Find the Other Trig Values in Quadrant IV cos(x)=24/25

Problem

cos(x)=24/25,x∈Quadrant IV

Solution

1. Identify the given trigonometric ratio and the quadrant. In Quadrant IV, x coordinates are positive and y coordinates are negative. This means 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 sin(x)

sin2(x)+(24/25)2=1

sin2(x)+576/625=1

sin2(x)=1−576/625

sin2(x)=49/625

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

sin(x)=−√(,49/625)

sin(x)=−7/25

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

tan(x)=(−7/25)/(24/25)

tan(x)=−7/24

5. Find the reciprocal identities for the remaining functions: sec(x)=1/cos(x) csc(x)=1/sin(x) and cot(x)=1/tan(x)

sec(x)=25/24

csc(x)=−25/7

cot(x)=−24/7

Final Answer

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

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