can someone help me with my math homework?

Prove, using the trigonometric identities,

that

cos((1/2)*θ)= sqrt((cos(θ )+1)/2)

for any 0 ≤ θ ≤ π. Remember to separate the left and right sides in

your proof. Start from the RIGHT side! Moreover: θ =(1/2)*θ+ (1/2)*θ

3 Answers

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  • 2 years ago
    Favorite Answer

     

    Angle sum identity:

    cos(x+y) = cos x cos y − sin x sin y

    cosθ = cos(θ/2 + θ/2)

    ........ = cos(θ/2) cos(θ/2) − sin(θ/2) sin(θ/2)

    ........ = cos²(θ/2) − sin²(θ/2)

    ........ = cos²(θ/2) − (1 − cos²(θ/2))

    ........ = 2cos²(θ/2)−1

    RHS = √[(cosθ + 1) / 2]

    = √[((2cos²(θ/2)−1) + 1) / 2]

    = √(2cos²(θ/2) / 2)

    = √(cos²(θ/2))

    = |cos(θ/2)|

    = cos(θ/2) ..... since cos(θ/2) > 0 for 0 ≤ θ ≤ π

    = LHS

  • 2 years ago

    Thank you for taking the time to answer me!!!

  • 2 years ago

    We have cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

    set a = b = x

    then cos(2x)

    = cos(x + x)

    = cos(x)cos(x) - sin(x)sin(x)

    = cos^2(x) - sin^2(x)

    = cos^2(x) - ( 1 - cos^2(x)

    = cos^2(x) - 1 + cos^2(x)

    = 2cos^2(x) - 1

    so cos(2x) = 2cos^2(x) - 1

    so 2cos^2(x) = cos(2x) + 1

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

    or cos(x) = sqrt( (cos(2x)+1)/2)

    Now set x = θ/2

    so cos(θ/2) = sqrt( (cos(2*θ/2)+1)/2)

    = sqrt( (cos(θ)+1)/2)

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