Find the Other Trig Values in Quadrant IV cos(x)=12/13

Problem

cos(x)=12/13,x∈Quadrant IV

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant IV. In Quadrant IV, cos(x) and sec(x) are positive, while sin(x) csc(x) tan(x) and cot(x) are negative.

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

sin2(x)+(12/13)2=1

3. Solve for sin(x) by isolating the term and taking the square root, choosing the negative root because the angle is in Quadrant IV.

sin2(x)=1−144/169

sin2(x)=25/169

sin(x)=−√(,25/169)=−5/13

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

tan(x)=(−5/13)/(12/13)=−5/12

5. Determine the reciprocal functions sec(x) csc(x) and cot(x) by inverting the values of cos(x) sin(x) and tan(x) respectively.

sec(x)=1/cos(x)=13/12

csc(x)=1/sin(x)=−13/5

cot(x)=1/tan(x)=−12/5

Final Answer

sin(x)=−5/13,tan(x)=−5/12,sec(x)=13/12,csc(x)=−13/5,cot(x)=−12/5

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