Find the Other Trig Values in Quadrant IV cos(x)=5/13

Problem

cos(x)=5/13,x∈Quadrant IV

Solution

1. Identify the given information and the signs of the 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)+(5/13)2=1

sin2(x)+25/169=1

sin2(x)=1−25/169

sin2(x)=144/169

3. Solve for sin(x) by taking the square root. Since x is in Quadrant IV, sin(x) must be negative.

sin(x)=−√(,144/169)

sin(x)=−12/13

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

tan(x)=(−12/13)/(5/13)

tan(x)=−12/5

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

csc(x)=1/sin(x)=−13/12

sec(x)=1/cos(x)=13/5

cot(x)=1/tan(x)=−5/12

Final Answer

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

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