Find Exact Value of cosθ Given sinθ=19/51

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SUMMARY

The discussion focuses on finding the exact value of cosθ given that sinθ = 19/51. Participants clarify that using the Pythagorean identity sin²θ + cos²θ = 1 is essential for deriving the exact value of cosθ. By substituting sinθ into the equation, one can calculate cos²θ as 1 - (19/51)², leading to the exact value of cosθ being √(1 - (19/51)²). This method eliminates the need for approximations and provides a definitive solution.

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cosmictide
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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 [itex]\theta[/itex] is an acute angle with sin[itex]\theta[/itex]=19/51 find the exact value of cos[itex]\theta[/itex]

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[itex]\theta[/itex] to get [itex]\theta[/itex]=21.872...degree. Therefore the cos[itex]\theta[/itex]=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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