Trigonometry

• Trigonometry involves calculating angles and sides in triangles
• The three sides of a right-angled triangle have special names:

  • Hypotenuse (\(h\)): the longest side, opposite the right angle
  • Opposite side (\(o\)): opposite the angle in question
  • Adjacent side (\(a\)): next to the angle in question
• Three trigonometric ratios are sine, cosine, and tangent, abbreviated as \(\sin\), \(\cos\), and \(\tan\)
• The trigonometric ratios are calculated by finding the ratio of two sides of a right-angled triangle:

  • \(\sin{x} = \frac{\text{opposite}}{\text{hypotenuse}}\)
  • \(\cos{x} = \frac{\text{adjacent}}{\text{hypotenuse}}\)
  • \(\tan{x} = \frac{\text{opposite}}{\text{adjacent}}\)
• Exact trigonometric ratios for 0°, 30°, 45°, 60°, and 90°:

  • \(\sin{30} = \frac{1}{2}\), \(\cos{30} = \frac{\sqrt{3}}{2}\), \(\tan{30} = \frac{1}{\sqrt{3}}~\text{or}~\frac{\sqrt{3}}{3}\)
  • \(\sin{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\), \(\cos{45} = \frac{1}{\sqrt{2}}~\text{or}~\frac{\sqrt{2}}{2}\), \(\tan{45} = 1\)
  • \(\sin{60} = \frac{\sqrt{3}}{2}\), \(\cos{60} = \frac{1}{2}\), \(\tan{60} = \sqrt{3}\)
  • \(\sin{0} = 0\), \(\cos{0} = 1\), \(\tan{0} = 0\)
  • \(\sin{90} = 1\), \(\cos{90} = 0\), \(\tan{90}\) is undefined