Half-Angle Identities

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Half-Angle Results

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1 of 2 videos by Eddie W.

Key Questions

  • The half-angle identities are defined as follows:

    #\mathbf(sin(x/2) = pmsqrt((1-cosx)/2))#

    #(+)# for quadrants I and II
    #(-)# for quadrants III and IV

    #\mathbf(cos(x/2) = pmsqrt((1+cosx)/2))#

    #(+)# for quadrants I and IV
    #(-)# for quadrants II and III

    #\mathbf(tan(x/2) = pmsqrt((1-cosx)/(1+cosx)))#

    #(+)# for quadrants I and III
    #(-)# for quadrants II and IV

    We can derive them from the following identities:

    #sin^2x = (1-cos(2x))/2#

    #sin^2(x/2) = (1-cos(x))/2#

    #color(blue)(sin(x/2) = pmsqrt((1-cos(x))/2))#

    Knowing how #sinx# is positive for #0-180^@# and negative for #180-360^@#, we know that it is positive for quadrants I and II and negative for III and IV.

    #cos^2x = (1+cos(2x))/2#

    #cos^2(x/2) = (1+cos(x))/2#

    #color(blue)(cos(x/2) = pmsqrt((1+cos(x))/2))#

    Knowing how #cosx# is positive for #0-90^@# and #270-360^@#, and negative for #90-270^@#, we know that it is positive for quadrants I and IV and negative for II and III.

    #tan(x/2) = sin(x/2)/(cos(x/2)) = (pmsqrt((1-cos(x))/2))/(pmsqrt((1+cos(x))/2))#

    #color(blue)(tan(x/2) = pmsqrt((1-cos(x))/(1+cos(x))))#

    We can see that if we take the conditions for positive and negative values from #sinx# and #cosx# and divide them, we get that this is positive for quadrants I and III and negative for II and IV.

  • Half-angle identities give exact answers for trig function values only when the original angle whose trig function you are finding is 1/2 of a "familiar" angle, e.g., 30 degrees, 60 degrees, 45 degrees, or an angle for which one of those measures is its reference angle.

  • Common Half angle identity:
    1. #sin a = 2 sin (a/2)* cos (a/2)#

    Half angle Identities in term of t = tan a/2.
    2. #sin a = (2t)/(1 + t^2)#

    3.#cos a = (1 - t^2)/(1 + t^2)#

    1. #tan a = (2t)/(1 - t^2).#

    Use of half angle identities to solve trig equations.

    Example. Solve #cos x + 2*sin x = 1 + tan (x/2).#
    Solution. Call #t = tan (x/2)#. Use half angle identities (2) and (3) to transform the equation.

    #(1 - t^2)/4 + (1 + t^2)/4 = 1 + t.#

    #1 - t^2 + 4t = (1 + t)(1 + t^2)#

    #t^3 + 2t^2 - 3t = t*(t^2 + 2t - 3) = 0.#

    Next, solve the #3# basic trig equations: #tan (x/2) = t = 0; tan (x/2) = -3;# and #tan (x/2) = 1.#

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