SOLUTION: given that θ is an acute angle and cot θ=p, e express cosecθ in terms of p

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Question 814237: given that θ is an acute angle and cot θ=p, e express cosecθ in terms of p
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
For problems like this:
  1. Draw a right triangle.
  2. Choose one of the acute angles in the triangle. This will be the angle the problem is talking about.
  3. If the given function value is not a fraction, make it a fraction. We were given cot%28theta%29+=+p. So we rewrite it as a fraction: cot%28theta%29+=+p%2F1
  4. Use the fraction/ratio and the particular function of the given function value to label the lengths of the appropriate sides. Since cot is adjacent over opposite, label the side adjacent to the chosen angle as "p" and label the side opposite the chosen angle as "1".
  5. If you need the third side of the triangle to find the desired function value, then use the Pythagorean Theorem to find it. Since we are looking for cos%28theta%29 and since cos is adjacent over hypotenuse, we will need the hypotenuse.
  6. Find the desired function value by creating the appropriate ratio and then simplifying.
So to find cos%28theta%29 we will need to find the hypotenuse:
h%5E2+=+p%5E2%2B1%5E2
which simplifies to:
h%5E2+=+p%5E2%2B1
Solving for h:
sqrt%28h%5E2%29+=+sqrt%28p%5E2%2B1%29
h+=+sqrt%28p%5E2%2B1%29
(Note: We ignore the negative square root because we've been told that theta is in the first quadrant where everything is positive.)

Since cos is adjacent over hypotenuse:
cos%28theta%29+=+p%2Fsqrt%28p%5E2%2B1%29