If cos(θ)=24/25, and θ is in Quadrant I, then what is cos(θ/2)?

Question

give answer as fraction

Answer 1

Given: cos θ = 24/25, and θ is in Quadrant 1.

Remember the half angle formula?

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cos (θ/2) = √ [(1+ cos θ)/2]

Let's substitute the value of cos θ on the right side of the formula.

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cos (θ/2) = √ [(1+ 24/25)/2]

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cos (θ/2) = √ [(49/25)/2]

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cos (θ/2) = √49/50 = 7/(5√2)

Rationalize by multiplying √2 on both numerator and denominator.

cos (θ/2) =(7√2 )/ 10

Answer 2

Since θ lies in quadrant 1, 0° < θ < 90°. So, 0° < θ/2 < 45°. Therefore, θ must also lie in the first quadrant. So, cos(θ/2) > 0.

Using the half-angle formula for cosine, we have cos(θ/2) = √[(1 + cosθ) / 2] = √(49/50) = 7 / (5√2) = 7√2/10

Answer 3

I assume that we are only allowing θ to be in [0°, 360°)

Then, if θ is in the first quadrant, so is θ/2.

With this problem, we use the cos half-angle formula

cos(θ/2) = √((1+cos(θ))/2) =

√((1+24/25)/2) = √49/50 = 7/5√2 or 7√2/10