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

Problem

cos(x)=5/13,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=5/13 in Quadrant I, where all trigonometric functions are positive.

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

sin2(x)+(5/13)2=1

sin2(x)+25/169=1

sin2(x)=1−25/169

sin2(x)=144/169

3. Solve for sin(x) by taking the square root. Since x is in Quadrant I, we take the positive root.

sin(x)=√(,144/169)

sin(x)=12/13

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

tan(x)=(12/13)/(5/13)

tan(x)=12/5

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)=13/12

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

Want more problems? Check here!