Find Exact Value of cosθ Given sinθ=19/51

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In summary, the conversation discusses finding the exact value of cos\theta when given that \theta is an acute angle with sin\theta=19/51. The person asking for help initially attempted to calculate the value using inverse sine and a right-angled triangle, but only obtained approximations. Another person suggests using the equation ##\sin^2 + \cos^2 = 1##, which leads to an exact value.
  • #1
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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  • #2
What about ##\sin^2 + \cos^2 = 1##, exactly ? It means you get a square root in your answer, but it definitely is exact.
 
  • #3
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.

:rofl: Genius.

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

1. How do you find the exact value of cosθ given sinθ=19/51?

To find the exact value of cosθ, we can use the Pythagorean identity: cos^2θ + sin^2θ = 1. Therefore, we can calculate cosθ by first finding sin^2θ (using the given value of sinθ), and then taking the square root of 1-sin^2θ. In this case, cosθ = √(1-19^2/51^2) = √(1-361/2601) = √(2240/2601) = 8√5/51.

2. Can you find the exact value of cosθ if sinθ is given in a decimal form?

Yes, you can still find the exact value of cosθ if sinθ is given in decimal form. Simply follow the same steps as in the previous question, but use the decimal value for sinθ in the calculation.

3. Is there more than one exact value of cosθ given sinθ=19/51?

Yes, there are two exact values of cosθ when sinθ=19/51. This is because the sine function has a positive and negative value in different quadrants of the unit circle, and the cosine function is the reciprocal of the sine function. Therefore, for every value of sinθ, there are two corresponding values of cosθ.

4. How can this exact value of cosθ be used in real-life applications?

The exact value of cosθ can be used in many real-life applications, such as engineering, physics, and astronomy. For example, in engineering, it can be used to calculate the angle of forces acting on a structure. In physics, it can be used to calculate the work done by a force acting at an angle. In astronomy, it can be used to calculate the distance and position of celestial objects.

5. Are there any other methods to find the exact value of cosθ given sinθ=19/51?

Yes, there are other methods to find the exact value of cosθ. One method is to use the trigonometric ratio table, which provides the exact values of the trigonometric functions for common angles. Another method is to use a calculator that has a "sin^-1" or "arcsin" function, which can directly give the angle θ when sinθ is known.

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