Find the Other Trig Values in Quadrant III cot(x)=12/5

Problem

cot(x)=12/5,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) are negative, while tan(x) and cot(x) are positive.

2. Determine the tangent value using the reciprocal identity.

tan(x)=1/cot(x)

tan(x)=5/12

3. Find the hypotenuse r using the relationship tan(x)=y/x In Quadrant III, we let x=−12 and y=−5

r=√(,(−12)2+(−5)2)

r=√(,144+25)

r=13

4. Calculate the sine value using sin(x)=y/r

sin(x)=−5/13

5. Calculate the cosine value using cos(x)=x/r

cos(x)=−12/13

6. Calculate the cosecant value using the reciprocal identity csc(x)=1/sin(x)

csc(x)=−13/5

7. Calculate the secant value using the reciprocal identity sec(x)=1/cos(x)

sec(x)=−13/12

Final Answer

sin(x)=−5/13,cos(x)=−12/13,tan(x)=5/12,csc(x)=−13/5,sec(x)=−13/12

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