Find the Other Trig Values in Quadrant I cot(theta)=3

Problem

cot(θ)=3,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. Since cot(θ)=3 and θ is in Quadrant I, all trigonometric functions will be positive.

2. Express the cotangent in terms of x and y using the definition cot(θ)=x/y We can let x=3 and y=1

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

r=√(,3+1)

r=√(,9+1)

r=√(,10)

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

sin(θ)=1/√(,10)=√(,10)/10

csc(θ)=√(,10)/1=√(,10)

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

cos(θ)=3/√(,10)=(3√(,10))/10

sec(θ)=√(,10)/3

6. Determine the tangent value using tan(θ)=y/x or the reciprocal of cotangent.

tan(θ)=1/3

Final Answer

sin(θ)=√(,10)/10,cos(θ)=(3√(,10))/10,tan(θ)=1/3,csc(θ)=√(,10),sec(θ)=√(,10)/3

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