TRIGONOMETRY TRIGONOMETRY SANGEETH.S SANGEETH.S XI.B. XI.B. Roll.No:50 Roll.No:50 K.V.PATTOM K.V.PATTOM
Mar 26, 2015
TRIGONOMETRYTRIGONOMETRY
SANGEETH.SSANGEETH.S
XI.B.XI.B.
Roll.No:50Roll.No:50
K.V.PATTOMK.V.PATTOM
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)
AC
B
a
b
If you know two of the sides, you can use the Pythagorean theorem to find the other side
22
22
22
BAC
ACB
BCA
A = 3C
B = 4
a
b
525
43
4,3
22
22
C
C
BAC
BAif
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
AC
B
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(A)
k(C)
k(B)
a
b
AC
B a
b
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 1:2
There are ratios for every
shape of triangle!
A = 1
C = 2
B
30º
60 º
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 sinecos…short for cosinetan…short or tangent and the ratios are already calculated, you just
need to use them
So what are the formulas?
hyp
oppsin
hyp
adjcos SOH
adj
opptan
CAHTOA
Sin is Opposite over HypotenuseCos is Adjacent over HypotenuseTan is Opposite over Adjacent
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
AC
B
b
adjhyp
opp
b C is always the hypotenuse
A is the adjacent (near the angle)
B is the opposite (across from the angle)
LongestNear
Across
But if we switch angles…
looking at the triangle in terms of angle a
AC
B
a
opphyp
adja
C is always the hypotenuse
A is the opposite (across from the angle)
B is the adjacent (near the angle)
LongestAcross
Near
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
adj
opptan
A = 3C
B = 4
a
b
Lets solve it
adj
opptan
A = 3C
B = 4
a
b
75.04
3tan a
scalculatorour check
36.87º a
Where did the numbers for the ratio come from?
Each shape of triangle has three ratios These ratios are stored your scientific
calculator In the last question, tanθ = 0.75 On your calculator try 2nd, Tan 0.75 = 36.87 °
Another tangent example…
we want to find angle b B is the opposite A is the adjacent so we use tan
adj
opptan
A = 3C
B = 4
a
b
13.53
33.1tan3
4tan
b
b
b
Calculating a side if you know the angle you know a side (adj) and an angle (25°) we want to know the opposite side
adj
opptan
A C
B = 6
25°
b
80.2
647.0
625tan6
25tan
A
A
A
A
Another tangent example
If you know a side and an angle, you can find the other side.
adj
opptan
CA = 6
25°
b
B87.1247.0
625tan
6
625tan
B
B
B
B
An application
65°
10m
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
4.21
14.210
65tan1010
65tan
opp
opp
opp
opp
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 = 10A
25°
b
B
Lets do sin first
we want to find angle a since we have opp and hyp we
use sin
hyp
oppsin
C = 10
a
b
B
A = 5
30
5.0sin10
5sin
a
a
a
And one more sin example
find the length of side A We have the angle and
the hyp, and we need the opp
hyp
oppsin
C = 20
25°
b
B
A 45.8
2042.0
2025sin20
25sin
A
A
A
A
And finally cos
We use cos when we need to work with the hyp and adj
so lets find angle bhyp
adjcos
C = 10
a
b
B
A = 4
42.66
4.0cos10
4cos
b
b
b
23.58 a
66.42 - 90 a
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
C = 5
65°
B
So lets calculate
so Spike will have fallen 4.53m
C = 5
65°
B
53.4
591.0
565sin5
65sin
65sin
opp
opp
opp
opp
hyp
opp
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
2m
88°
B
So lets calculate
Lucretia’s walkman fell 57.27m
2m
88°
B
27.57
264.28
288tan2
88tan
88tan
opp
opp
opp
opp
adj
opp
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?
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:
The second triangle
note that they share a side 45m long
a and b are heights!
60°
45m
40°
b
a
The First Triangle
We are dealing with an angle, the opposite and the adjacent
this gives us Tan
60°
45m
a
77.94m a
451.73a
4560tan45
60tan
a
a
The second triangle
We are dealing with an angle, the opposite and the adjacent
this gives us Tan
45m
40°
b
37.76mb
450.84b
4540tan45
40tan
b
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