Find the Other Trig Values in Quadrant I cos(x)=5/6

Problem

cos(x)=5/6,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=5/6 in Quadrant I, where all trigonometric functions are positive.

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

sin2(x)+(5/6)2=1

3. Solve for sin(x) by subtracting the squared value from 1 and taking the positive square root.

sin2(x)=1−25/36

sin2(x)=11/36

sin(x)=√(,11)/6

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

tan(x)=√(,11)/6/5/6

tan(x)=√(,11)/5

5. Determine the reciprocal functions sec(x) csc(x) and cot(x) by inverting the values of cos(x) sin(x) and tan(x)

sec(x)=6/5

csc(x)=6/√(,11)

cot(x)=5/√(,11)

6. Rationalize the denominators for csc(x) and cot(x)

csc(x)=(6√(,11))/11

cot(x)=(5√(,11))/11

Final Answer

sin(x)=√(,11)/6,tan(x)=√(,11)/5,csc(x)=(6√(,11))/11,sec(x)=6/5,cot(x)=(5√(,11))/11

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