Find the Other Trig Values in Quadrant II tan(theta)=-1

Problem

tan(θ)=−1,θ∈Quadrant II

Solution

1. Identify the given information and the constraints of the quadrant. In Quadrant II, the xcoordinate is negative and the ycoordinate is positive. This means sin(θ) and csc(θ) are positive, while cos(θ) sec(θ) and cot(θ) are negative.

2. Determine the values of x and y using the definition tan(θ)=y/x Since tan(θ)=−1 and we are in Quadrant II, we can set y=1 and x=−1

3. Calculate the radius r using the Pythagorean identity r=√(,x2+y2)

r=√(,(−1)2+(1)2)

r=√(,2)

4. Find the sine value using sin(θ)=y/r

sin(θ)=1/√(,2)

sin(θ)=√(,2)/2

5. Find the cosine value using cos(θ)=x/r

cos(θ)=(−1)/√(,2)

cos(θ)=−√(,2)/2

6. Find the cotangent value using cot(θ)=1/tan(θ)

cot(θ)=1/(−1)

cot(θ)=−1

7. Find the cosecant value using csc(θ)=1/sin(θ)

csc(θ)=√(,2)/1

csc(θ)=√(,2)

8. Find the secant value using sec(θ)=1/cos(θ)

sec(θ)=√(,2)/(−1)

sec(θ)=−√(,2)

Final Answer

sin(θ)=√(,2)/2,cos(θ)=−√(,2)/2,cot(θ)=−1,csc(θ)=√(,2),sec(θ)=−√(,2)

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