Find the Other Trig Values in Quadrant II sin(a)=12/13

Problem

sin(a)=12/13, Quadrant II

Solution

1. Identify the given information and the constraints of the quadrant. In Quadrant II, the ycoordinate (sine) is positive, while the xcoordinate (cosine) and the tangent are negative.

2. Use the Pythagorean identity to find the value of cos(a)

sin2(a)+cos2(a)=1

3. Substitute the known value of sin(a) into the identity.

(12/13)2+cos2(a)=1

4. Solve for cos2(a) by squaring the fraction and subtracting it from 1.

144/169+cos2(a)=1

cos2(a)=1−144/169

cos2(a)=25/169

5. Take the square root and choose the negative sign because the angle is in Quadrant II.

cos(a)=−√(,25/169)

cos(a)=−5/13

6. Calculate tan(a) using the ratio of sine to cosine.

tan(a)=sin(a)/cos(a)

tan(a)=(12/13)/(−5/13)

tan(a)=−12/5

7. Determine the reciprocal functions by flipping the values of sine, cosine, and tangent.

csc(a)=1/sin(a)=13/12

sec(a)=1/cos(a)=−13/5

cot(a)=1/tan(a)=−5/12

Final Answer

cos(a)=−5/13,tan(a)=−12/5,csc(a)=13/12,sec(a)=−13/5,cot(a)=−5/12

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