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

Problem

sin(x)=5/13, Quadrant II

Solution

1. Identify the given information and the signs of the 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. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to find the value of cos(x)

(5/13)2+cos2(x)=1

25/169+cos2(x)=1

cos2(x)=1−25/169

cos2(x)=144/169

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

cos(x)=−√(,144/169)

cos(x)=−12/13

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

tan(x)=(5/13)/(−12/13)

tan(x)=−5/12

5. Determine 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

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

Want more problems? Check here!