

    Work it out without a calculator.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 3

      sin⁡α=13\displaystyle {\sin\alpha}={\frac{1}{3}}, and the angle α\displaystyle \alpha is acute: α=sin⁡−113\displaystyle {\alpha}={\sin^{-1}\frac{1}{3}}.
      Work out cos⁡2α\displaystyle \cos2\alpha.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Exercise 4

        tan⁡α=3\displaystyle {\tan\alpha}={3}, and the angle α\displaystyle \alpha is acute.
        Work out tan⁡2α\displaystyle \tan2\alpha.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

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          Exercise 5

          Simplify until you reach tan⁡x\displaystyle \tan x.

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            Exercise 6

            Simplify until you reach cos⁡2x\displaystyle \cos2x. A hint: a difference of two squares.

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              Exercise 7

              Write sin⁡3x\displaystyle \sin3x with sin⁡x\displaystyle \sin x alone, until you reach 3sin⁡x−4sin⁡3x\displaystyle 3\sin x-4\sin^3x. A hint: 3x=2x+x\displaystyle {3x}={2x+x}.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power