Transcript Right Triangle Trigonometry

Digital Lesson
Right Triangle
Trigonometry
The six trigonometric functions of a right triangle, with an acute
angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are:
hyp
 the side opposite the acute angle ,
opp
 the side adjacent to the acute angle ,
θ
 and the hypotenuse of the right triangle.
adj
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
sin  =
cos  = adj
tan  = opp
hyp
hyp
adj
csc  =
hyp
opp
sec  = hyp
adj
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cot  = adj
opp
2
Calculate the trigonometric functions for  .
5
4

3
The six trig ratios are
4
sin  =
5
4
tan  =
3
5
sec  =
3
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3
cos  =
5
3
cot  =
4
5
csc  =
4
3
Example
adj.
cos A 
hyp.
x
cos 55  .5735 
9
x  9(0.5735)
x  5.16

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4
Sines, Cosines, and Tangents of
Special Angles
sin 30  sin
sin 45  sin
sin 60  sin

6

4

3
1

3

3
 , cos 30  cos 
, tan 30  tan 
2
6
2
6
3
2

2


, cos 45  cos 
, tan 45  tan  1
2
4
2
4
3
 1


, cos 60  cos  , tan 60  tan  3
2
3 2
3
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5
Trigonometric Identities are trigonometric
equations that hold for all values of the variables.
Example: sin  = cos(90  ), for 0 <  < 90
Note that  and 90  are complementary
angles.
Side a is opposite θ and also
adjacent to 90○– θ .
hyp
θ
a
a
sin  =
and cos (90  ) = .
hyp
hyp
So, sin  = cos (90  ).
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90○– θ a
b
6
Fundamental Trigonometric Identities for 0 <  < 90.
Cofunction Identities
sin  = cos(90  )
tan  = cot(90  )
sec  = csc(90  )
cos  = sin(90  )
cot  = tan(90  )
csc  = sec(90  )
Reciprocal Identities
sin  = 1/csc 
cot  = 1/tan 
cos  = 1/sec 
sec  = 1/cos 
tan  = 1/cot 
csc  = 1/sin 
Quotient Identities
tan  = sin  /cos 
cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1
tan2  + 1 = sec2 
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cot2  + 1 = csc2 
7
Example:
Given sin  = 0.25, find cos , tan , and sec .
Draw a right triangle with acute angle , hypotenuse of length
one, and opposite side of length 0.25.
Use the Pythagorean Theorem to solve for
the third side.
0.9682
= 0.9682
1
tan  = 0.25 = 0.258
0.9682
1
sec  =
= 1.033
0.9682
cos  =
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1
0.25
θ
0.9682
8
Example: Given sec  = 4, find the values of the
other five trigonometric functions of  .
Draw a right triangle with an angle  such
4
4
hyp
that 4 = sec  =
= .
adj 1
Use the Pythagorean Theorem to solve
for the third side of the triangle.
15
4
cos  = 1
4
tan  = 15 = 15
1
sin  =
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15
θ
1
1 = 4
sin 
15
1
sec  =
=4
cos 
1
cot  =
15
csc  =
9
Angle of Elevation and Angle of Depression
When an observer is looking upward, the angle formed
by a horizontal line and the line of sight is called the:
angle of elevation.
line of sight
object
angle of elevation
horizontal
observer
When an observer is looking downward, the angle formed
by a horizontal line and the line of sight is called the:
horizontal
angle of depression
line of sight
object
observer
angle of depression.
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10
Example 2:
A ship at sea is sighted by an observer at the edge of a cliff
42 m high. The angle of depression to the ship is 16. What
is the distance from the ship to the base of the cliff?
observer
cliff
42 m
horizontal
16○ angle of depression
line of sight
16○
d
ship
42
= 146.47.
tan 16
The ship is 146 m from the base of the cliff.
d=
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11
Text Example
Sighting the top of a building, a surveyor measured the angle of elevation to
be 22º. The transit is 5 feet above the ground and 300 feet from the building.
Find the building’s height.
Solution Let a be the height of the portion of the building that lies above
the transit in the figure shown. The height of the building is the transit’s
height, 5 feet, plus a. Thus, we need to identify a trigonometric function that
will make it possible to find a. In terms of the 22º angle, we are looking for
the side opposite the angle.
a
Transit
5 feet
22º
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300 feet
h
Text Example cont.
Solution
The transit is 300 feet from the building, so the side adjacent to the 22º angle
is 300 feet. Because we have a known angle, an unknown opposite side, and
a known adjacent side, we select the tangent function.
a
tan 22º = 300
Length of side opposite the 22º angle
Length of side adjacent to the 22º angle
a = 300 tan 22º 300(0.4040) 121
Multiply both sides of the equation by 300.
The height of the part of the building above the transit is approximately 121
feet. If we add the height of the transit, 5 feet, the building’s height is
approximately 126 feet.
a
Transit
5 feet
22º
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300 feet
h
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with two sides of
length 1.
45
2
1
12  12  2
45
1
The Pythagorean Theorem implies that the hypotenuse
is of length 2 .
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14
Calculate the trigonometric functions for a 45 angle.
2
1
45
1
sin 45 =
opp
1
2
=
=
hyp
2
2
1
2
adj
cos 45 =
=
=
2
hyp
2
tan 45 =
opp 1
= = 1
1
adj
cot 45 =
adj 1
= = 1
opp 1
sec 45 =
2
hyp
=
=
1
adj
csc 45 =
2
hyp
=
= 2
opp
1
2
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15
Geometry of the 30-60-90 triangle
Consider an equilateral triangle with
each side of length 2.
30○ 30○
The three sides are equal, so the
angles are equal; each is 60.
2
The perpendicular bisector
of the base bisects the
opposite angle.
60○
2
3
1
60○
2
1
Use the Pythagorean Theorem to
find the length of the altitude, 3 .
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16
Calculate the trigonometric functions for a 30 angle.
2
1
30
3
opp 1
sin 30 =
=
hyp
2
cos 30 =
3
1
opp
tan 30 =
=
=
adj
3
3
3
adj
cot 30 =
=
= 3
1
opp
2
2 3
hyp
sec 30 =
=
=
3
3
adj
hyp 2
csc 30 =
=
= 2
opp
1
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3
adj
=
2
hyp
17
Calculate the trigonometric functions for a 60 angle.
2
3
60○
opp
3
sin 60 =
=
hyp
2
tan 60 =
1
3
opp
=
= 3
1
adj
hyp 2
sec 60 =
= = 2
adj 1
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1
adj
cos 60 =
=
2
hyp
3
1
cot 60 = adj =
=
opp
3
3
csc 60 =
2
2 3
hyp
=
=
opp
3
3
18