Find the Other Trig Values in Quadrant I cot(theta)=1

Problem

cot(θ)=1,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since cot(θ)=1 and θ is in Quadrant I, both sin(θ) and cos(θ) must be positive.

2. Use the definition of the cotangent function. We know that cot(θ)=cos(θ)/sin(θ)=1 which implies cos(θ)=sin(θ)

3. Apply the Pythagorean identity sin2(θ)+cos2(θ)=1 Substituting cos(θ)=sin(θ) gives:

sin2(θ)+sin2(θ)=1

2*sin2(θ)=1

sin2(θ)=1/2

4. Solve for the sine value. Taking the square root and choosing the positive value for Quadrant I:

sin(θ)=1/√(,2)=√(,2)/2

5. Determine the cosine value. Since cos(θ)=sin(θ)

cos(θ)=√(,2)/2

6. Calculate the tangent value using the reciprocal identity tan(θ)=1/cot(θ)

tan(θ)=1/1=1

7. Calculate the cosecant value using the reciprocal identity csc(θ)=1/sin(θ)

csc(θ)=1/√(,2)/2=√(,2)

8. Calculate the secant value using the reciprocal identity sec(θ)=1/cos(θ)

sec(θ)=1/√(,2)/2=√(,2)

Final Answer

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

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