Find the Other Trig Values in Quadrant II sin(x)=3/4

Problem

sin(x)=3/4, Quadrant II

Solution

1. Identify the given information and the constraints of the quadrant. In Quadrant II, the sine value is positive, while cosine and tangent values are negative.

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

(3/4)2+cos2(x)=1

9/16+cos2(x)=1

cos2(x)=1−9/16

cos2(x)=7/16

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

cos(x)=−√(,7/16)

cos(x)=−√(,7)/4

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

tan(x)=(3/4)/(−√(,7)/4)

tan(x)=−3/√(,7)

tan(x)=−(3√(,7))/7

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

csc(x)=1/sin(x)=4/3

sec(x)=1/cos(x)=−4/√(,7)=−(4√(,7))/7

cot(x)=1/tan(x)=−√(,7)/3

Final Answer

cos(x)=−√(,7)/4,tan(x)=−(3√(,7))/7,csc(x)=4/3,sec(x)=−(4√(,7))/7,cot(x)=−√(,7)/3

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