Find the Other Trig Values in Quadrant I cos(x)=1/14

Problem

cos(x)=1/14,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=1/14 in Quadrant I, where all trigonometric functions are positive.

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

sin2(x)+(1/14)2=1

3. Solve for sin(x) by subtracting the squared value and taking the square root.

sin2(x)=1−1/196

sin2(x)=195/196

sin(x)=√(,195/196)

sin(x)=√(,195)/14

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

tan(x)=√(,195)/14/1/14

tan(x)=√(,195)

5. Determine the reciprocal functions sec(x) csc(x) and cot(x)

sec(x)=1/cos(x)=14

csc(x)=1/sin(x)=14/√(,195)

cot(x)=1/tan(x)=1/√(,195)

6. Rationalize the denominators for the reciprocal functions.

csc(x)=(14√(,195))/195

cot(x)=√(,195)/195

Final Answer

sin(x)=√(,195)/14,tan(x)=√(,195),sec(x)=14,csc(x)=(14√(,195))/195,cot(x)=√(,195)/195

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