Find Exact Value of cosθ Given sinθ=19/51

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To find the exact value of cosθ given sinθ = 19/51, the relationship sin²θ + cos²θ = 1 can be applied. By substituting sinθ into the equation, cos²θ can be calculated as 1 - (19/51)². This results in cos²θ = 1 - 361/2601, leading to cos²θ = 2240/2601. Taking the square root gives cosθ = √(2240)/51, which is the exact value. This method provides a precise solution rather than an approximation.
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Hi guys,

This has caused me some confusion. Any help in this regard would be greatly appreciated.

Homework Statement



Given that \theta is an acute angle with sin\theta=19/51 find the exact value of cos\theta

Homework Equations


The Attempt at a Solution



All I seem to be getting based on my calculations are approximations. I have no idea how to actually obtain the exact value. Initially I took the inverse of sin\theta to get \theta=21.872...degree. Therefore the cos\theta=0.928...I also tried using a right-angled triangle and finding the adjacent which I calculated to be 47.328... and dividing adj by hyp to see whether I get an exact value but got the same answer as before.

Any help in this regard would be greatly appreciated.

Thanks in advance.
 
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What about ##\sin^2 + \cos^2 = 1##, exactly ? It means you get a square root in your answer, but it definitely is exact.
 
BvU said:
What about ##\sin^2 + \cos^2 = 1##, exactly ? It means you get a square root in your answer, but it definitely is exact.

:smile: Genius.

Thank you so much, I can't believe I didn't think of that.
 

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