To find the value of sin(8) using the point (3, 4) on its terminal side, we first need to determine the hypotenuse of the right triangle formed by this point.
Using the Pythagorean theorem, we can calculate the hypotenuse (r):
r = √(x² + y²)
Substituting the coordinates:
r = √(3² + 4²) = √(9 + 16) = √25 = 5
Now, the sine function is defined as the ratio of the opposite side to the hypotenuse:
sin(θ) = opposite/hypotenuse
In our triangle, the opposite side corresponds to the y-coordinate (4), and the hypotenuse we’ve calculated is 5:
sin(θ) = 4/5
Thus, the value of sin(8) is:
sin(8) = 0.8
This means that given the point (3, 4), the sine of the angle whose terminal side passes through this point is 0.8.