SOLUTION: Find the values of the six trigonometric functions of θ with the given constraint. (If an answer is undefined, enter UNDEFINED.) Function Value Constraint cos θ =

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Question 719189: Find the values of the six trigonometric functions of θ with the given constraint. (If an answer is undefined, enter UNDEFINED.)
Function Value Constraint
cos θ = 4/5 tan θ < 0
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
We will do this in two parts: finding the ratios and then figuring out the signs (which are positive and which are negative).
  1. Finding the ratios:
    1. Draw a right triangle and pick one of the acute angles to be angle .
    2. Since the ratio for cos is adjacent over hypotenuse, make the side adjacent to be a 4 (the numerator of 4/5) and make the hypotenuse a 5 (the denominator of 4/5)
    3. Use the Pythagorean Theorem to find the missing side, the side which is opposite adjacent to . (Hint: It will be a square root.)
    4. Now that you have all three sides, the hypotenuse and the sides which are opposite to and adjacent to , you can now set up the ratios for the other five trig functions.
    5. Some of the ratios from the previous step will have the square root from step 3 in the denominator. Rationalize these fractions by multiplying the numerator and denominator of the fraction by the square root.
  2. Determining the signs. We were given that the cos of is positive (specifically 4/5) and we were given that the tan of is negative. There is only one quadrant where cos's are positive and tan's are negative: the 4th quadrant. So terminates somewhere in the 4th quadrant. In the 4th quadrant sin, csc, tan and cot are all negative. So take the ratios you found in the first part for these four functions and make them negative.
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