Find the Other Trig Values in Quadrant I sin(x)=( square root of 6)/5

Problem

sin(x)=√(,6)/5,0<x<π/2

Solution

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

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

(√(,6)/5)2+cos2(x)=1

6/25+cos2(x)=1

cos2(x)=1−6/25

cos2(x)=19/25

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

cos(x)=√(,19/25)

cos(x)=√(,19)/5

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

tan(x)=√(,6)/5/√(,19)/5

tan(x)=√(,6)/√(,19)

tan(x)=√(,114)/19

5. Calculate csc(x) by taking the reciprocal of sin(x)

csc(x)=5/√(,6)

csc(x)=(5√(,6))/6

6. Calculate sec(x) by taking the reciprocal of cos(x)

sec(x)=5/√(,19)

sec(x)=(5√(,19))/19

7. Calculate cot(x) by taking the reciprocal of tan(x)

cot(x)=√(,19)/√(,6)

cot(x)=√(,114)/6

Final Answer

cos(x)=√(,19)/5,tan(x)=√(,114)/19,csc(x)=(5√(,6))/6,sec(x)=(5√(,19))/19,cot(x)=√(,114)/6

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