Final answer:
To find the remaining trigonometric ratios when cotθ = 3 and θ < π/2, we can determine the values of sinθ, cosθ, and tanθ. By taking the reciprocal of cotθ, we find tanθ = 1/3. The Pythagorean identity allows us to find sinθ, giving us sinθ = 1/√10. Using the trigonometric identity tanθ = sinθ/cosθ, we can find cosθ = (3√10)/10.
Step-by-step explanation:
To find the remaining trigonometric ratios, we need to determine the values of sinθ, cosθ, and tanθ. Since cotθ is the reciprocal of tanθ, we can find the value of tanθ by taking the reciprocal of cotθ. So, tanθ = 1/3.
Next, we can use the Pythagorean identity to find the value of sinθ. Since cotθ = adj/opp, we can write cotθ = cosθ/sinθ. Squaring both sides of the equation, we get cos^2θ = sin^2θ. Since θ is less than π/2, sinθ will be positive. Therefore, we can take the square root of both sides to find sinθ = √(1/(1+3^2)) = √(1/10) = 1/√10.
Lastly, we can use the trigonometric identity tanθ = sinθ/cosθ to find cosθ. Rearranging the equation, we get cosθ = sinθ/tanθ = (1/√10)/(1/3) = 3/√10 = (3√10)/10.