Trigonometry
description
Transcript of Trigonometry
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Trigonometry
Obj: I can to use trigonometry to find unknown sides and unknown angles in a triangle.
Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.
Angle
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A
A
The sides of a right -angled triangle are given special names:
The hypotenuse, the opposite and the adjacent.
The hypotenuse is the longest side and is always opposite the right angle.
The opposite and adjacent sides refer to another angle, other than the 90o.
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The Trigonometric Functions we will be
looking at
SINE
COSINE
TANGENT
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The Trigonometric Functions
SINE
COSINE
TANGENT
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SINE
Prounounced “sign”
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Prounounced “co-
sign”
COSINE
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Prounounced “tan-
gent”
TANGENT
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Prounounced
“theta”
Greek Letter
Represents an unknown angle
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oppositehypotenuse
adjacent
hypotenuseopposite
adjacent
Hyp
OppSin
Hyp
AdjCos
Adj
OppTan
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We need a way to remember all of these ratios…
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Old Hippie
Old Hippie
SomeOldHippieCameAHoppin’ThroughOurApartment
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SOHCAHTOA
Old Hippie
Old Hippie
SinOppHypCosAdjHypTanOppAdj
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Using trigonometry on the calculator
All individual angles have different sine, cosine and tangent ratios (or decimal values).
Scientific calculators store information about every angle.
We need to be able to access this information in the correct manner.
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Finding the ratios
The simplest form of question is finding the decimal value of the ratio of a given angle.
Find:
sin 32 =
sin 32 = 0.5514
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Using ratios to find angles
We have just found that a scientific calculator holds the ratio information for sine (sin), cosine (cos) and tangent (tan) for all angles.
It can also be used in reverse, finding an angle from a ratio.
To do this we use the sin-1, cos-1 and tan-1 function keys.
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Example:1. sin x = 0.1115 find angle x.
x = sin-1 (0.1115)x = 6.4o
2. cos x = 0.8988 find angle x
x = cos-1 (0.8988)x = 26o
sin-1 0.1115 =
shift sin( )
cos-1 0.8988 =
shift cos( )
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Finding an angle from a triangle
To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.
Find angle C
a) Identify/label the names of the sides.
b) Choose the ratio that contains BOTH of the letters.
14 cm
6 cmC
1.
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C = cos-1 (0.4286)
C = 64.6o
14 cm
6 cmC
1.H
A
We have been given the adjacent and hypotenuse so we use COSINE:
Cos A = hypotenuseadjacent
Cos A = ha
Cos C =146
Cos C = 0.4286
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Find angle x2.
8 cm
3 cmx
A
O
Given adj and oppneed to use tan:
Tan A = adjacentopposite
x = tan-1 (2.6667)
x = 69.4o
Tan A = ao
Tan x =38
Tan x = 2.6667
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3.
12 cm10 cm
y
Given opp and hypneed to use sin:
Sin A = hypotenuseopposite
x = sin-1 (0.8333)
x = 56.4o
sin A = ho
sin x =1210
sin x = 0.8333
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Finding a side from a triangle
To find a missing side from a right-angled triangle we need to know one angle and one other side.
Cos45 = 13x
To leave x on its own we need to move the ÷ 13.
It becomes a “times” when it moves.
Note: If
Cos45 x 13 = x
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Cos 30 x 7 = k
6.1 cm = k
7 cm
k30o
4.
H
A
We have been given the adj and hyp so we use COSINE:
Cos A = hypotenuseadjacent
Cos A = ha
Cos 30 =7k
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Tan 50 x 4 = r
4.8 cm = r
4 cm
r
50o
5.
A
O
Tan A = ao
Tan 50 = 4r
We have been given the opp and adj so we use TAN:
Tan A =
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Sin 25 x 12 = k
5.1 cm = k
12 cm k
25o
6.
H
O
sin A = ho
sin 25 =12k
We have been given the opp and hyp so we use SINE:
Sin A =
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Finding a side from a triangle
There are occasions when the unknown letter is on the bottom of the fraction after substituting.
Cos45 = u13
Move the u term to the other side.
It becomes a “times” when it moves.
Cos45 x u = 13
To leave u on its own, move the cos 45 to other side, it becomes a divide.
u = 45 Cos13
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When the unknown letter is on the bottom of the fraction we can simply swap it with the trig (sin A, cos A, or tan A) value.
Cos45 = u13
u = 45 Cos13
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x =
x
5 cm30o
7.
H
A
Cos A = ha
Cos 30 = x5
30 cos5
m 8 cm
25o
8.
H
O m =
sin A = ho
sin 25 = m8
sin258
x = 5.8 cm
m = 18.9 cm
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sin 30 = 0.5
cos 30 = 0.866
tan 30 = 0.5774
sin 50 = 0.766
cos 50 = 0.6428
tan 50 = 0.1.1917