Find the Other Trig Values in Quadrant I cot(x)=1/9

Problem

cot(x)=1/9,0<x<π/2

Solution

1. Identify the relationship between the cotangent and the sides of a right triangle. Since cot(x)=adjacent/opposite we can let the adjacent side a=1 and the opposite side b=9

2. Calculate the hypotenuse c using the Pythagorean theorem a2+b2=c2

1+9=c2

1+81=c2

c=√(,82)

3. Determine the sine and cosecant values. Since x is in Quadrant I, all trigonometric values are positive.

sin(x)=opposite/hypotenuse=9/√(,82)

csc(x)=1/sin(x)=√(,82)/9

4. Determine the cosine and secant values.

cos(x)=adjacent/hypotenuse=1/√(,82)

sec(x)=1/cos(x)=√(,82)

5. Determine the tangent value.

tan(x)=1/cot(x)=9

6. Rationalize the denominators for the sine and cosine values.

sin(x)=(9√(,82))/82

cos(x)=√(,82)/82

Final Answer

sin(x)=(9√(,82))/82,cos(x)=√(,82)/82,tan(x)=9,sec(x)=√(,82),csc(x)=√(,82)/9

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