Find the Other Trig Values in Quadrant I csc(x)=4

Problem

csc(x)=4, Quadrant I

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity csc(x)=1/sin(x)

sin(x)=1/4

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

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

3. Solve for cos2(x) by subtracting the squared sine value from 1.

cos2(x)=1−1/16

cos2(x)=15/16

4. Take the square root of both sides, choosing the positive root because x is in Quadrant I, where all trigonometric functions are positive.

cos(x)=√(,15)/4

5. Find the tangent using the ratio tan(x)=sin(x)/cos(x)

tan(x)=(1/4)/(√(,15)/4)

tan(x)=1/√(,15)

tan(x)=√(,15)/15

6. Find the cotangent using the reciprocal identity cot(x)=1/tan(x)

cot(x)=√(,15)

7. Find the secant using the reciprocal identity sec(x)=1/cos(x)

sec(x)=4/√(,15)

sec(x)=(4√(,15))/15

Final Answer

sin(x)=1/4,cos(x)=√(,15)/4,tan(x)=√(,15)/15,sec(x)=(4√(,15))/15,cot(x)=√(,15)

Want more problems? Check here!