Download Trig Packet Notes #1.notebook

Trig Packet Notes #1.notebook
February 15, 2017
Trigonometry Packet #1
Objectives: Students will be able to solve triangles using trig
ratios and find trig ratios of a given angle.
hy
po
te
opposite
side
Name: ________________
S O H
nu
se
C A H
T O A
θ
adjacent side
Right Triangle Definitions of Trig Functions
c
sinθ = ______
cscθ = ______
cosθ = ______
secθ = ______
tanθ = ______
cotθ = ______
b
a
Pythagorean Theorem:
________________
Mar 26­1:51 PM
Examples Evaluate the six trig functions of the angle θ.
1.)
θ
sinθ = ____
cscθ = ____
cosθ = ____
secθ = ____
tanθ = ____
cotθ = ____
sinθ = ____
cscθ = ____
cosθ = ____
secθ = ____
tanθ = ____
cotθ = ____
5
13
2.)
θ
5√2
5
Mar 26­6:39 PM
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Trig Packet Notes #1.notebook
February 15, 2017
Example: Let θ be an acute angle of a right triangle. Find the
values of the other five trig functions of θ.
tanθ = 7
sinθ = ____
cscθ = ____
3
cosθ = ____
secθ = ____
cotθ = ____
Example: Find x and y.
x
4
30
o
y
Mar 26­6:41 PM
Example: Solve ΔABC.
Note: This means to find all of the missing angles measures and side
lengths.
B
c
a
28
o
A
15
C
Example: A tree casts a shadow as shown. What is the
height of the tree?
31o
25 ft
Mar 26­7:02 PM
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February 15, 2017
Objectives: Students will be able to work with angles in standard position, convert between
radians and degrees and use the unit circle to solve problems.
standard position:
Examples: Draw an angle with the given measure in standard position.
o
o
1.) 240
o
2.) 500
3.) -50
Apr 7­9:55 AM
coterminal angles:
Examples: Find one positive angle and one negative angle that are
coterminal with the given angles.
o
1.) 45
o
2.) -380
Angles can also be measured in __________.
There are ____ radians in a full circle.
o
o
_____ radians = 360 , so ____ radians = 180 .
-To convert degrees to radians, multiply by π .
180
-To convert radians to degrees, multiply by 180 .
π
Apr 7­10:18 AM
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February 15, 2017
Examples:
o
1.) Convert 125 to radians.
Degree measure
o
0
o
30
2.) Convert -π to degrees.
Radian measure
π/4
60
o
π/2
2π/3
o
135
o
150
o
180
7π/6
5π/4
o
240
o
270
5π/3
315
o
11π/6
360
o
Apr 7­10:30 AM
Fill in the ratios using O = opposite, A = adjacent and H = hypotenuse.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
General Definitions of Trig Functions
Let θ be an angle in standard position, and let (x,y) be the point where the terminal
2
2
2
side of θ intersects the circle x + y = r . The six trig functions of θ are as follows:
(x,y)
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
r
θ
Apr 7­10:44 AM
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Trig Packet Notes #1.notebook
February 15, 2017
Example: Let (-4,3) be a point on the terminal side of an angle θ
in standard position. Evaluate the six trig functions of θ.
2
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
2
The Unit Circle : the circle x + y = 1, which has center (0,0) and
radius 1.
sinθ =
cscθ =
(x,y)
1
cosθ =
secθ =
tanθ =
cotθ =
θ
Apr 7­10:53 AM
Example Use the unit circle to evaluate the six trig functions of
o
θ=270 .
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Reference Angles Acute angles formed by the terminal side of θ
and the x-axis.
Recall:
o
30 =
o
45 =
o
60 =
60o
45o
2
1
1
√2
30o
√3
45o
1
Apr 7­11:02 AM
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Trig Packet Notes #1.notebook
February 15, 2017
Examples: Evaluate the six trig functions of θ. Simplify and
rationalize.
1.) θ = π/3
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
2.) θ = 7π/6
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­11:12 AM
3.) θ=7π/4
4.) θ=2π/3
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­11:15 AM
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Trig Packet Notes #1.notebook
February 15, 2017
Objectives: Students will be able use inverse trig functions to solve for angles.
So far, we've learned how to evaluate trig functions of a given angle.
Now, we'll study how to reverse the problem - find an angle that
corresponds to a given value of a trig function.
Example sinθ = 1
Note: There are many θs that could satisfy the above equation.
For this reason, we must make some restrictions.
Inverse Trig Functions :
o
o
-Sine Inverse: -90 ≤θ≤90
o
o
o
Cosine Inverse: 0 ≤θ≤180
o
-Tangent Inverse: -90 ≤θ≤90
Apr 7­11:24 AM
Examples Evaluate the expression in both radians and degrees.
