Find the Other Trig Values in Quadrant III tan(x)=4/3

Problem

tan(x)=4/3,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. Since x is in Quadrant III, both sin(x) and cos(x) must be negative, while tan(x) and cot(x) are positive.

2. Determine the ratio of the sides of a reference triangle. Since tan(x)=opposite/adjacent=4/3 we let the opposite side be −4 and the adjacent side be −3 to account for the third quadrant.

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,(−3)2+(−4)2)

r=√(,9+16)

r=5

4. Find the value of sin(x) using the ratio opposite/hypotenuse

sin(x)=−4/5

5. Find the value of cos(x) using the ratio adjacent/hypotenuse

cos(x)=−3/5

6. Find the value of cot(x) by taking the reciprocal of tan(x)

cot(x)=3/4

7. Find the value of sec(x) by taking the reciprocal of cos(x)

sec(x)=−5/3

8. Find the value of csc(x) by taking the reciprocal of sin(x)

csc(x)=−5/4

Final Answer

sin(x)=−4/5,cos(x)=−3/5,cot(x)=3/4,sec(x)=−5/3,csc(x)=−5/4

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