Find the Other Trig Values in Quadrant II sin(x)=5/6

Problem

sin(x)=5/6,x∈Quadrant II

Solution

1. Identify the given values for the reference triangle. Since sin(x)=opposite/hypotenuse we let the opposite side y=5 and the hypotenuse r=6

2. Apply the Pythagorean theorem x2+y2=r2 to find the adjacent side x

x2+5=6

x2+25=36

x2=11

x=±√(,11)

3. Determine the sign of x based on the quadrant. In Quadrant II, the xcoordinate is negative.

x=−√(,11)

4. Calculate the remaining trigonometric ratios using x=−√(,11) y=5 and r=6

cos(x)=x/r=−√(,11)/6

tan(x)=y/x=−5/√(,11)=−(5√(,11))/11

csc(x)=1/sin(x)=6/5

sec(x)=1/cos(x)=−6/√(,11)=−(6√(,11))/11

cot(x)=1/tan(x)=−√(,11)/5

Final Answer

cos(x)=−√(,11)/6,tan(x)=−(5√(,11))/11,csc(x)=6/5,sec(x)=−(6√(,11))/11,cot(x)=−√(,11)/5

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