Trigonometry

1
Trigonometry

• Introduction

2
Instant Trig

• Trigonometry is math, so many people find it
  scary
• Its usually taught in a one-semester high-school
  course
• However, 95 of all the trig youll ever need
  to know can be covered in 15 minutes
• And thats what were going to do now

3
Angles add to 180

• The angles of a triangle always add up to 180

4
Right triangles

• We only care about right triangles
• A right triangle is one in which one of the
  angles is 90
• Heres a right triangle

Heres the right angle
hypotenuse
opposite
adjacent

• We call the longest side the hypotenuse
• We pick one of the other angles--not the right
  angle
• We name the other two sides relative to that angle

5
The Pythagorean Theorem

• If you square the length of the two shorter sides
  and add them, you get the square of the length of
  the hypotenuse
• adj2 opp2 hyp2

• 32 42 52, or 9 16 25
• hyp sqrt(adj2 opp2)
• 5 sqrt(9 16)

6
5-12-13

• There are few triangles with integer sides that
  satisfy the Pythagorean formula
• 3-4-5 and itsmultiples (6-8-10, etc.)are the
  best known
• 5-12-13 and its multiples form another set

• 25 144 169

7
Ratios

• Since a triangle has three sides, there are six
  ways to divide the lengths of the sides
• Each of these six ratios has a name (and an
  abbreviation)
• Three ratios are most used
• sine sin opp / hyp
• cosine cos adj / hyp
• tangent tan opp / adj
• The other three ratios are redundant with these
  and can be ignored

• The ratios depend on the shape of the triangle
  (the angles) but not on the size

8
Using the ratios

• With these functions, if you know an angle (in
  addition to the right angle) and the length of a
  side, you can compute all other angles and
  lengths of sides

• If you know the angle marked in red (call it A)
  and you know the length of the adjacent side,
  then
• tan A opp / adj, so length of opposite side is
  given by opp adj tan A
• cos A adj / hyp, so length of hypotenuse is
  given by hyp adj / cos A

9
The Hard Part

• If you understood this lecture, youre in great
  shape for doing all kinds of things with basic
  graphics
• Heres the part Ive always found the hardest
• Memorizing the names of the ratios
• sin opp / hyp
• cos adj / hyp
• tan opp / adj

10
Mnemonics from wikiquote

• The formulas for right-triangle trigonometric
  functions are
• Sine Opposite / Hypotenuse
• Cosine Adjacent / Hypotenuse
• Tangent Opposite / Adjacent
• Mnemonics for those formulas are
• Some Old Horse Caught Another Horse Taking Oats
  Away
• Saints On High Can Always Have Tea Or Alcohol