Trigonometry Fundamentals: Angles and Functions

Trigonometry is the branch of mathematics that explores the relationships between angles and sides in triangles. It’s a critical tool in fields like navigation, engineering, physics, and even astronomy, helping us measure distances and model periodic phenomena.

Basic Trigonometric Functions

In a right triangle, the three primary trigonometric functions relate the angles to the ratios of the sides:

  • Sine (\( \sin(\theta) \)): \( \frac{\text{opposite}}{\text{hypotenuse}} \) - measures the vertical component.
  • Cosine (\( \cos(\theta) \)): \( \frac{\text{adjacent}}{\text{hypotenuse}} \) - measures the horizontal component.
  • Tangent (\( \tan(\theta) \)): \( \frac{\text{opposite}}{\text{adjacent}} \) - the ratio of sine to cosine.

These functions are often remembered with the mnemonic SOH-CAH-TOA.

Example: The 3-4-5 Right Triangle

Consider a right triangle with legs of lengths 3 and 4, and a hypotenuse of 5 (a Pythagorean triple). For the angle \( \theta \) opposite the side of length 3:

\[ \sin(\theta) = \frac{3}{5} = 0.6 \]
\[ \cos(\theta) = \frac{4}{5} = 0.8 \]
\[ \tan(\theta) = \frac{3}{4} = 0.75 \]

These values can be verified using a calculator: \( \sin^{-1}(0.6) \approx 36.87^\circ \), which is the angle \( \theta \).

Graphical Representation

The sine function is periodic, oscillating between -1 and 1. Here’s a graph of \( y = \sin(x) \) over \( x \) from -5 to 5 radians:

Applications of Trigonometry

Trigonometry has vast real-world uses:

  • Navigation: Calculating distances and angles for ships or planes using triangulation.
  • Engineering: Designing bridges or buildings by analyzing forces and angles.
  • Physics: Modeling waves (sound, light) with sine and cosine functions.

For example, the height of a building can be found using the angle of elevation and the distance from its base: \( h = d \cdot \tan(\theta) \).