Transcript C - Hazlet.org

Trigonometry 10.1
• Define trigonometry.
• Label the sides and angles of a right triangle correctly.
• Find the ratio of the sides in a right triangle.
• Use trigonometry to find the measures of unknown sides
and angles in right triangles.
• Use a graphing calculator to find the measures of
unknown sides and angles.
• Solve a right triangle.
Definition
Trigonometry is concerned with the
relationship between the angles and
sides of triangles. An understanding of
these relationships enables unknown
angles and sides to be calculated
without recourse to direct measurement.
Triangle Labeling
All angles of a triangle are uppercase
letters and the sides opposite them are
the corresponding lower case letters.
Calculator for Homework
Make sure DEG is shown in the top
left corner.
Graphing Calculator
Using the calculator to evaluate trig functions
To evaluate trig functions of acute angles other than 30, 45, and 60,
you will use the calculator.
Your calculator has keys marked Sin, Cos, and Tan.
**Make sure the MODE is set to the correct unit of angle measure.
(Degree vs. Radian)
Calculator Work
Find each value using a calculator. Round
to the nearest ten-thousandths degrees.
a. Sin 43° .6820
b. Cos 84° .1045
c. Tan 15° .2679
d. Sin 36° .5878
e. Cos 50° .6428
f. Tan 38° .7813
g. Sin 17° .2924
h. Cos 75° .2588
i. Tan 26° .4877
j. Sin 56° .8290
k. Cos 22° .9272
l. Tan 43° .9325
Calculator Work
Using Inverse Trigonometric Functions to Find Angles
Use a calculator to find an angle A in degrees that satisfies
sin A  .9677091705.
Solution
With the calculator in degree mode, we find that an angle having a sine
value of .9677091705 is 75.4º. Write this as sin-1 .9677091705  75.4º.
Calculator Work
Find each value using a calculator. Round
to the degree.
a. Sin A = .8829 62
d. Sin B = .2588 15
b. Cos A = .5 60
e. Cos B =.5592 56
c. Tan A = .4663 25
f. Tan B = 2.0503 64
Trig Definitions
• Sin (angle) =
Opposite
---------------Hypotenuse
S-O-H
• Cos (angle) =
Adjacent
---------------Hypotenuse
C-A-H
• Tan (angle) =
Opposite
---------------Adjacent
T-O-A
In a right triangle, the shorter sides are called legs and the longest side
(which is the one opposite the right angle) is called the hypotenuse
A
We’ll label them a, b, and c and the angles A,B
and C. Trigonometric functions are defined by
taking the ratios of sides of a right triangle.
First let’s look at the three basic functions.
adjacent
b
C
opp. a
sin A 

hyp. c
SINE
a
COSINE
B
TANGENT
They are abbreviated using their first 3 letters
adj. b
cos A 

hyp. c
opp. a
tan A 

adj. b
Opp Leg
Sin 
Hyp
hypotenuse
Adj Leg
Cos 
Hyp
Opp Leg
Tan 
Adj Leg

adjacent
opposite
opposite
Sin, Cos, or Tan?
Answer: Tan
You know the adjacent
and want the opposite.
x
35o
7
O
S
H
A
O
C H
T A
Sin, Cos, or Tan?
Answer: Sin
You know the opposite
and want the hypotenuse.
10
40o
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Cos
You know the adjacent
and want the hypotenuse.
35o
20
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Sin
x
You know the hypotenuse
and want the opposite.
38o
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Cos
You know the hypotenuse
and want the adjacent.
21o
x
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Sin
You know the opposite
and the hypotenuse. You
want to find the angle.
10
o
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Tan
You know the opposite
and want the adjacent.
24
37o
x
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Sin
You know the opposite
and the hypotenuse. And
want to know the angle
10
o
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Tan
You know the opposite
and want the adjacent.
20
42o
x
O
S
A
H
C
O
H
T A
Sin, Cos, or Tan?
Answer: Sin
You know the opposite
and the hypotenuse. You
want to find the angle.
200
o
O
S
A
H
C
O
H
T A
SOH CAH TOA
Find the values of sin A, cos A, and tan A;
sin B, cos B, and tan B in the right triangle.
Solution
24
7 77777
24
2424
24
77777
A, ,, , , ,cos
cos
AA
tan
cos
 
