When you have a right triangle there are 5 things you can know about it..

1
When you have a right triangle there are 5 things
you can know about it..

• the lengths of the sides (A, B, and C)
• the measures of the acute angles (a and b)
• (The third angle is always 90 degrees)

b
C
A
a
B
2
If you know two of the sides, you can use the
Pythagorean theorem to find the other side
b
C
A 3
a
B 4
3
And if you know either angle, a or b, you can
subtract it from 90 to get the other one a b
90

• This works because there are 180º in a triangle
  and we are already using up 90º
• For example
• if a 30º
• b 90º 30º
• b 60º

b
C
A
a
B
4
But what if you want to know the angles?

• Well, here is the central insight of
  trigonometry
• If you multiply all the sides of a right triangle
  by the same number (k), you get a triangle that
  is a different size, but which has the same
  angles

k(C)
b
C
b
k(A)
A
a
a
B
k(B)
5
How does that help us?

• Take a triangle where angle b is 60º and angle a
  is 30º
• If side B is 1unit long, then side C must be 2
  units long, so that we know that for a triangle
  of this shape the ratio of side B to C is 12
• There are ratios for every
• shape of triangle!

C 2
60 º
A 1
30º
B
6
But there are three pairs of sides possible!

• Yes, so there are three sets of ratios for any
  triangle
• They are mysteriously named
• sinshort for sine
• cosshort for cosine
• tanshort or tangent
• and the ratios are already calculated, you just
  need to use them

7
So what are the formulas?
Tan is Opposite over Adjacent
Sin is Opposite over Hypotenuse
Cos is Adjacent over Hypotenuse
SOH
CAH
TOA
You can use this word if you need to memorize
the formulas!
8
Some terminology

• Before we can use the ratios we need to get a few
  terms straight
• The hypotenuse (hyp) is the longest side of the
  triangle it never changes
• The opposite (opp) is the side directly across
  from the angle you are considering
• The adjacent (adj) is the side right beside the
  angle you are considering

9
A picture always helps

• looking at the triangle in terms of angle b

b

• A is the adjacent (near the angle)

C
A

• B is the opposite (across from the angle)

B
b
Near
hyp

• C is always the hypotenuse

Longest
adj
opp
Across
10
But if we switch angles

• looking at the triangle in terms of angle a

• A is the opposite (across from the angle)

C
A
a

• B is the adjacent (near the angle)

B
Across
hyp

• C is always the hypotenuse

Longest
opp
a
adj
Near
11
Lets try an example

• Suppose we want to find angle a
• what is side A?
• the opposite
• what is side B?
• the adjacent
• with opposite and adjacent we use the
• tan formula

b
C
A 3
a
B 4
12
Lets solve it
b
C
A 3
a
B 4
13
Another tangent example

• we want to find angle b
• B is the opposite
• A is the adjacent
• so we use tan

b
C
A 3
a
B 4
14
Calculating a side if you know the angle

• you know a side (adj) and an angle (25)
• we want to know the opposite side

b
C
A
25
B 6
15
Another tangent example

• If you know a side and an angle, you can find the
  other side.

b
C
A 6
25
B
16
An application

• You look up at an angle of 65 at the top of a
  tree that is 10m away
• the distance to the tree is the adjacent side
• the height of the tree is the opposite side

65
10m
17
Why do we need the sin cos?

• We use sin and cos when we need to work with the
  hypotenuse
• if you noticed, the tan formula does not have the
  hypotenuse in it.
• so we need different formulas to do this work
• sin and cos are the ones!

b
C 10
A
25
B
18
Lets do sin first

• we want to find angle a
• since we have opp and hyp we use sin

b
C 10
A 5
a
B
19
And one more sin example

• find the length of side A
• We have the angle and the hyp, and we need the
  opp

b
C 20
A
25
B
20
And finally cos

• We use cos when we need to work with the hyp and
  adj
• so lets find angle b

b
C 10
A 4
a
B
21
Here is an example

• Spike wants to ride down a steel beam
• The beam is 5m long and is leaning against a tree
  at an angle of 65 to the ground
• His friends want to find out how high up in the
  air he is when he starts so they can put add it
  to the doctors report at the hospital
• How high up is he?

22
How do we know which formula to use???

• Well, what are we working with?
• We have an angle
• We have hyp
• We need opp
• With these things we will use the sin formula

C 5
B
65
23
So lets calculate
C 5
B

• so Spike will have fallen 4.53m

65
24
One last example

• Lucretia drops her walkman off the Leaning Tower
  of Pisa when she visits Italy
• It falls to the ground 2 meters from the base of
  the tower
• If the tower is at an angle of 88 to the ground,
  how far did it fall?

25
First draw a triangle

• What parts do we have?
• We have an angle
• We have the Adjacent
• We need the opposite
• Since we are working with the adj and opp, we
  will use the tan formula

B
88
2m
26
So lets calculate
B

• Lucretias walkman fell 57.27m

88
2m
27
What are the steps for doing one of these
questions?

1. Make a diagram if needed
2. Determine which angle you are working with
3. Label the sides you are working with
4. Decide which formula fits the sides
5. Substitute the values into the formula
6. Solve the equation for the unknown value
7. Does the answer make sense?

28
Two Triangle Problems

• Although there are two triangles, you only need
  to solve one at a time
• The big thing is to analyze the system to
  understand what you are being given
• Consider the following problem
• You are standing on the roof of one building
  looking at another building, and need to find the
  height of both buildings.

29
Draw a diagram

• You can measure the angle 40 down to the base of
  other building and up 60 to the top as well.
  You know the distance between the two buildings
  is 45m

60
40
45m
30
Break the problem into two triangles.

• The first triangle
• The second triangle
• note that they share a side 45m long
• a and b are heights!

a
60
45m
40
b
31
The First Triangle

• We are dealing with an angle, the opposite and
  the adjacent
• this gives us Tan

a
60
45m
32
The second triangle

• We are dealing with an angle, the opposite and
  the adjacent
• this gives us Tan

45m
40
b
33
What does it mean?

• Look at the diagram now
• the short building is 37.76m tall
• the tall building is 77.94m plus 37.76m tall,
  which equals 115.70m tall

77.94m
60
40
37.76m
45m