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
1
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
30o
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
28o
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
2
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.
1.) 240o
2.) 500o
3.) -50o
Apr 7­9:55 AM
coterminal angles:
Examples: Find one positive angle and one negative angle that are
coterminal with the given angles.
1.) 45o
2.) -380o
Angles can also be measured in __________.
There are ____ radians in a full circle.
_____ radians = 360o , so ____ radians = 180o.
-To convert degrees to radians, multiply by π .
-To convert radians to degrees, multiply by 180 .
Apr 7­10:18 AM
3
Examples:
1.) Convert 125o to radians.
Degree measure
0o
30o
2.) Convert -π to degrees.
Radian measure
π/4
60
o
π/2
2π/3
135o
150o
180o
7π/6
5π/4
240o
270o
5π/3
315o
11π/6
360o
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
side of θ intersects the circle x2 + y2 = r2. The six trig functions of θ are as follows:
(x,y)
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
r
θ
Apr 7­10:44 AM
4
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.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
The Unit Circle : the circle x 2 + y2 = 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
θ=270o.
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Reference Angles Acute angles formed by the terminal side of θ
and the x-axis.
Recall:
30o =
45o =
60o =
60o
45o
2
1
1
√2
30o
√3
45o
1
Apr 7­11:02 AM
5
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
6
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 θ's that could satisfy the above equation.
For this reason, we must make some restrictions.
Inverse Trig Functions :
-Sine Inverse: -90o≤θ≤90o
Cosine Inverse: 0 o≤θ≤180o
-Tangent Inverse: -90o≤θ≤90o
Apr 7­11:24 AM
Examples Evaluate the expression in both radians and degrees.
1.) cos -1 √3
2
2.) sin -1 -√2
2
Apr 7­11:40 AM
7
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
Los Angeles has an angle of elevation of 30o. 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
8
Homework #1
Name: ______________
1.) Find all 6 trig functions for 30o, 45o and 60o 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
30o
45o
60o
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
9
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
is 48o. Estimate the parasailer's height
above the boat.
300 ft
48o
Mar 26­7:46 PM
10
Homework #2
Name: ______________
Draw an angle with the given measure in standard position.
1.) 110o
2.) 450o
3.) -3π/2
ange to degrees f
Find one positive angle and one negative angle that are coterminal with
the given angles.
4.) -87o
5.) 120o
Apr 7­12:44 PM
6.) 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 =
7.) 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
11
Evaluate the six trig functions of θ. Simplify and rationalize.
8.) θ = π
9.) θ = 4π/3
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
sinθ =
cscθ =
cosθ =
secθ =
tanθ =
cotθ =
Apr 7­12:53 PM
Evaluate the expressions in both radians and degrees.
10.) cos-1 (1/2)
11.) tan-1 (-1)
12.) 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
12