Trigonometric Functions and Applets

Trigonometry studies the relationships between the angles and side lengths of triangles, and more generally the periodic functions that arise from rotation around a circle. The three primary functions — sine, cosine and tangent — can be defined as ratios of the sides of a right triangle and extended to all angles using the unit circle, a circle of radius one centred at the origin. For a point on this circle, the cosine gives its horizontal coordinate and the sine its vertical coordinate, so as the angle increases the coordinates trace out the familiar wave shapes.

Interactive applets make the connection between the rotating radius and the resulting graphs explicit. Dragging a point around the unit circle while the corresponding sine or cosine value is plotted against the angle shows how the wave is generated, why its period is a full turn, and how amplitude and phase shifts transform the curve. The tangent, defined as sine divided by cosine, has its own distinctive graph with vertical asymptotes where the cosine is zero.

The functions satisfy many identities, such as the Pythagorean identity relating the squares of sine and cosine, and the angle-sum and double-angle formulas. These identities are essential for simplifying expressions and for solving trigonometric equations, where the periodicity of the functions means a single equation usually has infinitely many solutions. Equation-solving applets let users see all the angles that satisfy a given relation marked on the circle or the graph, reinforcing why solutions repeat at regular intervals.

Frequently asked questions

How does the unit circle define sine and cosine?

For a point on a unit-radius circle at a given angle, the cosine is its horizontal coordinate and the sine its vertical coordinate, which extends the definitions to all angles.

Why do trigonometric equations have many solutions?

Because sine, cosine and tangent are periodic, any angle that satisfies an equation does so again after each full period, giving infinitely many solutions.

What is the Pythagorean identity?

It states that the square of the sine of an angle plus the square of its cosine equals one, a direct consequence of the unit circle.