Find the Other Trig Values in Quadrant I cot(x)=12/5

Problem

cot(x)=12/5,0<x<π/2

Solution

1. Identify the given trigonometric ratio and the quadrant. Since cot(x)=12/5 and x is in Quadrant I, all trigonometric functions will be positive.

2. Define the relationship between the cotangent and the sides of a right triangle. We have cot(x)=adjacent/opposite so we let the adjacent side a=12 and the opposite side o=5

3. Calculate the hypotenuse h using the Pythagorean theorem a2+o2=h2

12+5=h2

144+25=h2

169=h2

h=13

4. Determine the sine and cosecant values using the definitions sin(x)=o/h and csc(x)=h/o

sin(x)=5/13

csc(x)=13/5

5. Determine the cosine and secant values using the definitions cos(x)=a/h and sec(x)=h/a

cos(x)=12/13

sec(x)=13/12

6. Determine the tangent value using the reciprocal identity tan(x)=1/cot(x)

tan(x)=5/12

Final Answer

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

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