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3.4.2: Double Angle Identities

  • Page ID
    4222
  • Simplifying trigonometric functions with twice a given angle.

    These identities are significantly more involved and less intuitive than previous identities.  By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase.  Each identity in this concept is named aptly.  Double angles work on finding \(\sin 80^{\circ} \) if you already know \(\sin 40^{\circ} \).  Half angles allow you to find \(\sin 15^{\circ} \) if you already know \(\sin 30^{\circ} \).  Power reducing identities allow you to find \(\sin ^2 15^{\circ} \) if you know the sine and cosine of \(30^{\circ} \). 

    What is \(\sin ^2 15^{\circ} \)? 

    Double Angle, Half Angle, and Power Reducing Identities

    Double Angle Identities

    The double angle identities are proved by applying the sum and difference identities.  They are left as review problems.  These are the double angle identities.

    • \(\sin 2x=2\sin x\cos x\)
    • \(\cos 2x=\cos ^2x−\sin ^2x\)
    • \(\tan 2x=\dfrac{2\tan x}{1−\tan ^2x}\)

    Half Angle Identities

    The half angle identities are a rewritten version of the power reducing identities.  The proofs are left as review problems.

    • \(\sin \dfrac{x}{2}=\pm \sqrt{\dfrac{1−\cos x}{2}}\)
    • \(\cos \dfrac{x}{2}=\pm \sqrt{\dfrac{1+\cos x}{2}}\)
    • \(\tan \dfrac{x}{2}=\pm \sqrt{\dfrac{1−\cos x}{1+\cos x}}\)

    Power Reducing Identities

    The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers.  The proofs are left as examples and review problems. 

    • \(\sin ^2x=\dfrac{1−\cos 2x}{2}\)
    • \(\cos ^2x=\dfrac{1+\cos 2x}{2}\)
    • \(\tan ^2x=\dfrac{1−\cos 2x}{1+\cos 2x}\)

    Power reducing identities are most useful when you are asked to rewrite expressions such as \sin 4x as an expression without powers greater than one. While \(\sin x\cdot \sin x\cdot \sin x\cdot \sin x\) does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.

    \(\begin{aligned} \sin ^4x=(\sin ^2x)^2 &=\left(\dfrac{1−\cos 2x}{2} \right)^2 \\&=\dfrac{1−2\cos 2x+\cos ^2 2x}{4} \\&= \dfrac{1}{4}\left(1−2\cos 2x+\dfrac{1+\cos 4x}{2}\right)\end{aligned}\)

    Example \(\PageIndex{1}\)

    Earlier, you were asked to find \(\sin ^2 15^{\circ} \). In order to fully identify \(\sin ^2 15^{\circ} \) you need to use the power reducing formula.

    Solution

    \(\begin{aligned} \sin ^2x&=\dfrac{1−\cos 2x}{2}=\dfrac{1}{2}−\dfrac{\sqrt{3}}{4} \\ \sin ^2 15^{\circ} &=\dfrac{1−\cos 30^{\circ}}{2} \\ &=\dfrac{2−\sqrt{3}}{4} \end{aligned}\)

    Example \(\PageIndex{2}\)

    Write the following expression with only \(\sin x\) and \(\cos x \): \(\sin 2x+\cos 3x\).

    Solution

    \(\begin{aligned} \sin 2x+\cos 3x &=2\sin x\cos x+\cos (2x+x) \\&=2\sin x\cos x+\cos 2x\cos x−\sin 2x\sin x \\&=2\sin x\cos x+(\cos ^2x−\sin ^2x)\cos x−(2\sin x\cos x)\sin x \\ &=2\sin x\cos x+\cos ^3x−\sin ^2x\cos x−2\sin ^2x\cos x \\ &=2\sin x\cos x+\cos ^3x−3\sin ^2x\cos x \end{aligned}\)

    Example \(\PageIndex{3}\)

    Use half angles to find an exact value of \tan 22.5^{\circ}  without using a calculator. 

    Solution

    \(\tan \dfrac{x}{2}=\pm \sqrt{\dfrac{1−\cos x}{1+\cos x}}\)

    \(\begin{aligned}
    \tan 22.5^{\circ} &=\tan \dfrac{45^{\circ}}{2}=\pm \sqrt{\dfrac{1-\cos 45^{\circ}}{1+\cos 45^{\circ}}}=\pm \sqrt{\dfrac{1-\dfrac{\sqrt{2}}{2}}{1+\dfrac{\sqrt{2}}{2}}}=\pm \sqrt{\dfrac{\dfrac{2}{2}-\dfrac{\sqrt{2}}{2}}{\dfrac{2}{2}+\dfrac{\sqrt{2}}{2}}}=\pm \sqrt{\dfrac{2-\sqrt{2}}{2+\sqrt{2}}} \\
    &=\pm \sqrt{\dfrac{(2-\sqrt{2})^{2}}{2}}
    \end{aligned}\)

    Example \(\PageIndex{4}\)

    Prove the power reducing identity for sine.

    \(\sin ^2x=\dfrac{1−\cos 2x}{2}\)

    Solution

    Using the double angle identity for cosine:

    \(\begin{aligned} \cos 2x&=\cos ^2x−\sin ^2x \\  \cos 2x&=(1−\sin ^2x)−\sin ^2x \\ \cos 2x &=1−2\sin ^2x \end{aligned}\)

    This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

    Example \(\PageIndex{5}\)

    Simplify the following identity: \(\sin ^4x−\cos ^4x\).

    Solution

    Here are the steps:

    \(\begin{aligned}\sin ^4x−\cos ^4x&=(\sin ^2x−\cos ^2x)(\sin ^2x+\cos ^2x)\\&=−(\cos ^2x−\sin ^2x)\\&=−\cos ^2x \end{aligned}\)

    Review

    Prove the following identities.

    1. \(\sin 2x=2\sin x\cos x\)
    2. \(\cos 2x=\cos^2 x−\sin^2 x\)
    3. \(\tan 2x=\dfrac{2\tan x}{1−\tan^2 x}\)
    4. \(\cos^2 x=\dfrac{1+\cos 2x}{2}\)
    5. \(\tan^2 x=\dfrac{1−\cos 2x}{1+\cos 2x}\)
    6. \(\sin \dfrac{x}{2}=\pm \sqrt{\dfrac{1−\cos x}{2}}\)
    7. \(\cos \dfrac{x}{2}=\pm \sqrt{\dfrac{1+\cos x}{2}}\)
    8. \(\tan \dfrac{x}{2}=\pm \sqrt{\dfrac{1−\cos x}{1+\cos x}}\)
    9. \(\csc 2x=\dfrac{1}{2} \csc x\sec x\)
    10. \(\cot 2x=\dfrac{\cot ^2 x−1}{2\cot x}\)

    Find the value of each expression using half angle identities.

    1. \(\tan 15^{\circ}\)
    2. \(\tan 22.5^{\circ}\)
    3. \(\sec 22.5^{\circ}\)
    4. Show that \(\tan \dfrac{x}{2}=\dfrac{1−\cos x}{\sin x}\).
    5. Using your knowledge from the answer to question 14, show that \(\tan \dfrac{x}{2}=\dfrac{\sin x}{1+\cos x}\).

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 6.4. 

    Vocabulary

    Term Definition
    Half Angle Identity A half angle identity relates a trigonometric function of one half of an argument to a set of trigonometric functions, each containing the original argument.
    identity An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
    power reducing identity A power reducing identity relates the power of a trigonometric function containing a given argument to a set of trigonometric functions, each containing the original argument.

     

    Additional Resources

    Interactive Element

     

    Video: Multiple Angle Formulas - Overview

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