Double Angle Identities

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Math Analysis - Trigonometry Identity - Double Angle - Sine

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1 of 2 videos by AJ Speller

Key Questions

  • Double Angle Identities

    #sin2theta=2sin theta cos theta#

    #cos2theta=cos^2theta-sin^2theta=2cos^2theta-1=1-2sin^2theta#

    #tan2theta={2tan theta}/{1-tan^2theta}#


    I hope that this was helpful.

  • Substitute in the one angle you get into the identity.

    For example, if you are asked to find #sin2a# and #a = 30#, then you substitute in #sin2(30) = 2sin30cos30# into your calculator, and that's how you solve equations with double angle identities.

  • You can use the double angle ID to evaluate the exact value of a trigonometric expression when you can relate the expression of one of the double angle formulas to one of the special angles.

    Ex.1 Evaluate #2sin(15)cos(15)#

    using #sin(2x)=2sinxcosx#, we observe that the left had side of this ID resembles the given question (with #x=15#).
    Thus we can convert the expression to
    #2sin(15)cos(15)=sin(2*15)=sin(30)=1/2#


    It is also possible to use known values of #sin(x), cos(x), and tan(x)# to evaluate the exact value of a given trigonometric function of a double angle.

    Ex.2 Given #sin(x)=3/5#, find #cos(2x)#

    To answer this question, we can then use the ID #cos(2x)=1-2sin^2(x)#

    Since #sin(x)=3/5#, we can find #sin^2(x)=9/25#

    Therefore #cos(2x)=1-2sin^2(x)=1-2(9/25)=1-18/25=7/25#

  • You would need an expression to work with.

    For example:
    Given #sinalpha=3/5# and #cosalpha=-4/5#, you could find #sin2 alpha# by using the double angle identity
    #sin2 alpha=2sin alpha cos alpha#.

    #sin2 alpha=2(3/5)(-4/5)=-24/25#.

    You could find #cos2 alpha# by using any of:
    #cos2 alpha=cos^2 alpha -sin^2 alpha#
    #cos2 alpha=1 -2sin^2 alpha#
    #cos2 alpha=2cos^2 alpha -1#

    In any case, you get #cos alpha=7/25#.

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