Trigonometry Review

A structured review of angle measure, unit-circle definitions, graphs, identities, equations, triangles, and verification.

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Trigonometric functions connect angle, unit-circle coordinates, and right-triangle ratios. A radius at angle theta meets the unit circle at a labelled cosine-sine coordinate. Its horizontal and vertical projections form a reference triangle that links circular and triangle definitions.

Precise definition

Trigonometry relates angles, rotation, and ratios. On the unit circle, $\cos\theta$ is the $x$-coordinate, $\sin\theta$ the $y$-coordinate, and $\tan\theta=\sin\theta/\cos\theta$ where $\cos\theta\ne0$. Radians measure arc length divided by radius.

Notation and mathematical language

Core identities include $\sin^2\theta+\cos^2\theta=1$, reciprocal and quotient identities, and angle-sum formulas. Sine and cosine have period $2\pi$; tangent has period $\pi$. In right triangles, ratios use acute reference angles and labelled sides.

Conceptual picture

The unit circle unifies triangle ratios with all real angles and determines quadrant signs. Graph transformations encode amplitude, period, phase shift, and vertical shift. Exact values expose structure that decimals conceal.

Fully worked example

Interpretation and application

Trigonometry models waves, rotation, surveying, vectors, and periodic data. A sinusoidal model can predict within an observed regime, but a periodic correlation does not prove a causal mechanism.