Find the value of sin if tan = 5/4
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.