Master the Unit Circle, explore trigonometric ratios, and find exact values for special angles.
Trigonometry allows us to connect angles to lengths. It is the foundation of everything from building roofs (triangles) to analyzing sound waves (circles).
This Trigonometry Studio brings the abstract formulas to life. drag the point on the Unit Circle to see how Sine (Height), Cosine (Width), and Tangent (Slope) evolve as the angle rotates.
The classic mnemonic for right triangles. If you have an angle θ:
In the Unit Circle, the Hypotenuse is always 1. So Sin becomes just the Opposite (y) and Cos becomes just the Adjacent (x).
This rule tells you which functions are Positive in each quadrant:
By forcing the radius to be exactly 1, we simplify the math drastically. The length of the hypotenuse is 1, so dividing by the hypotenuse disappears.
This transforms trigonometry from "Triangle Math" into "Circle Math", allowing us to define sin/cos for angles bigger than 90° (which is impossible in a right triangle).
Every primary function has a "reciprocal" partner (1 divided by the value).
A radian is a measure of angle based on the radius. One radian is the angle formed when the arc length is equal to the radius. There are exactly 2π radians (approx 6.28) in a full circle (360°). 1 radian ≈ 57.3°.
Tan(θ) = Sin(θ) / Cos(θ). At 90°, Cos(90°) = 0. Dividing by zero is impossible (undefined). Geometrically, the tangent line becomes parallel to the y-axis and never intersects it.
Use the 'Hand Trick' or remember the pattern for sine: √0/2, √1/2, √2/2, √3/2, √4/2. This simplifies to 0, 0.5, 0.707, 0.866, 1.
360° or 2π radians. This means the values repeat exactly every full rotation. Sin(30°) is the same as Sin(390°).
No (for real arguments). The hypotenuse is the longest side of a right triangle. Since Sine = Opposite/Hypotenuse, the ratio can never exceed 1. (It can be greater than 1 in complex number theory, but not in basic trig).
Sin²(x) means (Sin(x))² — you find the sine first, then square the result. Sin(x²) means you square the angle first, then find the Sine. They are completely different values.
The most famous one is sin²(θ) + cos²(θ) = 1. This is literally the Pythagorean Theorem (a² + b² = c²) applied to the Unit Circle where the hypotenuse (c) is 1.
Most calculators don't have a 'csc' button. You compute Sin(x) first, and then press the '1/x' (reciprocal) button.
Because they are defined by the coordinates of a point moving around a circle. X is Cosine, Y is Sine.
The amplitude is the height of the wave from the center line to the peak. For y = sin(x), the amplitude is 1. For y = 5sin(x), the amplitude is 5.