Trigonometry

Definitions and Key Concepts in Trigonometry

  • Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
  • A right-angled triangle consists of two shorter sides (the adjacent and opposite sides) and the longest side (the hypotenuse).
  • The three primary trigonometric functions are the sine (sin), cosine (cos), and tangent (tan) functions.
  • The sine of an angle in a right-angled triangle is equal to the length of the opposite side divided by the length of the hypotenuse.
  • The cosine of an angle in a right-angled triangle is equal to the length of the adjacent side divided by the length of the hypotenuse.
  • The tangent of an angle in a right-angled triangle is equal to the length of the opposite side divided by the length of the adjacent side.
  • The acronyms SOH (Sine equals Opposite over Hypotenuse), CAH (Cosine equals Adjacent over Hypotenuse), and TOA (Tangent equals Opposite over Adjacent) are useful in remembering these relationships.

Trigonometric Functions and Their Graphs

  • The unit circle is a powerful tool in understanding the graphs of sine, cosine, and tangent functions.
  • The sin function has a wave-like shape, peaking at 1 and dipping to -1. It starts at 0 when the angle is 0.
  • The cos function also has a wave-like shape but starts at 1 when the angle is 0.
  • The tan function has vertical asymptotes where the function is undefined, that is at odd multiples of 90 degrees.

Trigonometric Identities and Equations

  • Trigonometric identities are equations involving trigonometric functions that hold true for all angles.
  • The Pythagorean identity is one of the most fundamental and states that in any right-angled triangle, (sin x)² + (cos x)² = 1.
  • The double-angle formulas express the trigonometric functions of double angles in terms of the single angle measure.
  • To solve trigonometric equations, you identify the angle measures that make the equation true.

Applications of Trigonometry

  • Trigonometry has a multitude of applications in different fields. For example, it’s fundamental in physics for resolving force vectors, modelling periodic phenomena, and analyzing wave motion.
  • In architecture and engineering, trigonometry is essential for calculating structural load and bridge designs.
  • Trigonometry is important in geology for calculating plate tectonic movement and the behaviour of waves.
  • It’s also crucial in computer science for graphics, virtual reality, and game development.
  • Understanding and performing trigonometric computations is a critical skill for further study in science and mathematics at higher education levels.