Find the Other Trig Values in Quadrant IV sin(x)=-7/25

Problem

sin(x)=−7/25, Quadrant IV

Solution

1. Identify the given information and the quadrant. We are given sin(x)=−7/25 and told that x is in Quadrant IV, where 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 cos(x)

(−7/25)2+cos2(x)=1

49/625+cos2(x)=1

cos2(x)=1−49/625

cos2(x)=576/625

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

cos(x)=√(,576/625)

cos(x)=24/25

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

tan(x)=(−7/25)/(24/25)

tan(x)=−7/24

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)=−25/7

sec(x)=1/cos(x)=25/24

cot(x)=1/tan(x)=−24/7

Final Answer

cos(x)=24/25,tan(x)=−7/24,csc(x)=−25/7,sec(x)=25/24,cot(x)=−24/7

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