-1
1.) cos √3
2
-1
2.) sin -√2
2
Apr 7­11:40 AM
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Trig Packet Notes #1.notebook
February 15, 2017
Examples Find the measure of angle θ.
1.)
θ
9
4
2.) A monster truck drives off a ramp in order to jump onto a row
of cars. The ramp has a height of 8 feet and a horizontal length of
20 feet. What is the angle θ of the ramp?
Apr 7­11:43 AM
Some More Application Problems
1.) The escalator at the Wilshire/Vermont Metro Rail Station in
o
Los Angeles has an angle of elevation of 30 . The length of the
escalator is 152 feet. What is the height of the escalator?
2.) A fire truck has a 100 ft. ladder whose base is 10 feet
above the ground. A firefighter extends a ladder toward a
burning building to reach a window 90 ft. above the ground. Draw
a diagram. At what angle should the firefighter set the ladder?
Apr 7­11:55 AM
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Trig Packet Notes #1.notebook
February 15, 2017
Find the length of the arc.
Feb 25­8:44 AM
Feb 25­8:55 AM
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Feb 25­8:56 AM
Feb 25­8:56 AM
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Trig Homework #1
Name: ______________
o
o
o
1.) Find all 6 trig functions for 30 , 45 and 60 and fill in the
table below. Make sure to rationalize all values.
60o
45o
2
1
1
√2
30o
45o
√3
θ
sinθ
cosθ
tanθ
cscθ
1
secθ
cotθ
o
30
o
45
o
60
Mar 26­7:19 PM
2.) Evaluate the six trig functions of θ.
sinθ = ____
cscθ = ____
cosθ = ____
secθ = ____
tanθ = ____
cotθ = ____
θ
17
15
3.) Let θ be an acute angle of a right triangle. Find the values
of the other 5 trig functions of θ. sinθ = ____
cscθ = ____
cotθ = 6
11
cosθ = ____
secθ = ____
tanθ = ____
cotθ = ____
Mar 26­7:38 PM
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4.) Solve ΔABC.
A
35o
16
b
B
a
C
B = ____
b = ____
a = ____
Mar 26­7:43 PM
5.) Find the length, x, of the prop holding open the piano.
x
25o
150 cm
6.) A parasailer is attached to a boat
with a rope 300 feet long. The angle of
elevation from the boat to the parasailer
o
is 48 . Estimate the parasailer's height
above the boat.
300 ft
48o
Mar 26­7:46 PM
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Trig Packet Notes #1.notebook
February 15, 2017
o
7.) On an analog clock, the minute hand has moved 128 from the
hour. What number will it pass next?
8.) It is 2:46 p.m. What is the measure of the angle that the
minute hand swept through since 2:00 p.m?
Feb 25­8:41 AM
Draw an angle with the given measure in standard position.
9.) 110
o
o
10.) 450
11.) -3π/2
(Hint: change to degrees f
Find one positive angle and one negative angle that are coterminal with
the given angles.
o
12.) -87
13.) 120
o
Apr 7­12:44 PM
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Trig Packet Notes #1.notebook
February 15, 2017
14.) Let (-3,-4) be a point on the terminal side of an angle θ in
standard position. Evaluate the six trig functions of θ.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
15.) Let (-6,9) be a point on the terminal side of an angle θ. Find all
the trig ratios. Simplify and rationalize all values.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­12:50 PM
Evaluate the six trig functions of θ. Simplify and rationalize.
16.) θ = π
17.) θ = 4π/3
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­12:53 PM
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Trig Packet Notes #1.notebook
February 15, 2017
Evaluate the six trig functions of θ. Simplify and rationalize.
18.) θ = -7π/6
19.) θ = -π/4
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­12:53 PM
Evaluate the expressions in both radians and degrees.
-1
-1
-1
20.) cos (1/2)
21.) tan (-1)
22.) sin (√2/2)
23.) A crane has a 200 ft. arm with a lower end that is 5 ft.
off the ground. The arm has to reach to the top of the building
that is 160 ft. high. At what angle θ should the arm be set?
Apr 7­12:56 PM
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February 15, 2017
24.) A geostationary satellite is positioned 35,800 km above the
Earth's surface. It takes 24 hours to complete one orbit. The
radius of the Earth is about 6,400 km. What distance does the
satellite travel in 1 hour? 3 hours?
After how many hours has the satellite travelled 200,000 km?
Feb 25­8:50 AM
25.) Suppose a windshield wiper arm has a length of 22 inches and
o
rotates through an angle of 110 . What distance does the tip of the
wiper travel as it moves once across the windshield?
26.) A CD with a diameter of 12 cm spins in a CD player. Calculate
how much farther a point on the outside edge of the CD travels in
one revolution than a point 1 cm closer to the center of the CD.
Feb 25­8:52 AM
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