tan
sin
Asin
AA
A
,, ,,, tan
AAA
AA 
AA
A
cos
,tan
tan
sinsin
Asin
cos
A
tan
A
2525
25
24
25
25
24
25
2525
24
24
2525
25
24
7 24
24
7 24
24
7
7
24
7
7
24
24
7
sin
ABA, ,, ,cos
cos
AB
A
, ,, tan
tan
BA
  A ,
A
cos

tan
, sin
cos
tan

sin
A

cos
A
cos
A

,
tan
sin
Asin

A

,
A

25
25
24
25
25
24
25 25
25 25
725
5
25
2425
7
24
7
24
SOH CAH TOA
Opp
Sin A 
Hyp
Adj 6
3
Cos A 

Hyp 10
5
Opp
Tan A 
Adj
B
8 4

10 5
8 4

6 3 A
10
6
8
C
SOH CAH TOA
Find c. a2 + b2 = c2
242 + 702 = c2
70
5476 = c2
c = 74
74
θ
24
SOH CAH TOA
Find the values of the trigonometric functions for θ.
Find a. a2 + 102 = 262
a2 + 100 = 676
24
576 = c2
c = 24
To Solve Any Trig Word Problem
– Step 1: Draw a triangle to fit problem
– Step 2: Label sides from angle’s view
– Step 3: Identify trig function to use
– Step 4: Set up equation
– Step 5: Solve for variable
Assignment
8.3 Practice 1 – 15
Solve the triangle.
B
16 ft
y
A
55 °
x
C
Solve means to find all angles and all sides.
y
a. Sin 55 =
16
y  13.11 ft
x
b. Cos 55 =
16
x  9.18 ft
c. mB = 35
From a point 80m from the base of a tower, the angle
from the ground is 28˚. How tall is the tower?
x
28˚
80
Using the 28˚ angle as a reference, we use opposite and adjacent sides.
opp
Use
adj
tan
tan 28˚ =
x
80
80 (tan 28˚) = x
80 (.5317) = x
x ≈ 42.5 m
A ladder that is 20 ft is leaning against the side of a
building. If the angle formed between the ladder and ground
is 75˚, how far is the bottom of the ladder from the base of
the building?
building
20
75˚
x
Using the 75˚ angle as a reference, we use hypotenuse and adjacent side.
adj
x
Use
cos 75˚ =
cos
hyp
20
20 (cos 75˚) = x
20 (.2588) = x
x ≈ 5.2 ft
Find the missing value.
Find the measure of the missing side or hypotenuse for the triangle.
x
a. 13.95
b. 37
c. 184.08
d. 41.04
e. 14.14
f. 42.43
Find the missing value
Find the measure of the missing side or hypotenuse for the triangle.
a. 6651.87 ft
b. 15.45 ft
x
c. 137.97 ft
d. 16.48 ft
Find the missing value
Find the measure of the missing side or hypotenuse for the triangle.
a. 106.48 ft
b. 135.32 ft
c. 4.95 ft
d. 8398.54 ft
Find the missing value
Find the measure of the missing side or hypotenuse for the triangle.
a. 72.79 m
b. 74.89 ft
c. 445.38 ft
d. 524.46 m
e. 355.77 m
f. 3090.96 ft
Find the missing value.
Find the measure of the missing angle.
b. 4.76
a. 60
Angle A
6 ft
c. 15.95
Things to remember.
To solve a triangle find all missing sides an angles.
Use inverse trigonometric functions to find a missing angle.
Assignment
Geometry:
8.3 Practice 16 – 23
Back 13, 14
Angles of Elevation & Depression
10.2
Definitions
Angle of elevation is the angle between the
horizontal and the line of sight to an object
above the horizontal.
Angle of depression is the angle between
the horizontal and the line of sight to an
object below the horizontal.
Angles of Elevation and Depression
Top Horizontal
Angle of Depression
Angle of Elevation
Bottom Horizontal
Since the two horizontal lines are parallel, by Alternate Interior Angles
