SOLUTION: find the other five functions of acute angle A,given that cscA=4 over 3

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Question 623876: find the other five functions of acute angle A,given that cscA=4 over 3
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To better understand how this works, let's draw a diagram:
  1. Draw a right triangle. (It doesn't matter how it looks.)
  2. Choose one of the acute angles in the triangle and label it A.
  3. Since the csc ratio is hypotenuse/opposite and since we want A to have a csc of 4/3, label the hypotenuse of the triangle (opposite the right angle) with a length of 4 and label the length of the side opposite angle A with a length of 3.
To find all five of the other functions, we will need the adjacent side, too. Since we have two of the three sides of the triangle, we can use the Pythagorean Theorem to find it. Let's call the adjacent side "b". Inserting "b" and 4 and 3 into the Pythagorean equation we get:
4%5E2+=+3%5E2+%2B+b%5E2
Solving for b:
16+=+9+%2B+b%5E2
7+=+b%5E2
sqrt%287%29+=+b (ignoring the negative square root since lengths of the sides of triangles cannot be negative)

Now we can find the other 5 functions, using 4 for the hypotenuse, 3 for the opposite side and sqrt%287%29 for the adjacent side:
sin%28A%29+=+3%2F4
cos%28A%29+=+sqrt%287%29%2F4
tan%28A%29+=+3%2Fsqrt%287%29
sec%28A%29+=+4%2Fsqrt%287%29
cot%28A%29+=+sqrt%287%29%2F3

We don;t usually leave square roots in denominators so we should rationalize tan and sec: