Find the Other Trig Values in Quadrant II sin(x)=2/5

Problem

sin(x)=2/5, Quadrant II

Solution

1. Identify the given values for the sine function. In a right triangle or on the unit circle, sin(x)=y/r where y=2 and r=5

2. Apply the Pythagorean theorem to find the missing side x The relationship is x2+y2=r2

x2+2=5

x2+4=25

x2=21

x=±√(,21)

3. Determine the sign of x based on the quadrant. In Quadrant II, the xcoordinate is negative, so x=−√(,21)

4. Calculate the remaining trigonometric ratios using x=−√(,21) y=2 and r=5

cos(x)=x/r=−√(,21)/5

tan(x)=y/x=2/(−√(,21))=−(2√(,21))/21

csc(x)=r/y=5/2

sec(x)=r/x=5/(−√(,21))=−(5√(,21))/21

cot(x)=x/y=−√(,21)/2

Final Answer

cos(x)=−√(,21)/5,tan(x)=−(2√(,21))/21,csc(x)=5/2,sec(x)=−(5√(,21))/21,cot(x)=−√(,21)/2

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