Transcript Dr. Math Does Trigonometry
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)
a C B b A
If you know two of the sides, you can use the Pythagorean theorem to find the other side
A
C
2
B
2
B
C
C
2
A
2
A
2
B
2
if A
3 ,
B
4
C
C
C
A
3 2 25 2
B
2 4 2 5
a C B = 4 b A = 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 180 º in a triangle and we are already using up 90º For example: if a = 30 º b = 90 º – 30º b = 60 º
C b A a B
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 A k(A) a a B k(B)
How does that help us?
Take a triangle where angle b is 60 º and angle a is 30
º
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 1:2 There are ratios for every shape of triangle!
C = 2 60 º A = 1 30 º B
But there are three pairs of sides possible!
Yes, so there are three sets of ratios for any triangle They are mysteriously named: sin …short for sine cos …short for cosine tan …short or tangent and the ratios are already calculated, you just need to use them
So what are the formulas?
sin tan cos
opp hyp
adj hyp opp adj
SOH CAHTOA
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
A picture always helps…
looking at the triangle in terms of angle b
b
A is the adjacent (near the angle) B is the opposite (across from the angle) C is always the hypotenuse Longest
hyp C B A b
Near
adj opp
Across
But if we switch angles…
looking at the triangle in terms of angle a A is the opposite (across from the angle) B is the adjacent (near the angle) C is always the hypotenuse Longest
a hyp C B a adj
Near
A
Across
opp
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
a
tan
opp adj
C b B = 4 A = 3
Lets solve it
tan tan
a
opp
adj
3 4 0 .
75 check our calculator s a 36.87º
C a B = 4 b A = 3
Another tangent example…
we want to find angle b B is the opposite A is the adjacent so we use tan tan
b
tan
b
4 3 1 .
33
b
53 .
13
a C
tan
opp adj
B = 4 b A = 3
Calculating a side if you know the angle
you know a side (adj) and an angle (25
°
) we want to know the opposite side tan
A
25 tan
A
25 6 6
A
0 .
47 6
A
2 .
80
25 ° C
tan
opp adj
b A B = 6
Another tangent example
If you know a side and an angle, you can find the other side. tan 25 6
B
tan
opp adj B
6 tan 25
b
B B
6 0 .
47 12 .
87
25 ° C B A = 6
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 tan
opp
65 10
opp
10 tan 65
opp
10 2 .
14
opp
21 .
4
65 ° 10m
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!
C = 10 b A 25 ° B
Lets do sin first
we want to find angle a since we have opp and hyp we use sin sin sin
a
5 10 sin
a
0 .
5
C = 10
a
30
a B
opp hyp
b A = 5
And one more sin example
find the length of side A We have the angle and the hyp, and we need the opp sin
A
25 sin
A
25 20 20
A
0 .
42 20
A
8 .
45
25 °
sin
C = 20
opp hyp
B b A
And finally cos
We use cos when we need to work with the hyp and adj so lets find angle b cos
adj hyp
cos
b
4 10
C = 10 b
cos
b
b
0 .
4 66 .
42 a 90 -
a
66.42
B
a 23.58
A = 4
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?
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
B C = 5 65 °
So lets calculate
sin 65
opp hyp
sin 65
opp
opp
sin 5 65 5
opp
0 .
91 5
opp
4 .
53 so Spike will have fallen 4.53m
B C = 5 65 °
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?
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
88 ° 2m B
So lets calculate
tan 88
opp adj
tan 88
opp
opp
tan 2 88 2
opp
28 .
64 2
opp
57 .
27 Lucretia’s walkman fell 57.27m
88 ° 2m B
What are the steps for doing one of these questions?
3.
4.
5.
6.
7.
1.
2.
Make a diagram if needed Determine which angle you are working with Label the sides you are working with Decide which formula fits the sides Substitute the values into the formula Solve the equation for the unknown value Does the answer make sense?
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.
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
Break the problem into two triangles.
The first triangle:
a
The second triangle
60 ° 45m 40 ° b
note that they share a side 45m long a and b are heights!
The First Triangle
We are dealing with an angle, the opposite and the adjacent this gives us Tan tan
a
60 tan
a
60 45 45 a 1.73
a 77.94m
45
60 ° 45m a
The second triangle
We are dealing with an angle, the opposite and the adjacent this gives us Tan tan
b
40 tan
b
40 45 45 b 0.84
45 b 37.76m
45m 40 ° b
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
60 ° 40 ° 45m 77.94m
37.76m