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

Problem

cos(x)=−7/25,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given cos(x)=−7/25 in Quadrant III, where both sin(x) and cos(x) are negative, while tan(x) and cot(x) are positive.

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 III, 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 trigonometric functions csc(x) sec(x) and cot(x) using their definitions.

csc(x)=1/sin(x)=−25/24

sec(x)=1/cos(x)=−25/7

cot(x)=1/tan(x)=7/24

Final Answer

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

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