Question

Angle θ is an acute angle of a right triangle. Determine the value of cot θ given the value of sin θ below.

Answer 1

To determine the value of cot θ, we need to find the values of the adjacent and opposite sides of the triangle.

We know that sin θ = opposite/hypotenuse.

Since θ is an acute angle of a right triangle, the hypotenuse is the longest side, and the opposite side is the side opposite to angle θ. Therefore, we can represent the opposite side as x and the hypotenuse as 1. So, we have:

sin θ = x/1

Simplifying the equation, we have:

sin θ = x

To find the value of cot θ, we need to find the adjacent side. We can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides:

(adjacent side)^2 + (opposite side)^2 = (hypotenuse)^2

Let the adjacent side be represented by y. Substituting the values we have:

y^2 + x^2 = 1^2

y^2 + x^2 = 1

Solving for y, we have:

y = sqrt(1 - x^2)

Now, we can find the value of cot θ:

cot θ = adjacent/opposite

cot θ = y/x

Substituting the values we have:

cot θ = sqrt(1 - x^2)/x

Therefore, the value of cot θ given the value of sin θ is sqrt(1 - x^2)/x.