A ratio can be expressed 3
different ways:
As a fraction …
With words …
1_
3
1 to 3
With a colon …
1:3
Cross Multiplication
_5 = _x_
8
12
5 · 12 = 8 · X
60 = 8X
60 ÷ 8 = 8X ÷ 8 or 60/8 = 8x/8
7.5 = X
Proportion
• If ratios are equal, they are proportionate.
• If ratios are unequal, they are disproportionate.
½ = 4/ 8
proportionate
1:2 = 4:8
1 to 2 = 4 to 8
½ ⅗
≠
1:2 ≠ 3:5
1 to 2 ≠ 3 to 5
disproportionate
Similar
~
9
3
12
4
5
15
Corresponding angles are equal (congruent),
but corresponding sides are not equal,
Corresponding sides are proportional.
How do I solve for X?
9
15
~
5
3
X
21
Step 1 - Set up the ratios (Remember you
must use corresponding sides!)
9/3= 21/x
How to use ratios to solve for X?
Similar Shapes
9
15
~
5
3
X
21
Step 1 - Set up the ratios (Remember you
must use corresponding sides!)
9/3= 21/x (You could also use 15/5=21/X
Step 2- Cross Multiply
9
15
5
3
21
9/3= 21/x
9 *X = 3*21
9X = 63
X
Step 3- Division
9
15
5
3
21
X
9X = 63
Divide by both sides by 9 to get X by itself
9X/9 = 63/9
X=7
What happens if X ends up being
squared?
9X
15
5
3
21
•
•
•
•
X
9x/3= 21/x
9x2 = 63
X2 = 63
Then you need to find the square root of each side!
X
2 continued...
X2 = 63
The square root of X2 is X.
The square root of 63 is 7.937
Therefore X= 7.937
Introduction to Trigonometry
• Trigonometry ... is all about triangles
Right Angled Triangle
A right-angled triangle
(the right angle is shown
by the little box in the
corner) has names for
each side:



Adjacent is adjacent to
the angle "θ",
Opposite is opposite
the angle "θ", and
the longest side is the
Hypotenuse; it is the
side opposite the right
angle.
"Sine, Cosine and Tangent"
The three most common functions in trigonometry are
Sine, Cosine and Tangent.
We will use them a lot!
They are simply one side of a triangle divided by another.
For any angle "θ":
Sine Function: SOH
sin(θ) = Opposite / Hypotenuse
Cosine Function: CAH
cos(θ) = Adjacent / Hypotenuse
Tangent Function: TOA
tan(θ) = Opposite / Adjacent
SOH
Sine
ᶿ
=
opposite
hypotenuse
O
S=
H
CAH
Cosine
ᶿ
=
adjacent
hypotenuse
A
C=
H
TOA
Tan
ᶿ=
opposite
Adjacent
O
T=
H
Example: What is the sine of 35°?
Using this triangle
sin(35°) = Opposite / Hypotenuse
sin(35°) = 2.8/4.9
sin(35°) = 0.57...
Angles
Angles (such as the angle "θ" below) can be in Degrees or Radians.
Here are some examples:
Angle
Degrees Radians
Right Angle
90°
π/2
Straight Angle
180°
π
Full Rotation
360°
2π