Transcript Sides and Angles day

Introduction to
Trigonometry
This section presents the 3 basic trigonometric
ratios sine, cosine, and tangent. The concept of
similar triangles and the Pythagorean Theorem
can be used to develop the trigonometry of
right triangles.
Engineers and scientists have
found it convenient to formalize
the relationships by naming the
ratios of the sides.
You will memorize these
3 basic ratios.
The Trigonometric Functions
SINE
COSINE
TANGENT
SINE
Pronounced like “sign”
COSINE
Pronounced like “co-sign”
TANGENT
Pronounced “tan-gent”
B
Opp
toRespect
A
With
to angle A,
Sin(A) 
Hypthe three sides
label
Adj to A
Cos(A) 
Hyp
Opp to A
Tan(A) 
Adj to A
A
C
We need a way
to remember
all of these
ratios…
SOHCAHTOA
Sin
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj
Finding sin, cos, and tan.
(Just writing a ratio or decimal.)
Find the sine, the cosine, and the tangent of M.
Give a fraction and decimal answer (round to 4 places).
opp
9
sin M 

hyp
10.8  .8333
N
9
P
10.8
6
M
adj  6
cos M 
hyp 10.8  .5556
9
opp

tan M 
adj
6
 1.5
B
Find the sine,
cosine, and the
tangent of angle A
24.5
8.2
C
23.1
Give a fraction
and decimal
answer.
Round to 4 decimal
places
A
8 .2
opp

sin A 
24.5  .3347
hyp
adj
cos A 
hyp
23.1

24.5  .9429
opp
tan A 
adj
8 .2


.
3550
23.1
Finding a side.
(Figuring out which ratio to use and
getting to use a trig button.)
Ex: 1 Find x. Round to the nearest tenth.
Figure out which ratio to use.
What we’re looking for…
What we know…
55
20 m
opp
tan 
adj
tan 55 
adj
x
opp
x
20
20 tan 55  x
We can find the tangent of 55
using a calculator
20
tan
55
x  28.6 m
)
Ex: 2 Find the missing side.
Round to the nearest tenth.
x
283 m
24
x
sin 24  
283
283sin 24  x
x  115.1 m
Ex: 3 Find the missing side.
Round to the nearest tenth.
40
20 m
x
x
cos40  
20
20 cos40  x
x  15.3 m
Ex: 4 Find the missing side.
Round to the nearest tenth.
x tan 72  80
80 m
Note: When the variable is
in the denominator,
you end up dividing
72
x
80
80


tan 72  x

tan
72
)
=
x
80
tan( 72)
x  26.0 m
Sometimes the right triangle is hiding
ABC is an isosceles triangle as
marked. Find sin C.
Answer as a fraction.
A
13
13
12
B
C
10
5
opp 12
sin C 

hyp 13
Ex. 5
A person is 200 yards from a river. Rather than walk
directly to the river, the person walks along a straight
path to the river’s edge at a 60° angle. How far must
the person walk to reach the river’s edge?
cos 60°
x (cos 60°) = 200
200
60°
x
x
X = 400 yards
Ex: 6
A surveyor is standing 50 metres from the
base of a large tree. The surveyor measures
the angle of elevation to the top of the tree
as 71.5°. How tall is the tree?
Opp
tan 71.5° 
Adj
y
tan 71.5° 
50
?
71.5°
50 m
50 (tan 71.5°) = y
y  149.4 m
For some applications of trig,
we need to know these meanings:
angle of elevation and
angle of depression.
Angle of Elevation
If an observer looks UPWARD toward
an object, the angle the line of sight
makes with the horizontal.
Angle of
elevation
Angle of Depression
If an observer looks DOWNWARD toward
an object, the angle the line of sight
makes with the horizontal.
Angle of
depression
Finding an angle.
(Figuring out which ratio to use and getting to
use the 2nd button and one of the trig buttons.
These are the inverse functions.)
Ex. 1: Find . Round to four decimal places.
17.2
tan  
9
1  17.2 
  tan 

 9 
17.2

  62.4
9
nd
2
tan
17.2

9
)
Make sure you are in degree mode (not radians).
Ex. 2: Find . Round to three decimal places.

7
cos  
23
1  7 
  cos  
 23 
7
23
  72.3
nd
2
cos
7

23
)
Make sure you are in degree mode (not radians).
Ex. 3: Find . Round to three decimal places.

200
sin  
400
1  200 
  sin 

 400 
200
  30
nd
2
sin
200

400 )
Make sure you are in degree mode (not radians).
When we are trying to find a side
we use sin, cos, or tan.
When we need to find an angle
we use sin-1, cos-1, or tan-1.