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

Problem

cos(x)=12/13, Quadrant I

Solution

1. Identify the given information and the quadrant. In Quadrant I, all trigonometric functions (sin() cos() tan() csc() sec() cot() are positive.

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

sin2(x)+(12/13)2=1

3. Solve for sin(x) by squaring the fraction and subtracting it from 1.

sin2(x)+144/169=1

sin2(x)=1−144/169

sin2(x)=25/169

sin(x)=√(,25/169)=5/13

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

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

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

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