Find the Other Trig Values in Quadrant II tan(x)=-3/8

Problem

tan(x)=−3/8, Quadrant II

Solution

1. Identify the given information and the quadrant. In Quadrant II, x is negative and y is positive.

2. Relate the tangent function to the coordinates on a circle using tan(x)=y/x Since the angle is in Quadrant II, we let y=3 and x=−8

3. Calculate the radius r using the Pythagorean theorem r=√(,x2+y2)

r=√(,(−8)2+3)

r=√(,64+9)

r=√(,73)

4. Determine the sine value using sin(x)=y/r

sin(x)=3/√(,73)

sin(x)=(3√(,73))/73

5. Determine the cosine value using cos(x)=x/r

cos(x)=(−8)/√(,73)

cos(x)=−(8√(,73))/73

6. Determine the cosecant value by taking the reciprocal of sine.

csc(x)=√(,73)/3

7. Determine the secant value by taking the reciprocal of cosine.

sec(x)=−√(,73)/8

8. Determine the cotangent value by taking the reciprocal of tangent.

cot(x)=−8/3

Final Answer

sin(x)=(3√(,73))/73,cos(x)=−(8√(,73))/73,csc(x)=√(,73)/3,sec(x)=−√(,73)/8,cot(x)=−8/3

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