Half-Angle Identities

Key Questions

  • How do you use half angle identities to solve equations?

    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.

    Nghi N. · 1 · Apr 12 2015
  • What is the Half-Angle Identities?

    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.

    Mika · 1 · Dec 18 2014

Questions

Trigonometric Identities and Equations