Find the Other Trig Values in Quadrant III tan(x)=8/15

Problem

tan(x)=8/15,π<x<(3*π)/2

Solution

1. Identify the given information and the quadrant. We are given tan(x)=8/15 in Quadrant III, where both sin(x) and cos(x) are negative.

2. Determine the values of y and x using the definition tan(x)=y/x Since we are in Quadrant III, both coordinates must be negative, so y=−8 and x=−15

3. Calculate the hypotenuse r using the Pythagorean theorem r=√(,x2+y2)

r=√(,(−15)2+(−8)2)

r=√(,225+64)

r=√(,289)

r=17

4. Evaluate the sine function using sin(x)=y/r

sin(x)=−8/17

5. Evaluate the cosine function using cos(x)=x/r

cos(x)=−15/17

6. Evaluate the cosecant function by taking the reciprocal of sine.

csc(x)=−17/8

7. Evaluate the secant function by taking the reciprocal of cosine.

sec(x)=−17/15

8. Evaluate the cotangent function by taking the reciprocal of tangent.

cot(x)=15/8

Final Answer

sin(x)=−8/17,cos(x)=−15/17,csc(x)=−17/8,sec(x)=−17/15,cot(x)=15/8

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