Transcript Trigonometry Introduction

Introduction To Trigonometry.
h
37 o
28m
Calculate the height of the tree:
opp
tan x 
adj
o
……………….
What Does Trigonometry Do ?
Without trigonometry we wouldn’t have sailed
the world , satellites in space wouldn’t know
where they were and planes would be lost.
Without trigonometry architects couldn’t design
buildings and engineers couldn’t make cars,
planes…..
Without trigonometry millions of calculations made
all over the world to keep things moving couldn’t be
made.
Naming Sides Of Triangles.
Consider the right angled triangle shown below:
Hyp’
Opp’
xo
Adj’
It has an angle of size xo marked in the right hand corner.
The side opposite the angle you are looking at is called the
opposite side. Opp’ for short.
The side opposite the right angle is called the hypotenuse.
Hyp’ for short.
The remaining side is called the adjacent. Adj’ for short.
The Ratio Of Two Sides.
Look at the right angled triangle ABC below:
If AB = BC = 6cm what type of triangle is ABC. Isosceles.
Calculate the ratio of opp’  adj’ for angle xo .
Opp’ = 6
C
Opp’ = 6
Adj,
6
6cm
Opp’
xo
A Adj’ 6cm
B
Adj’ = 6
= 1
What size is must the angle xo ?
xo = 45o
Now consider the same calculation again for the triangle
below:
We can see at once that the
opposite divided by the
adjacent is 1 and that the
angle xo is still 45o
C
The same result will occur for all
Right angled isosceles triangles.
10cm
xo
A
10cm
B
The Tangent Of An Angle.
C
The ratio of opposite divided
by adjacent is called the
tangent of the angle xo .
Opp’
Normally written as tan xo for
short.
xo
A
Adj’
B
Key Result.
opp
tan x 
adj
o
For all right angled triangles
the value of the tangent ratio
will determine the size of the
angle xo regardless of the size
of the triangle.
Using The Tangent Ratio.
Consider once more the triangle we started with:
C
To calculate xo on a calculator
follow the steps below:
6cm
tan x o 
Opp’
6
tan x 
6
xo
o
A Adj’ 6cm
B
On your calculator select the
following buttons:
INV
Tan-1
opp
adj
1
=
tan x o  1
x o  tan 1 1
Xo = 45o
Calculating Angles Using The Tangent Ratio.
We have now found how to calculate angles using the
tangent ratio as the following example shows.
Calculate the angle ao below.
adj’
11cm
ao
(1) Identify the opposite side.
(2) Identify the adjacent side.
16cm opp’
opp
tan x 
adj
(3) Write down the tangent
ratio.
o
tan x o 
16
11
tan x o  1.455
x o  tan 1 1.455
xo = 55.5o
(4) Substitute your values .
(5) Divide the ratio giving
your answer to 3d.p.
(6) Use your calculator to find
the angle ao to 1d.p.
Further Example.
12m Opp’
adj’
Calculate the angle bo.
tan x o 
16m
bo
opp
adj
12
tan x 
16
o
tan x o  0.75
x o  tan 1 0.75
xo = 36.9 o
Always follow the routine !!
What Goes In The Box ?
Find the size of each of the unknown angles below using
the tangent ratio.
ans:
37.9 o
(1)
ans:
(2)
51.5 o
bo
14cm
27cm
ao
18cm
(3)
2.7cm
c
o
ans:
33.7 o
34cm
34.7m
(3)
do
29.1m
ans:
40 o
1.8cm