Find the Other Trig Values in Quadrant I sin(x)=( square root of 2)/6

Problem

sin(x)=√(,2)/6, Quadrant I

Solution

1. Identify the given information and the quadrant. In Quadrant I, all trigonometric functions (sin(x) cos(x) tan(x) csc(x) sec(x) cot(x) are positive.

2. Find cos(x) using the Pythagorean identity sin2(x)+cos2(x)=1

cos2(x)=1−sin2(x)

cos2(x)=1−(√(,2)/6)2

cos2(x)=1−2/36

cos2(x)=34/36

cos(x)=√(,34/36)

cos(x)=√(,34)/6

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

tan(x)=√(,2)/6/√(,34)/6

tan(x)=√(,2)/√(,34)

tan(x)=1/√(,17)

tan(x)=√(,17)/17

4. Find csc(x) by taking the reciprocal of sin(x)

csc(x)=6/√(,2)

csc(x)=3√(,2)

5. Find sec(x) by taking the reciprocal of cos(x)

sec(x)=6/√(,34)

sec(x)=(6√(,34))/34

sec(x)=(3√(,34))/17

6. Find cot(x) by taking the reciprocal of tan(x)

cot(x)=√(,17)

Final Answer

cos(x)=√(,34)/6,tan(x)=√(,17)/17,csc(x)=3√(,2),sec(x)=(3√(,34))/17,cot(x)=√(,17)

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