Find the Other Trig Values in Quadrant II tan(x)=-5/4

Problem

tan(x)=−5/4, Quadrant II

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant II. In Quadrant II, sin(x) and csc(x) are positive, while cos(x) sec(x) tan(x) and cot(x) are negative.

2. Define the relationship between the tangent function and the sides of a right triangle. Since tan(x)=opposite/adjacent we let the opposite side y=5 and the adjacent side x=−4 to satisfy the conditions of Quadrant II.

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

r=√(,(−4)2+5)

r=√(,16+25)

r=√(,41)

4. Determine the sine and cosecant values using sin(x)=y/r and csc(x)=r/y

sin(x)=5/√(,41)=(5√(,41))/41

csc(x)=√(,41)/5

5. Determine the cosine and secant values using cos(x)=x/r and sec(x)=r/x

cos(x)=(−4)/√(,41)=−(4√(,41))/41

sec(x)=−√(,41)/4

6. Determine the cotangent value using cot(x)=1/tan(x)

cot(x)=−4/5

Final Answer

sin(x)=(5√(,41))/41,cos(x)=−(4√(,41))/41,cot(x)=−4/5,sec(x)=−√(,41)/4,csc(x)=√(,41)/5

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