the angle of depression must be equal to the angle of elevation.
Classify the angles as an
angle of elevation or an
angle of depression.
1
1 is formed by a horizontal line and a line of
sight to a point below the line. It is an angle of
depression.
4
4 is formed by a horizontal line and a line of sight
to a point above the line. It is an angle of elevation.
Use the diagram to classify
the angles as an angle of
elevation or depression.
5
5 is formed by a horizontal line and a line of
sight to a point below the line. It is an angle of
depression.
6
6 is formed by a horizontal line and a line of sight
to a point above the line. It is an angle of elevation.
Classify the angles as an angle of elevation
or depression.
6 angle of depression
9 angle of elevation
When the sun is 62˚ above the horizon, a building
casts a shadow 18 m long. How tall is the building?
x
62˚
18 shadow
Using the 62˚ angle as a reference, we use opposite and adjacent side.
opp
x
Use
tan 62˚ =
tan
adj
18
18 (tan 62˚) = x
18 (1.8807) = x
x ≈ 33.9 m
A kite is flying at an angle of elevation of about 55˚. Find
the height of the kite if 85m of string has been let out.
kite
85
x
55˚
Using the 55˚ angle as a reference, we use hypotenuse and opposite side.
opp
x
Use
sin 55˚ =
sin
hyp
85
85 (sin 55˚) = x
85 (.8192) = x
x ≈ 69.6 m
A 5.50 foot person standing 10 feet from a street light casts a
14 foot shadow. What is the height of the streetlight?
5.5
10
tan x˚ =
5.5
14
x° ≈ 21.45°
14 shadow
x˚
tan 21.45 
height
24
About 9.4 ft.
The angle of depression from the top of a tower to a
boulder on the ground is 38º. If the tower is 25m high,
how far from the base of the tower is the boulder?
38º
angle of depression
Alternate Interior Angles are congruent
25
38º
x
Using the 38˚ angle as a reference, we use opposite and adjacent side.
opp
Use
tan
tan 38˚ = 25/x
adj
(.7813) = 25/x
x = 25/.7813
x ≈ 32.0
Jody sees a plane above the airport at an angle of elevation of
32°. She is 2 miles from the airport where it is circling. How
high is the airplane above the airport?
The plane is
approximately 1.25
miles above the airport.
P
x
32°
A
opp
tan B 
adj
2 mi
x
tan 32 
2
2 tan 32  x
J
1.25  x
A forestry service has two fire towers located 8,000 feet apart. If
the first is located 1,000 feet above on a mountain, what is the
angle of depression from the first to the second tower?
T1
X°
1,000 ft.
8,000 ft
opp
sin X 
hyp
1000
sin X 
8000
sin X  .125
X  sin 1 (.125)
T2
X  7.18
The angle of depression from the first
tower to the second is about 7.18°.
Suppose the ranger sees another fire and the angle of
depression to the fire is 3°. What is the horizontal
distance to this fire? Round to the nearest foot.
3
3°
By the Alternate Interior Angles Theorem, mF = 3°.
Write a tangent ratio.
x  1717 ft
Multiply both sides by x and
divide by tan 3°.
Simplify the expression.
Solve for x.
x
x
a. Tan 12 =
3
y  .6377 km
b. Tan 3 =
x
6
x  .3144 mi
Assignment
Geometry:
8.4 Angle Elevation & Depression
1 – 13