Sine and cosine on the unit circle
The two coordinates of one point going round a circle.
Draw a circle of radius 1 around the origin and put a point on it. Turn the point by an angle θ, counted counter-clockwise from the positive x-axis.
The point's x-coordinate is cos θ and its y-coordinate is sin θ. That is the definition, and it works for every angle, not only for the acute ones of a right triangle.
Everything else follows from the picture. The point is always at distance 1 from the centre, so cos²θ + sin²θ = 1. Going once round brings it back, so both functions repeat every 360°.
The signs come from the quadrant: above the x-axis the sine is positive, right of the y-axis the cosine is. An angle and its mirror image across the y-axis, θ and 180° − θ, have the same sine.
In radians a full turn is 2π, the length of the circle's edge. An angle in radians is simply the length of the arc the point has travelled.
Answer it first
Without a calculator: what is sin 150°?
Show the answer
0.5. 150° is the mirror image of 30° across the y-axis, so the point is at the same height, and sin 30° = 0.5.
Reading is not learning it
Leo teaches this at your pace, asks for your attempt before he shows his, and checks a week later what stayed.