5.9 Chapter summary

5.9 Chapter summary (EMA3Y)

Presentation: 2FT2

• We can define three trigonometric ratios for right-angled triangles: sine (\(\sin\)), cosine (\(\cos\)) and tangent (\(\tan\))

  These ratios can be defined as:

  • \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}\)
  • \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}\)
  • \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}\)

• Each of these ratios have a reciprocal: cosecant (\(\text{cosec }\)), secant (\(\sec\)) and cotangent (\(\cot\)).

  These ratios can be defined as:

  • \(\text{cosec } \theta = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{r}{y}\)
  • \(\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{r}{x}\)
  • \(\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{x}{y}\)

• We can use the principles of solving equations and the trigonometric ratios to help us solve simple trigonometric equations.

• For some special angles (0°, 30°, 45°, 60° and 90°), we can easily find the values of \(\sin\), \(\cos\) and \(\tan\) without using a calculator.

• We can extend the definitions of the trigonometric ratios to any angle.