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45 votes
Given that tan(θ)=5/8 and θ is in quadrant i find and cos (θ/2)

User Iamisti
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1 Answer

27 votes
27 votes

Recall the half-angle identity for cosine,

cos²(θ/2) = (1 + cos(θ))/2

Since θ lies in quadrant I, we also have θ/2 in quadrant I, since

0 < θ < π/2 ⇒ 0 < θ/2 < π/4

Then for this θ, we have

cos(θ/2) = + √((1 + cos(θ))/2)

Also recall the Pythagorean identity,

cos²(θ) + sin²(θ) = 1

Multiplying through both sides of this identity by cos²(θ) gives another form of it,

1 + tan²(θ) = sec²(θ)

Because θ belongs to quadrant I, we know cos(θ) > 0, so we also have sec(θ) = 1/cos(θ) > 0.

It follows that

sec(θ) = + √(1 + tan²(θ)) = √89/8

⇒ cos(θ) = 8/√89

and so

cos(θ/2) = + √((1 + 8/√89)/2) = √(1/2 + 4/√89)

User Cangrejo
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