Find the Other Trig Values in Quadrant II tan(theta)=- square root of 3

Problem

tan(θ)=−√(,3),θ∈Quadrant II

Solution

1. Identify the given information and the quadrant. In Quadrant II, x is negative and y is positive.

2. Determine the reference angle. Since tan(θ)=−√(,3) the reference angle (θ_ref) is 60^∘ or π/3 because tan(60^∘)=√(,3)

3. Define the ratio for tangent. Since tan(θ)=y/x we can set y=√(,3) and x=−1 to satisfy the quadrant conditions.

4. Calculate the radius r using the Pythagorean identity r=√(,x^2+y^2)

r=√(,(−1)^2+(√(,3))^2)

r=√(,1+3)

r=2

5. Evaluate the sine function using sin(θ)=y/r

sin(θ)=√(,3)/2

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

cos(θ)=−1/2

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

csc(θ)=2/√(,3)

csc(θ)=(2√(,3))/3

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

sec(θ)=−2

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

cot(θ)=−1/√(,3)

cot(θ)=−√(,3)/3

Final Answer

sin(θ)=√(,3)/2,cos(θ)=−1/2,csc(θ)=(2√(,3))/3,sec(θ)=−2,cot(θ)=−√(,3)/3

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