1.2 Trigonometry
SWBAT use the Pythagorean Theorem to
find the length of sides of a right triangle
SWBAT use sine, cosine, and tangent to find
the angle and/or length of sides of a right
triangle
The highest posted speed limit in the world is
140 km/hr, which applies to some roads in
Poland and Bulgaria. Convert this to miles/hour
(1 mile = 1.6 km)
1) 0.145 kg
2) 5x10-5 m
3) a. 1x10-8 m
-5
b. 1x10 mm
4) 341.1 m/s
5) B
6) B
7) C
8) C
Definition: The sum of the squares of the
lengths of the two other sides of any right
triangle will equal the square of the length of
the hypotenuse.
Formula:
a2 + b2 = c 2
Where is the Pythagorean Theorem?
A firefighter needs to reach a 2nd story
window 4 meters high using a ladder.
The ladder needs to be at least 2
meters away from the building to be
safe. Should he use his 4 m or 8 m
ladder to reach the window?
Opposite’
Opposite
θ
θ
Adjacent
Adjacent’
For
Example:
Two
right triangles with the same angle (θ) are similar triangles.
This means
their sides
are proportional
to each
Opp
Opp′
Adj
Adj′
Oppother.
Opp′
= the ratio of one side
= to another is equal=in both
This means
Hyp Hyp′
Hyp Hyp′
Adj
Adj′
triangles…
If we substitute in values, we can see what this ratio might look like
8
4
θ
θ
3
6
These ratios change depending on θ.
Opp
4 Opp′
8
=
=
Hyp
Hyp′
5 10
0.8 = 0.8
Adj
Adj′
3
6
=
=
Hyp
Hyp′
5 10
0.6 = 0.6
Opp4 Opp′
8
=
=
Adj3 6
Adj′
1.33 = 1.33
Opposite’
Opposite
θ
θ
Adjacent
Adjacent’
Mathematicians studied these relationships, and came up with the following
conclusions:
• If you know the length of 2 sides, you can find the angle
• If you know the angle and the length of 1 side, you can find the length of
all the other sides
They did this by coming up with relationships called sine, cosine, and tangent
Where are sine, cosine, and tanget?
Opposite’
Opposite
θ
θ
Adjacent
Opp
sin 𝜃 =
Hyp
SOH
Adjacent’
Adj
cos 𝜃 =
Hyp
CAH
Opp
tan 𝜃 =
Adj
TOA
4
θ
What is the measure of angle θ?
θ
15
What is the measure of angle θ?
30o
12
What are the lengths of the other two sides?
30
250
What are the lengths of the other two sides?
People had to look up the
values of sine, cosine, and
tangent in tables like this
one to the right. Now we
can just plug the numbers
into our calculators! (If
you remember to bring it
to class)
Sin(θ)
Cos(θ)
Tan(θ)
0o
0
1
0
30o
45o
60o
90o
0.5 0.707 0.866
1
0.866 0.707 0.5
0
0.577
1
1.732 Undef
X
15o
10
What are the lengths of sides X and Y?