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

Problem

sin(x)=1/4, Quadrant I

Solution

1. Identify the given information and the goal. We are given sin(x)=1/4 in Quadrant I, where all trigonometric functions are positive.

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

(1/4)2+cos2(x)=1

1/16+cos2(x)=1

cos2(x)=1−1/16

cos2(x)=15/16

3. Solve for cosine by taking the square root. Since x is in Quadrant I, cos(x) is positive.

cos(x)=√(,15/16)

cos(x)=√(,15)/4

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

tan(x)=(1/4)/(√(,15)/4)

tan(x)=1/√(,15)

tan(x)=√(,15)/15

5. Find cosecant by taking the reciprocal of sin(x)

csc(x)=1/sin(x)

csc(x)=4

6. Find secant by taking the reciprocal of cos(x)

sec(x)=1/cos(x)

sec(x)=4/√(,15)

sec(x)=(4√(,15))/15

7. Find cotangent by taking the reciprocal of tan(x)

cot(x)=1/tan(x)

cot(x)=√(,15)

Final Answer

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

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