Find the Other Trig Values in Quadrant I csc(x)=6

Problem

csc(x)=6, in Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since csc(x)=6 the reciprocal identity gives sin(x)=1/6 In Quadrant I, all trigonometric values are positive.

2. Determine the side lengths of a right triangle. Let the opposite side be 1 and the hypotenuse be 6

3. Apply the Pythagorean theorem to find the adjacent side a

a^2+1^2=6^2

a^2+1=36

a^2=35

a=√(,35)

4. Calculate cos(x) using the ratio of the adjacent side to the hypotenuse.

cos(x)=√(,35)/6

5. Calculate tan(x) using the ratio of the opposite side to the adjacent side.

tan(x)=1/√(,35)

tan(x)=√(,35)/35

6. Calculate sec(x) by taking the reciprocal of cos(x)

sec(x)=6/√(,35)

sec(x)=(6√(,35))/35

7. Calculate cot(x) by taking the reciprocal of tan(x)

cot(x)=√(,35)

Final Answer

sin(x)=1/6,cos(x)=√(,35)/6,tan(x)=√(,35)/35,sec(x)=(6√(,35))/35,cot(x)=√(,35)

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