Basics of Trigonometry
Introduction to right triangles, ratios, and terminology.
Trigonometry studies the relationships between the side lengths and angles of triangles. In a Right-Angled Triangle (a triangle with one 90° angle), the sides relative to a reference angle θ (Theta) are named:
- Hypotenuse: The longest side opposite the 90° angle.
- Opposite: The side directly across from the angle θ.
- Adjacent: The side next to the angle θ that is not the hypotenuse.
Sine (sin θ) Simulator
Observe how vertical coordinate heights map directly to the sine waveform.
In a unit circle (radius = 1), the Sine of an angle represents the vertical component of the intersecting coordinate. As the angle θ rotates from 0° to 360°, the value of sin θ oscillates between -1.0 and +1.0, creating a continuous sine wave.
Cosine (cos θ) Simulator
Observe how horizontal coordinate widths map directly to the cosine waveform.
The Cosine of an angle in the unit circle represents the horizontal component of the intersecting coordinate. It starts at +1.0 at 0°, drops to 0 at 90°, reaches -1.0 at 180°, and climbs back to +1.0 at 360°. The cosine wave is 90° out of phase with the sine wave.
Tangent (tan θ) Simulator
Explore ratios of Sine to Cosine and trace the vertical asymptotes.
The Tangent (tan θ) is defined as the ratio of Sine to Cosine (y/x). Geometrically, it is the distance along the vertical line tangent to the circle at (1, 0) up to the extended radius line. Tangent approaches positive infinity at 90° and negative infinity at 270°, resulting in vertical asymptotes.
Reciprocal Ratios (Cosec, Sec, Cot)
Visualize reciprocal functions which are the inverted versions of standard ratios.
The reciprocal functions represent the ratios flipped upside down:
- Cosecant (csc θ) = 1 / sin θ (Hypotenuse / Opposite)
- Secant (sec θ) = 1 / cos θ (Hypotenuse / Adjacent)
- Cotangent (cot θ) = 1 / tan θ (Adjacent / Opposite)
Trigonometric Identities
Interactive mathematical proofs demonstrating algebraic invariants.
Why are they constant?
These equations are derived from the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to triangles in the coordinate grid. Since the radius is always 1, the triangle formed has legs of length $x$ (cosine) and $y$ (sine), guaranteeing that $x^2 + y^2 = 1^2$ for any angle.
Finding Height & Distance
Solve practical real-world problems using trigonometry calculations.
By measuring the Angle of Elevation (θ) from a observer to the top of an object, and knowing the Distance (d) to the base, the object's Height (h) can be found using the Tangent formula:
=> Height = Distance * tan(θ)