Trigonometry

Trigonometry begins with the right triangle. Given a right triangle with angle theta, three fundamental ratios are defined. Sine (abbreviated sin) is the ratio of the side opposite theta to the hypotenuse. Cosine (cos) is the ratio of the side adjacent to theta to the hypotenuse. Tangent (tan) is the ratio of the opposite side to the adjacent side. The mnemonic SOH CAH TOA helps remember: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. These ratios depend only on the angle, not on the size of the triangle — all similar right triangles with the same angle theta have the same trig ratios.

The unit circle is a circle of radius one centered at the origin. For any angle theta measured counterclockwise from the positive x-axis, the point where the angle ray intersects the circle has coordinates (cosine of theta, sine of theta). This elegant fact connects trigonometry to coordinate geometry. As theta increases past ninety degrees, the trig values extend beyond acute angles to all real numbers. Sine and cosine repeat every full rotation, which is two pi radians or 360 degrees — this is called periodicity. The unit circle reveals that sine squared plus cosine squared always equals one, because any point on the circle satisfies the equation x squared plus y squared equals one.

Trigonometric identities are equations that are true for all angle values. The angle sum identities express the sine or cosine of a sum in terms of sines and cosines of the individual angles. The double angle identities are special cases where both angles are equal. The Law of Sines relates side lengths to the sines of opposite angles in any triangle. The Law of Cosines generalizes the Pythagorean theorem to non-right triangles. These identities underpin everything from solving triangles to modeling waves in physics and engineering.

Trigonometry extends far beyond triangles. It is the language of periodic phenomena — anything that repeats regularly. Sound waves, light waves, alternating current, planetary orbits, and pendulums are all modeled with sine and cosine functions. In engineering, trigonometric functions describe vibrations, rotations, and harmonic motion. In computer graphics, rotation matrices use sine and cosine to rotate objects in 2D and 3D space. In GPS and navigation, triangulation (using the Law of Sines and Law of Cosines) determines positions from known reference points.

Drag the angle slider to see sine (vertical) and cosine (horizontal) change in real time. The point on the circle moves as the angle changes.