Find the Other Trig Values in Quadrant II sin(x)=8/9

Problem

sin(x)=8/9, Quadrant II

Solution

1. Identify the given information and the quadrant. We are given sin(x)=8/9 in Quadrant II, where sin(x) is positive, but cos(x) tan(x) cot(x) and sec(x) are negative.

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

(8/9)2+cos2(x)=1

64/81+cos2(x)=1

cos2(x)=1−64/81

cos2(x)=17/81

3. Solve for cos(x) by taking the square root. Since x is in Quadrant II, cos(x) must be negative.

cos(x)=−√(,17/81)

cos(x)=−√(,17)/9

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

tan(x)=(8/9)/(−√(,17)/9)

tan(x)=−8/√(,17)

tan(x)=−(8√(,17))/17

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

csc(x)=1/sin(x)

csc(x)=9/8

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

sec(x)=1/cos(x)

sec(x)=−9/√(,17)

sec(x)=−(9√(,17))/17

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

cot(x)=1/tan(x)

cot(x)=−√(,17)/8

Final Answer

cos(x)=−√(,17)/9,tan(x)=−(8√(,17))/17,csc(x)=9/8,sec(x)=−(9√(,17))/17,cot(x)=−√(,17)/8

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