# Find the missing measures. Write all answers in radical by Noodlezs

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```									Warm – up
Find the missing measures. Write all
y
60°
30                     10
45    y
x
60
45          z                         30°
z
x6
y3 3
z 5 3
z3 2                         y 5
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 to A
Sin(A)  With Respect to angle A,
Hyp
label the three sides

Cos(A) 
Hyp

A           C
Opp to A
Tan(A) 
We need a way
to remember
all of these
ratios…
SOHCAHTOA   Sin
Opp
Hyp
Cos
Hyp
Tan
Opp
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).

N                              opp    9
sin M 
hyp  10.8  .8333

9        10.8
cos M 
hyp 10.8  .5556
P        6       M

opp   9
tan M                 1.5
A
Find the sine,
24.5           cosine, and the
8.2
tangent of angle A
C           23.1         B

opp   23.1
sin A             .9429
Give a fraction                  hyp   24.5
and decimal
cos A               .3347
hyp     24.5
Round to 4 decimal
places
opp     23.1
tan A               2.8171
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.
55
What we’re looking for…
What we know…
20 m
opp
tan 
tan55 
x                              opp

20
20 tan55   x                     20        tan         55   )

x  28.6 m
We can find the tangent of 55
using a calculator
Ex: 2 Find the missing side.
Round to the nearest tenth.

283 m
sin 24 
x                                          x
24                        283
283 sin 24   x
x  115.1 m
Ex: 3 Find the missing side.
Round to the nearest tenth.

cos40 
x
40
20 m           x                      20
20 cos40   x

x  15.3 m
Ex: 4 Find the missing side.
Round to the nearest tenth.
tan72  80
x

80 m                   x tan72   80
Note: When the variable is     x     80
72                in the denominator,
tan(72 )
x               you end up dividing

80          tan       72     )      =
x  26 m
Sometimes the right triangle is hiding
ABC is an isosceles triangle as
marked. Find sin C.
A

13             13
12
opp 12
sin C     
B                       C           hyp 13
10   5
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°

200                                  x (cos 60°) = 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° 
y
?                                     tan 71.5° 
50
71.5°      50 (tan 71.5°) = y
50 m
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
17.2            2nd      tan 17.2               9       )


9                 62.3789

Make sure you are in degree mode (not radians).
Ex. 2: Find . Round to three decimal places.
7                          7
                        cos 
23
23
2nd      cos 7                23       )

  72.281
Make sure you are in degree mode (not radians).
Ex. 3: Find . Round to three decimal places.
         sin  
200
200                                      400

2nd       sin    200         400 )

  30

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.

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