Trigonometry: Ratios & the Unit Circle
Trigonometry connects angles to ratios of sides. From right triangles to the unit circle, these tools are essential for physics, engineering, and advanced math.
Key Concepts
SOH-CAH-TOA
In a right triangle: sin(θ) = Opposite/Hypotenuse, cos(θ) = Adjacent/Hypotenuse, tan(θ) = Opposite/Adjacent. Remember: SOH-CAH-TOA.
Special Angles
30°: sin=½, cos=√3/2. 45°: sin=cos=√2/2. 60°: sin=√3/2, cos=½. 90°: sin=1, cos=0. These values appear constantly in math and physics.
The Unit Circle
A circle with radius 1 centered at the origin. Any point on the circle is (cos θ, sin θ) where θ is the angle from positive x-axis. sin²θ + cos²θ = 1 always.
Inverse Trig & Applications
arcsin, arccos, arctan find angles from ratios. Used to find unknown angles in triangles. Law of Sines: a/sin A = b/sin B = c/sin C works for any triangle.
Live Python Practice
Interactive Lab
Check Your Understanding
In SOH-CAH-TOA, cos(θ) equals:
On the unit circle, every point is:
sin²θ + cos²θ always equals: