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

Problem

cos(x)=−5/13, 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. Find the sine using the Pythagorean identity sin2(x)+cos2(x)=1

sin2(x)+(−5/13)2=1

sin2(x)+25/169=1

sin2(x)=144/169

sin(x)=√(,144/169)

sin(x)=12/13

3. Find the tangent using the ratio tan(x)=sin(x)/cos(x)

tan(x)=(12/13)/(−5/13)

tan(x)=−12/5

4. Find the reciprocal functions by inverting the values of sin(x) cos(x) and tan(x)

csc(x)=1/sin(x)=13/12

sec(x)=1/cos(x)=−13/5

cot(x)=1/tan(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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