Find the Other Trig Values in Quadrant III cos(x)=-5/13

Problem

cos(x)=−5/13,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=−5/13 in Quadrant III, where both sin(x) and cos(x) are negative, while tan(x) and cot(x) are positive.

2. Use the Pythagorean identity sin2(x)+cos2(x)=1 to find the sine value.

sin2(x)+(−5/13)2=1

sin2(x)+25/169=1

sin2(x)=1−25/169

sin2(x)=144/169

3. Solve for sine by taking the square root. Since x is in Quadrant III, sin(x) must be negative.

sin(x)=−√(,144/169)

sin(x)=−12/13

4. Calculate tangent using the ratio tan(x)=sin(x)/cos(x)

tan(x)=(−12/13)/(−5/13)

tan(x)=12/5

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

csc(x)=1/(−12/13)

csc(x)=−13/12

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

sec(x)=1/(−5/13)

sec(x)=−13/5

7. Calculate cotangent using the reciprocal identity cot(x)=1/tan(x)

cot(x)=1/(12/5)

cot(x)=5/12

Final Answer

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

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