Trigonometry Fundamentals: A Comprehensive Guide

Trigonometry explores the relationships between angles and sides in triangles, extending to periodic phenomena like waves and oscillations. Originating from Greek words for "triangle" and "measure," it’s foundational in mathematics, physics, and engineering. This MathMultiverse guide covers sine, cosine, tangent functions, their applications, and visualizations.

Basic Trigonometric Functions

In a right triangle with angle \( \theta \), trigonometric functions are defined as ratios:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{opposite}}{\text{adjacent}} \]

Reciprocal functions include:

\[ \csc(\theta) = \frac{1}{\sin(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)} \]

Use the mnemonic SOH-CAH-TOA to recall definitions. On the unit circle (radius 1), \( \sin(\theta) \) and \( \cos(\theta) \) represent the y- and x-coordinates of a point at angle \( \theta \), with periodicity \( 2\pi \). Key identity:

\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]

These functions model periodic behavior and solve geometric problems.

Examples

Right Triangle (3-4-5)

For angle \( \theta \) opposite side 3:

\[ \sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4} \]
\[ \theta \approx \sin^{-1}(0.6) \approx 36.87^\circ \]

Angle of Elevation

Tree height, angle \( 30^\circ \), distance 50 m:

\[ \tan(30^\circ) = \frac{\text{height}}{50} \]
\[ \text{height} \approx 50 \times \tan(30^\circ) \approx 50 \times 0.577 \approx 28.85 \, \text{m} \]

Law of Sines

Triangle with \( \angle A = 40^\circ \), \( \angle B = 60^\circ \), side \( a = 10 \):

\[ \angle C = 180^\circ - 40^\circ - 60^\circ = 80^\circ \]
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \]
\[ b = \frac{10 \times \sin(60^\circ)}{\sin(40^\circ)} \approx 13.47 \]

Identities

Verify \( \sin^2(\theta) + \cos^2(\theta) = 1 \) at \( \theta = 45^\circ \):

\[ \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \]
\[ \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = 0.5 + 0.5 = 1 \]

Periodic Motion

Pendulum displacement: \( y = 5 \sin(2t) \), at \( t = \frac{\pi}{4} \):

\[ y = 5 \sin\left(2 \cdot \frac{\pi}{4}\right) = 5 \sin\left(\frac{\pi}{2}\right) = 5 \]
\[ \text{Amplitude} = 5, \quad \text{Period} = \frac{2\pi}{2} = \pi \]

Graphical Representation

Sine and Cosine Functions

Graphs of \( y = \sin(x) \) and \( y = \cos(x) \) over \( [-5, 5] \).

Applications

  • Navigation: Ship distance, bearings \( 30^\circ \), \( 120^\circ \):
    • \[ \text{Distance} \approx 180.28 \, \text{km} \]
  • Engineering: Bridge height, \( \angle 60^\circ \), base 20 m:
    • \[ \text{Height} \approx 20 \times \tan(60^\circ) \approx 34.64 \, \text{m} \]
  • Physics: Wave frequency \( y = 3 \sin(4\pi t) \):
    • \[ f = \frac{4\pi}{2\pi} = 2 \, \text{Hz} \]