Download Right Triangle Trigonometry

Preparing for the SAT II
Triangle Trigonometry
Topics
 Basis of Trigonometry
 The Six Ratios
 Solving Right Triangles
 Special Right Triangles
 Law of Sines
 Law of Cosines
©Carolyn C. Wheater, 2000
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Basis of Trigonometry
Trigonometry, or "triangle
measurement," developed as a means to
calculate the lengths of sides of right
triangles.
It is based upon similar triangle
relationships.
©Carolyn C. Wheater, 2000
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Right Triangle Trigonometry
You can quickly prove
that the two right
triangles with an acute
angle of 25°are similar
All right triangles
containing an angle of
25° are similar
You could think of this as
the family of 25° right
triangles. Every triangle in
the family is25similar.
We could imagine such a
family of triangles for any
acute angle.
©Carolyn C. Wheater, 2000
25
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Right Triangle Trigonometry
In any right triangle in the family, the ratio
of the side opposite the acute angle to the
hypotenuse will always be the same, and
the ratios of other pairs of sides will
remain constant.
©Carolyn C. Wheater, 2000
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The Six Ratios
the hypotenuse,
 the side opposite a particular
acute angle, A, and
 the side adjacent to the acute
angle A,

opposite
If the three sides of the right
angle are labeled as
A
adjacent
six different ratios are possible.
©Carolyn C. Wheater, 2000
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The Six Ratios
opposite
sin( A) 
hypotenuse
hypotenuse
csc( A) 
opposite
adjacent
cos( A) 
hypotenuse
hypotenuse
sec( A) 
adjacent
opposite
tan( A) 
adjacent
adjacent
cot( A) 
opposite
©Carolyn C. Wheater, 2000
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Solving Right Triangles
With these six ratios, it is possible
to solve for any unknown side of the right
triangle, if another side and an acute angle are
known, or
 to find the angle if two sides are known.

Once upon a time, students had to rely on
tables to look up these values. Now the sine,
cosine, and tangent of an angle can be found
on your calculator.
©Carolyn C. Wheater, 2000
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Sample Problem
In right triangle ABC, hypotenuse is 6 cm
long, and A measures 32. Find the length
of the shorter leg.
58
6
Make a sketch
32
 If one angle is 32, the other is 58
 The shorter leg is opposite the smaller angle, so
you need to find the side opposite the 32 angle.

©Carolyn C. Wheater, 2000
9
Choosing the Ratio
... Find the length of the
shorter leg.
You need a ratio that talks about
opposite and hypotenuse
 Can use sine (sin) or cosecant
(csc), but since your calculator
has a key for sin, sine is more
convenient.

©Carolyn C. Wheater, 2000
58
6
32
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Solving the Triangle
x
sin(32 ) 
6

58
6
32
From your calculator, you can
find that sin(32)  0.53, so
x
0.53 
6
x  32
.
©Carolyn C. Wheater, 2000
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Special Right Triangles
45 – 45 – 90 Triangle
The legs are of equal
length
 The length of the
hypotenuse is 2
times the leg
s 2

s
45°
s
2
sin(45 ) 

2
s 2

2
cos(45 ) 

2
s 2

s
s
tan(45 )   1
s

s
©Carolyn C. Wheater, 2000
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Special Right Triangles
30 – 60 – 90 Triangle
The side opposite the
30 angle is half the
hypotenuse
 The side opposite the
60 angle is half the
hypotenuse times 3

30°
1
2
h 1
sin(30 ) 

h 2

1
2
h 3
3
cos(30 ) 

h
2

1
2
h
3
tan(30 )  1

3
h 3
2
©Carolyn C. Wheater, 2000

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60°
Special Right Triangles
30 – 60 – 90 Triangle
The side opposite the
30 angle is half the
hypotenuse
 The side opposite the
60 angle is half the
hypotenuse times 3

1
2
h 3
3
sin(60 ) 

h
2

1
2
h 1
cos(60 ) 

h 2

tan(60 ) 
©Carolyn C. Wheater, 2000

1
2
h 3
1
2
h
 3
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Memory Work
Know these values as well as you know your
own name.
sin
cos
tan
30°
1
2
3
2
3
3
45°
2
2
2
2
1
60°
3
2
1
2
©Carolyn C. Wheater, 2000
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Non-Right Triangles
All these relationships are based on the
assumption that the triangle is a right
triangle.
It is possible, however, to use
trigonometry to solve for unknown
sides or angles in non-right triangles.
©Carolyn C. Wheater, 2000
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Law of Sines
a
b
c


sin( A) sin( B) sin(C)
In geometry, you learned that the largest
angle of a triangle was opposite the longest
side, and the smallest angle opposite the
shortest side.
The Law of Sines says that the ratio of a side
to the sine of the opposite angle is constant
throughout the triangle.
©Carolyn C. Wheater, 2000
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Sample Problem
In ABC, mA = 38, mB = 42,
and BC = 12 cm. Find the length of
side AC.
 Draw
a diagram to see the position of the
C
given angles and side.
 BC is opposite A
A
 You must find AC, the side opposite B.
©Carolyn C. Wheater, 2000
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B
Sample Problem
.... Find the length of side AC.
 Use
the Law of Sines with mA = 38,
mB = 42, and BC = 12
a
b

sin( A) sin( B)
12
b



sin(38 ) sin(42 )
12 sin(42  )  b sin(38 )
b g
12 sin(42  ) 12 .6691
b


sin(38 )
.6157
b  13.041
©Carolyn C. Wheater, 2000
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Warning
The Law of Sines is useful when you
know
 the
sizes of two sides and one angle or
 two angles and one side.
However, the results can be ambiguous if
the given information is two sides and an
angle other than the included angle (ssa).
©Carolyn C. Wheater, 2000
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Warning
The Law of Sines gives a unique solution
when the given information is
 sas
 asa
Remember that these are
all sufficient conditions
for congruent triangles.
 aas
The ambiguous case is ssa, which is not a
way of proving triangles congruent.
©Carolyn C. Wheater, 2000
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Law of Cosines
c  a  b  2ab cos(C)
2
2
2
If you apply the Law of Cosines to a right
triangle, that extra term becomes zero,
leaving just the Pythagorean Theorem.
The Law of Cosines is most useful
when you know the lengths of all three sides and
need to find an angle, or
 when you two sides and the included angle.

©Carolyn C. Wheater, 2000
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Largest angle
opposite
longest side
Sample Problem
Triangle XYZ has sides of lengths 15,
22, and 35. Find the measure of the
largest angle of the triangle.
c 2  a 2  b 2  2ab cos(C)
22
15
35
35  15  22  2  15  22  cos(C)
2
2
2
1225  225  484  660 cos(C)
1225  709  660 cos(C)
©Carolyn C. Wheater, 2000
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Sample Problem
... Find the measure of the largest angle
of the triangle.
516  660 cos(C )
15
22
35

516 
cos(C ) 
 .7818
660
C  cos1 (  .7818)  1414
. 
©Carolyn C. Wheater, 2000
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Caution
Many people, when they reach the line
1225  709  660 cos(C)
will mistakenly subtract 660 from 709.
Don’t be one of them.
The multiplication should be done
before any addition or subtraction.
©Carolyn C. Wheater, 2000
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