What is the value of sin 0 given that (-3,7) is a point on the terminal side 0

To find the value of sinθ for the angle θ whose terminal side passes through a point (-3,7), we need to use the definition of sine in the context of a right-angled triangle within the Cartesian coordinate system.

The sine of an angle in the unit circle (where the hypotenuse, r, is of length 1) is defined as the ratio of the opposite side (y-coordinate) to the hypotenuse (r). In the general case where the hypotenuse is not of length 1, the sine is defined as:

sinθ = y / r

Here, the point (-3,7) determines a right-angled triangle with the x-axis, where the opposite side of angle θ is 7 (the y-coordinate of the point), and the 'adjacent' side is -3 (the x-coordinate, which signifies the horizontal distance from the origin). The hypotenuse, r, is the distance from the origin to the point (-3,7), which can be found using the Pythagorean theorem:

r = √(x^2 + y^2) r = √((-3)^2 + 7^2) r = √(9 + 49) r = √58

Now, we can calculate sinθ as follows:

sinθ = y / r sinθ = 7 / √58

Since we are interested in the ratio, and not its simplified version, we can stop here. The value of sinθ given the point (-3,7) on the terminal side of θ is 7/√58