Find the value of sin if tan = 5/4

Answer 1

To find the value of sin, we need to use the relationship between sine and tangent in a right triangle:

sin(angle) = opposite/hypotenuse

Given that tan(angle) = 5/4, we can create a right triangle to find the values of the opposite and hypotenuse sides.

Let's assume the angle is θ. tan(θ) = opposite/adjacent = 5/4.

We can choose any values for the opposite and adjacent sides that satisfy this ratio. For simplicity, let's use 5 for the opposite side and 4 for the adjacent side.

Now we can use the Pythagorean theorem to find the hypotenuse:

hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 5^2 + 4^2
hypotenuse^2 = 25 + 16
hypotenuse^2 = 41

Taking the square root of both sides:
hypotenuse ≈ √41

Now we can find sin(θ):

sin(θ) = opposite/hypotenuse
sin(θ) = 5/√41

To simplify the expression, we multiply both the numerator and denominator by √41:

sin(θ) = (5/√41) * (√41/√41)
sin(θ) = (5√41)/41

Therefore, sin(θ) ≈ (5√41)/41.