SOLUTION: if tan(x) = -1/3 and sin(x)is less than 0, find the exact values of each of the remaining trig functions of x?

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Question 614136: if tan(x) = -1/3 and sin(x)is less than 0, find the exact values of each of the remaining trig functions of x?
Answer by jsmallt9(3758) (Show Source):
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I hope you've learned what ratios (in terms of opposite, adjacent and hypotenuse) each of the trig functions represent.

We'll start by finding the values for opposite, adjacent and hypotenuse. (For the time being we are going to to ignore negatives. We will deal with positive or negative later.)

It may help to draw a right triangle. Then pick one of the acute angles to be "x". Since we're given that the tan(x) = -1/3 and since tan is opposite over adjacent, we will use 3 for the opposite side and 1 for the adjacent side. (Remember, we're ignoring negatives for the time being.)

Using the Pythagorean Theorem we can find the hypotenuse. You should find that it is . Now that we have numbers for the opposite, adjacent and hypotenuse sides we are almost ready to write the ratios.

Next we will determine which functions will be positive and which ones will be negative. Since we were given that tan(x) = -1/3, a negative number, we know that x must terminate in either the 2nd or 4th quadrant (where tan's are negative). We are also told that sin(x) < 0. IOW, sin(x) is negative. sin's are negative for angles that terminate in the 3rd or 4th quadrants. For both tan and sin to be negative, x must terminate in the 4th quadrant. In the 4th quadrant only cos and sec are positive.

Finally we have all we need to answer the question. Using
opposite = 3
adjacent = 1
hypotenuse =
and using what we determined about which functions will have negative values we can say:

These are exact values for the other 5 trig functions. You may want to rationalize the denominators of the sin and cos values.