2 Todays Objective Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities What You Should Learn ID: 784278
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Slide1
Section 13.1
Right Triangle Trigonometry
Slide22
Today’s Objective
Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions
Begin learning some of the Trigonometric identities
Slide3What You Should Learn
Evaluate trigonometric functions of acute angles.
Use fundamental trigonometric identities.
Use a calculator to evaluate trigonometric
functions.
Use trigonometric functions to model and solve
real-life problems.
Slide44
Right Triangle Trigonometry
Trigonometry is based upon ratios of the sides of right triangles.
The ratio of sides in triangles with the same angles is consistent. The size of the triangle does not matter because the triangles are similar (same shape different size).
Slide55
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:
the side opposite the acute angle ,
the side adjacent to the acute angle ,
and the hypotenuse of the right triangle.
opp
adj
hyp
θ
Slide66
Trigonometric Functions
The trigonometric functions are
sine, cosine, tangent, cotangent, secant,
and
cosecant
.
opp
adj
hyp
θ
sin
= cos
= tan
=
csc
= sec
= cot =
opp
hyp
adj
hyp
hyp
adj
adj
opp
opp
adj
Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.
Slide77
Reciprocal Functions
Another way to look at it…
sin
= 1/csc
csc
= 1/sin cos = 1/sec sec = 1/cos tan
= 1/cot cot = 1/tan
Slide8Given 2 sides of a right triangle you should be able to find the value of all 6 trigonometric functions.
Example:
8
5
12
Slide99
Example: Six Trig Ratios
Calculate the trigonometric functions for
.
The six trig ratios are
4
3
5
sin
=
tan
=
sec
=
cos
=
cot
=
csc
=
cos
α
=
sin
α
=
cot
α
=
tan
α
=
csc
α
=
sec
α
=
What is the relationship of
α
and
θ
?
They are complementary (
α
= 90 –
θ
)
Calculate the trigonometric functions for
.
Slide1010
Example: Using Trigonometric Identities
Note : These functions of the complements are called cofunctions.
Note
sin
=
cos(90
), for 0 <
< 90
Note that and 90 are complementary angles.
Side
a is opposite θ and also adjacent to 90○– θ .
a
hyp
b
θ
90
○
–
θ
sin
= and cos
(90
) = .
So,
sin
= cos (90
)
.
Slide1111
Geometry of the
45-45-90 Triangle
Geometry of the 45-45-90 triangle
Consider an isosceles right triangle with two sides of length 1.
1
1
45
45
The
Pythagorean Theorem
implies that the hypotenuse is of length .
Slide1212
Example: Trig Functions for
45
Calculate the trigonometric functions for a 45
angle
.
1
1
45
csc
45
= = =
opp
hyp
sec
45
= = =
adj
hyp
cos
45
= = =
hyp
adj
sin
45
= = =
cot
45
= = =
1
opp
adj
tan
45
= = =
1
adj
opp
Slide1313
Example: Trig Functions for
30
Calculate the trigonometric functions for a 30
angle
.
1
2
30
csc
30
= =
=
2
opp
hyp
sec
30
= = =
adj
hyp
cos
30
= =
hyp
adj
tan
30
= = =
adj
opp
cot
30
= = =
opp
adj
sin
30
= =
Slide1414
Example: Trig Functions for
60
Calculate the trigonometric functions for a 60
angle
.
1
2
60
○
csc
60
= =
=
opp
hyp
sec
60
= = =
2
adj
hyp
cos
60
= =
hyp
adj
tan
60
= = =
adj
opp
cot
60
= = =
opp
adj
sin
60
= =
Slide1515
Some basic trig values
Sine
Cosine
Tangent
30
0
45
0
1
60
0
Slide16Find Missing Side Length
Find the value of x for the right triangle shown
sin 60° =
=
5
= x
Slide17Using a Calculator To Solve
Solve
∆ABC
a= 4.48 and c = 13.8
a/13 = tan 19 c/13 = sec 19
Slide18Applications Involving Right Triangles
The angle you are given is the
angle of elevation,
which represents the angle from the horizontal upward to an object.
For objects that lie below the horizontal, it is common to use the term
angle of depression.
Slide19Using Trigonometry to Solve a Right Triangle
A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3
.
How tall is the Washington Monument?
Figure 4.33
Slide20Solution
where
x
= 115 and
y
is the height of the monument. So, the height of the Washington Monument is
y
=
x
tan 78.3
115(4.82882) 555 feet.
Slide21An airplane flying at an altitude of 30,000 feet is headed toward an airport. To guide the airplane to safe landing, the airport’s landing system sends radar signals from the runway to the airplane at a 10 angle of elevation. How far is the airplane from the airport runway?
30,000
ft