Introduction to Trigonometry

1
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.
2
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.
3
The Trigonometric Functions
SINE
COSINE
TANGENT
4
SINE
Pronounced like sign
COSINE
Pronounced like co-sign
TANGENT
Pronounced tan-gent
5
B
With Respect to angle A, label the three sides
Hypotenuse
Opposite
A
C
Adjacent
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We need a way to remember all of these ratios
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Sin
SOHCAHTOA
Opp
Hyp
Cos
Adj
Hyp
Tan
Opp
Adj
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Finding sin, cos, and tan.(Just writing a ratio
or decimal.)
9
Find the sine, the cosine, and the tangent of
M. Give a fraction and decimal answer (round to 4
places).
N
10.8
9
6
P
M
10
B
Find the sine, cosine, and the tangent of angle A
24.5
8.2
A
C
23.1
Give a fraction and decimal answer. Round to 4
decimal places
11
Finding a side.(Figuring out which ratio to use
and getting to use a trig button.)
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Ex 1 Find x. Round to the nearest tenth.
Figure out which ratio to use.
What were looking for
What we know
adj
opp
We can find the tangent of 55? using a calculator
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Ex 2 Find the missing side.Round to the
nearest tenth.
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Ex 3 Find the missing side.Round to the
nearest tenth.
20 m
15
Ex 4 Find the missing side.Round to the
nearest tenth.
80 m
Note When the variable is in the
denominator, you end up dividing
x
16
hiding
Sometimes the right triangle is
?ABC is an isosceles triangle as marked. Find
sin ?C.
Answer as a fraction.
A
13 13
12
C
B
10
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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 rivers edge at a
60 angle. How far must the person walk to reach
the rivers edge?
cos 60
x (cos 60) 200
200
x
60
x

X 400 yards
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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?
tan 71.5
tan 71.5
?
50 (tan 71.5) y
71.5
50 m
y ? 149.4 m
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For some applications of trig, we need to know
these meaningsangle of elevation and angle
of depression.
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Angle of Elevation
Angle of Elevation
If an observer looks UPWARD toward an object, the
angle the line of sight makes with the horizontal.
Angle of elevation
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Angle of Depression
If an observer looks DOWNWARD toward an object,
the angle the line of sight makes with the
horizontal.
Angle of depression
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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.)
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Ex. 1 Find ?. Round to four decimal places.
17.2
9
Make sure you are in degree mode (not radians).
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Ex. 2 Find ?. Round to three decimal places.
7
23
Make sure you are in degree mode (not radians).
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Ex. 3 Find ?. Round to three decimal places.
200
400
Make sure you are in degree mode (not radians).
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When we are trying to find a sidewe use sin,
cos, or tan.
When we need to find an angle we use sin-1,
cos-1, or tan-1.