Unit Circle & Trig Graphs
Walk a point around a circle of radius 1. Its height is sin θ and its sideways position is cos θ: unroll them and you get the sine and cosine waves.
One circle, every angle
Right-angled triangles only define sin, cos and tan up to 90°. The unit circle extends them to every angle: for the point at angle θ, cos θ = x and sin θ = y, and tan θ = y/x is the slope of the radius. Since x² + y² = 1, you get sin²θ + cos²θ = 1 for free.
Signs, symmetry and radians
The signs depend on the quadrant: All positive in the first, Sine in the second, Tan in the third, Cos in the fourth ("All Students Take Chemistry"). Mirror symmetry gives sin(180° − θ) = sin θ. A radian is the angle whose arc is one radius long, so a full turn is 2π rad = 360°.
θ30°
sin θ1/2
cos θ√3/2
tan θ1/√3
QuadrantI (all +)
Checksin² + cos² = 1