A visually themed introduction to what Trigonometry is really all about.

1. Trigonometry: Similar Triangles in Disguise
1 1
1 2
2 2

2. “She can tell you the height of the attacker from
the trigonometry of the blood spatter, while I’m fuzzy
on what trigonometry is.”
— Ilona Andrews

3. Triangles by Sides
• Triangles can be classified as Scalene, Isosceles or Equilateral.
• Can you spot how each of these types differ in terms of their sides and
angles?

4. Triangles by Angles
• Triangles can also be classified by the presence of particular angles.
Right Triangles Acute Triangles Obtuse Triangles

5. Triangles inside Triangles
• All Triangles can be written as the sum or difference of Right Triangles
Right Triangle Sum Difference

6. Considering all Triangles
• Trigonometry focuses closely on Right Triangles as they form the building
blocks of all other Triangles.
• What is the main difference you notice between each of these Right
Triangles?

7. The Hypotensue
• The Hypotenuse (H) is the name given to the longest side in a Right
Triangle, which is also always facing the 90 degree angle.
• The angle of this line measured from the horizontal distinguishes one
Right Triangle from another.
H
H

8. The “Other” Sides
• The other sides in a Right Triangle are also given special names in relation
to the angle the Hypotenuse make with the horizontal
• The side in line with the horizontal is called the Adjacent side and the
remaining side, the Opposite side
H
H
A
O
A
O

9. Relationships
• An important consideration for Trigonometry is the relationships between
the sides of Right Triangles.
• Does does the relationship between the Hypotenuse (H), the Adjacent(A)
and the Opposite side (O) change for different Right Triangles?
• How could you best describe that change?
H
A
O
A
O
H
H
A
O

10. Ratios
• The way to keep track of how these sides change in relationship to each
other for different Right Triangles is to calculate their Ratios.
• We given special mathematical names to Ratios O / H , A / H and O / A
• As you move from left to the right above, which Ratio gets bigger?
H
A
O
A
O
H
H
A
O

11. Size and Ratios
• Do the Ratios of the sides in your Right Triangle depend on on how big your Right
Triangle is?
• How could you turn these equilateral Triangles into pairs of Right Triangles?
• How could Pythagoras Theorem help you figure out the remaining side?
1 1
1 2
2 2

12. Does size matter?
• Divide the Triangles by drawing the height at 90 degrees to the base.
• Find the remaining sides using Pythag.
O / H = 0.866/1 = 1.732/2 = 0.866
A / H = 0.5/1 = 1/2 = 0.5
O / A = 0.866/0.5 = 1.732/1 = 1.732
2
2 2
1 1
1
0.5 1
h = 0.866
h = 1.732

13. Summing Up Part 1
• The Ratios of the sides of a Right Triangle are not affected by Size.
• As bigger versions of a Right Triangle or “Similar” versions have an
embedded scale factor in each of the sides.
• This scale Factor cancels out when computing Ratios.
O / H = 0.866/1 = 1.732/2 = 0.866
A / H = 0.5/1 = 1/2 = 0.5
O / A = 0.866/0.5 = 1.732/1 = 1.732

14. Summing Up Part 2
• Every Triangle in existence can be expressed in Right Triangles.
• And each Triangle has in it, constant ratios independent of size and
only related to the shape of the underlying Right Triangles.
• This fact, that all Triangles are built out of a limited number of Right
Triangles whose ratio of sides we can tabulate, is the foundational
idea of Trigonometry.
• Using these Ratios such as 0.866 , 0.5 and 1.732 to solve for unknow
sides within Triangles and later relate these ratios to the angle that
created them, is essentially what Trigonometry is all about.

15. “So, when a forensic analyst works out the height of
the attacker from the trigonometry of the blood
spatter, they’re just using the ratios of sides in a
Triangle, together with the idea of Similar Triangles”
— Peter Schutte