Find the Other Trig Values in Quadrant II cos(x)=-12/13

Problem

cos(x)=−12/13, Quadrant II

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant II. In Quadrant II, sin(x) and csc(x) are positive, while cos(x) sec(x) tan(x) and cot(x) are negative.

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

sin2(x)+(−12/13)2=1

sin2(x)+144/169=1

sin2(x)=1−144/169

sin2(x)=25/169

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

sin(x)=√(,25/169)

sin(x)=5/13

4. Find tan(x) using the identity tan(x)=sin(x)/cos(x)

tan(x)=(5/13)/(−12/13)

tan(x)=−5/12

5. Find the reciprocal functions csc(x) sec(x) and cot(x) by inverting the values of sin(x) cos(x) and tan(x)

csc(x)=1/sin(x)=13/5

sec(x)=1/cos(x)=−13/12

cot(x)=1/tan(x)=−12/5

Final Answer

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

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