If tanθ = 2, find the values of other trigonometric ratios.

We know that:

tan θ = opposite / adjacent

Therefore, let’s draw a right triangle where the angle θ is one of the acute angles, and the opposite side is x and the adjacent side is y. Then we can use the Pythagorean theorem to find the hypotenuse h:

h² = x² + y²

We can also use the definition of the tangent function to write:

tan θ = x / y = 2

Solving for x and y, we get:

x = 2y

Substituting into the Pythagorean theorem, we get:

h² = (2y)² + y² = 5y²

h = √(5y²) = y√5

Now we can find the values of the other trigonometric ratios:

sin θ = opposite / hypotenuse = x / h = 2y / (y√5) = 2/√5

cos θ = adjacent / hypotenuse = y / h = y / (y√5) = 1/√5

csc θ = 1 / sin θ = √5 / 2

sec θ = 1 / cos θ = √5

cot θ = 1 / tan θ = 1/2

Therefore, the values of the other trigonometric ratios are:

sin θ = 2/√5

cos θ = 1/√5

csc θ = √5/2

sec θ = √5

cot θ = 1/2