Find the Other Trig Values in Quadrant I cos(x)=1/4

Problem

cos(x)=1/4,0<x<π/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=1/4 in Quadrant I, where all trigonometric functions (sin() cos() tan() csc() sec() cot() are positive.

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

sin2(x)+(1/4)2=1

sin2(x)+1/16=1

sin2(x)=15/16

sin(x)=√(,15/16)

sin(x)=√(,15)/4

3. Find the tangent using the identity tan(x)=sin(x)/cos(x)

tan(x)=√(,15)/4/1/4

tan(x)=√(,15)

4. Find the reciprocal functions sec(x) csc(x) and cot(x) using their definitions.

sec(x)=1/cos(x)=4

csc(x)=1/sin(x)=4/√(,15)

csc(x)=(4√(,15))/15

cot(x)=1/tan(x)=1/√(,15)

cot(x)=√(,15)/15

Final Answer

sin(x)=√(,15)/4,tan(x)=√(,15),sec(x)=4,csc(x)=(4√(,15))/15,cot(x)=√(,15)/15

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