Step-by-step explanation:
1.
the length of the terminal side from (0, 0) to (4, -7) is the radius of the circle, while
x = 4 = cos(theta)×radius
y = -7 = sin(theta)×radius
so,
sin(theta) = -7/radius
radius² = (4 - 0)² + (-7 - 0)² = 4² + 7² = 16 + 49 = 65
radius = sqrt(65)
sin(theta) = -7/ sqrt(65) = -0.868243142...
FYI
theta = -60.2551187...°
or 299.7448813...°
2.
tan(theta) = 3/2
so,
sin(theta)/cos(theta) = 3/2
2×sin(theta) = 3×cos(theta)
square both sides
4×sin²(theta) = 9×cos²(theta)
completing the square
4×sin²(theta) + 9×sin²(theta) = 9×cos²(theta) + 9×sin²(theta)
13×sin²(theta) = 9×(cos²(theta) + sin²(theta)) = 9
sin²(theta) = 9/13
sin(theta) = 3/sqrt(13)
to make it a rational denominator we multiply by
sqrt(13)/ sqrt(13)
and we get
sin(theta) = 3×sqrt(13)/13
3.
because the sum of all angles in a triangle is 180°.
one angle is theta.
then we have the 90° angle at the origin (or center of the circle).
and that leaves 180 - 90 - theta = 90 - theta for the third angle.
this triangle can be flipped with putting 90 - theta in place of theta.
the flipped triangle is equivalent to the original triangle (all angles and side lengths are identical between the 2 triangles).
the baseline (radius) is the same length in both triangles.
in that flipped triangle the side formerly sin(theta) is now cos(90-theta) and cos(theta) is now sin(90-theta).
