Find the Other Trig Values in Quadrant II sin(x)=3/10

Problem

sin(x)=3/10,x∈Quadrant II

Solution

1. Identify the given information and the constraints of Quadrant II. In Quadrant II, sin(x) and csc(x) are positive, while cos(x) sec(x) tan(x) and cot(x) are negative.

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

(3/10)2+cos2(x)=1

9/100+cos2(x)=1

cos2(x)=1−9/100

cos2(x)=91/100

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

cos(x)=−√(,91/100)

cos(x)=−√(,91)/10

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

tan(x)=(3/10)/(−√(,91)/10)

tan(x)=−3/√(,91)

tan(x)=−(3√(,91))/91

5. Find the reciprocal functions csc(x) sec(x) and cot(x)

csc(x)=1/sin(x)=10/3

sec(x)=1/cos(x)=−10/√(,91)=−(10√(,91))/91

cot(x)=1/tan(x)=−√(,91)/3

Final Answer

cos(x)=−√(,91)/10,tan(x)=−(3√(,91))/91,csc(x)=10/3,sec(x)=−(10√(,91))/91,cot(x)=−√(,91)/3

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