Trigonometric Functions: Right Triangle Approach

Definition

In the context of acute angles, trigonometric functions are defined as ratios of the sides of a right triangle. For an angle $\theta$ (or $\alpha, \beta$ ):

Sine Ratio: $\sin \theta = \frac{\text{opposite leg}}{\text{hypotenuse}}$ $sin θ = \frac{opposite leg}{hypotenuse}$
- How to read: “Sine theta equals opposite over hypotenuse.”
- Meaning: Vertical rise relative to the longest side—how “tall” the angle makes the triangle.
Cosine Ratio: $\cos \theta = \frac{\text{adjacent leg}}{\text{hypotenuse}}$ $cos θ = \frac{adjacent leg}{hypotenuse}$
- How to read: “Cosine theta equals adjacent over hypotenuse.”
- Meaning: Horizontal run relative to the hypotenuse—how “wide” the angle opens.
Tangent Ratio: $\tan \theta = \frac{\text{opposite leg}}{\text{adjacent leg}}$ $tan θ = \frac{opposite leg}{adjacent leg}$
- How to read: “Tangent theta equals opposite over adjacent.”
- Meaning: Slope of the terminal side—rise over run within the right triangle.

Why It Matters

The right-triangle approach is the ‘entry point’ for spatial reasoning. It provides the simple ratios (SOH CAH TOA) that allow anyone to solve for distances and angles in everyday life, from carpentry to basic surveying.

Core Concepts

SOH-CAH-TOA: The mnemonic for Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent).
Geometric Foundation: Trigonometry is built on the property of Similar Triangles. Because similar triangles have proportional sides (CSSTP), the ratio of any two sides depends only on the interior angles, not the absolute size of the triangle.
Uniqueness of Ratios: For a given acute angle measure, the numerical value of the trigonometric ratio is unique. Even if the side lengths change, the ratio remains constant as long as the angle measure is preserved.
Boundary Definitions:
- $\sin 0^\circ = 0$ $sin 0^{\circ} = 0$ , $\sin 90^\circ = 1$ $sin 9 0^{\circ} = 1$
  - How to read: “Sine of zero is zero; sine of ninety is one.”
  - Meaning: At $0^\circ$ the opposite side vanishes; at $90^\circ$ it equals the hypotenuse.
- $\cos 0^\circ = 1$ $cos 0^{\circ} = 1$ , $\cos 90^\circ = 0$ $cos 9 0^{\circ} = 0$
  - How to read: “Cosine of zero is one; cosine of ninety is zero.”
  - Meaning: At $0^\circ$ adjacent equals hypotenuse; at $90^\circ$ the adjacent side vanishes.
- $\tan 0^\circ = 0$ $tan 0^{\circ} = 0$ , $\tan 90^\circ = \text{undefined}$ $tan 9 0^{\circ} = undefined$
  - How to read: “Tangent of zero is zero; tangent of ninety is undefined.”
  - Meaning: At $90^\circ$ the adjacent side is zero, so division by zero makes tangent undefined.
Notation Convention: Measures are frequently designated by Greek letters: $\alpha$ (alpha), $\beta$ (beta), $\gamma$ (gamma), and $\theta$ (theta). Standard labeling uses $a, b, c$ for sides opposite vertices $A, B, C$ .
Complementary Angles (Cofunctions): $\sin A = \cos B$ $sin A = cos B$ and $\cos A = \sin B$ $cos A = sin B$ where $A + B = 90^\circ$ $A + B = 9 0^{\circ}$ . Also, $\tan A = \frac{1}{\tan B}$ $tan A = \frac{1}{t a n B}$ .
- How to read: “Sine A equals cosine B; A plus B equals ninety degrees; tangent A equals one over tangent B.”
- Meaning: In a right triangle, an acute angle and its complement swap opposite/adjacent roles, so sine and cosine are cofunctions.

Special Angle Ratios

Angle ( $\theta$ )	$\sin \theta$	$\cos \theta$	$\tan \theta$
30°	$1/2$	$\sqrt{3}/2$	$\sqrt{3}/3$
45°	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
60°	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$

Trigonometric Functions: Right Triangle Approach

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes