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

Problem

cos(x)=7/25,x∈Quadrant IV

Solution

1. Identify the given information and the quadrant. In Quadrant IV, the xcoordinate (cosine) is positive, and the ycoordinate (sine) is negative.

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

sin2(x)+(7/25)2=1

sin2(x)+49/625=1

sin2(x)=1−49/625

sin2(x)=576/625

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

sin(x)=−√(,576/625)

sin(x)=−24/25

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

tan(x)=(−24/25)/(7/25)

tan(x)=−24/7

5. Determine the reciprocal functions sec(x) csc(x) and cot(x)

sec(x)=1/cos(x)=25/7

csc(x)=1/sin(x)=−25/24

cot(x)=1/tan(x)=−7/24

Final Answer

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

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