Download Unit 4 - Basic Trigonometry Packet

Pre-AP/GT Pre-Calculus
Unit 4 – Basic Trigonometry
Review of Right Triangles and Basic Trigonometric Functions (Cosine, Sine, Tangent
cos 
sin 
tan 
Two-Special Right Triangles (From Geometry)
45-45-90
cos 45 
sin 45 
tan 45 
30-60-90
cos 30 
cos 60 
sin 30 
sin 60 
tan 30 
tan 60 
1
Angles on a Circle
Vocab
Parts of an Angle
Initial and Terminal Side and Vertex
Angle in Standard Position
The initial side is on the positive x-axis with the vertex at the origin
Positive Angles
Direction of the angles that are measured counterclockwise from the initial side.
Negative Angles
Direction of the angles that are measured clockwise from the initial side.
Co-Terminal Angles
Angles that have the same initial and terminal side.
Rotation
The direction and measure of an angle rotated around a circle.
Draw and label degrees of circle on the axis of the coordinate plane
State the Quadrant in which the terminal side of the given angle lies and draw the angle in standard
position.
1. 187°*
2. – 14.3°
3. 245°
4. – 120°
*5. 800°
7. – 460.5°
8. 315°
9. – 912°
10. 13°
11. 537°
6. 1075°
12. – 345.14°
2
Recall from problem # 5, when we had to graph 800°, we did so by subtracting 360 ° until the angle was between
0    360 . We got 80°. Therefore, we say, “800° and 80°” are Co-Terminal Angles. In other words, they have the
same angle measure; their initial and terminal sides are the same.
Find two angles, one positive and one negative angle, that are co-terminal with the given angle.
13. 74°
14. – 81°
15. 115.3°
Neg:
Neg:
Neg:
Pos:
Pos:
Pos:
16. 275°
17. – 180°
18. – 370°
Neg:
Neg:
Neg:
Pos:
Pos:
Pos:
More Vocab:
Complementary angles:
Supplementary angles:
Find the complement and supplement for the given angles:
19. 17.11°
20. 123.2°
Complement:
Complement:
Supplement:
Supplement:
Find the degree measure of the angle for each rotation. Draw the angle in standard position.
5
3
21.
rotation clockwise
22.
rotation counterclockwise
8
5
23.
7
rotation counterclockwise
9
*24.
17
rotation clockwise (what does this fraction mean?)
4
3
Notes on Angles in a Plane Radians
What is a Radian:
A Radian is the measure of the central arc of a circle that intercepts an arc on the circle. The length of the arc is
equal to the length of the radius of the circle (We will make the radius = 1, we call this the unit circle)
Draw and label Radians of circle on the axis of the coordinate Plane:
Determine the quadrant each angle lies and sketch the angle in standard position.

6

1.
2.
3.
9
5
7
5.
17
8
4.
 5
7
7.  4.2 r
6. 3.7 r
Find the Complement and Supplement of each angle (if possible)
Complement:
Supplement:
6
5
10. 3 r
11. 1.5 r
Comp:
Comp:
Comp:
Comp:
Supp:
Supp:
Supp:
Supp:
8.

7
9.
4
How to convert Degrees to Radians:
12. 330 
14. 145 
13. –72 
16. 36 
15. 765 
How to Convert Radians to Degrees
17. 
7
2
18.
5
6
19.
2
9
20.

12
21. 2 r
Review of Co-Terminal Angles:
Find two co-terminal angles for the following. One Positive and One Negative.
11

22.
23. 
24. -50 
6
4
Pos:
Pos:
Pos:
Neg:
Neg:
Neg:
Determine if the following are co-terminal.
25. -30  , 330 
26.
5 17
,
6
6
27.
32 11
,
3
3
5
Notes on Reference Angles and Special Right Triangles
Definition of a reference angle:
Find the reference angle  ' and sketch  and  ' in standard position.
2.   132
1.   32
6.  

7.  
7
11. 1.4 r
9
7
12. 2.8r
3.   132
8.  
 4
5
13.. 4.22r
5.   200
4.   232
9.  
 2
7
10.  
20
9
15.  1.7 r
14. 5.95r
Review of Right Triangles and Basic Trigonometric Functions (Cosine, Sine, Tangent)
Find the Values of the three trigonometric functions using the right triangles below.
cos 
4
1.
sin  
tan  
cos 
2.
2
4
sin  
tan  
3
6
Notes on Unit Circle (Quadrant I and Quadrantals)
The Unit Circle is the circle that is centered at the origin of the coordinate plane and has a radius of one. If an
angle, that forms a right triangle, is drawn inside the unit circle, with its vertex at the center of the circle and the
right angle perpendicular to the x-axis, we can use trigonometry to evaluate the values of the coordinates on
the unit circle itself. (Side note, recall…Radians will be the arc length on the unit circle)
Note: X becomes the cos 
Y becomes the sin 
y
 tan 
x
What would be the values of cos  and
sin  on the x and y axis? (These are called
the Quadrantals.) (Put these on your unit
circle)
Special Right Triangles:
45-45-90
What would be the values of values of the
coorniates for all angles whose reference
angle is 45o (or
cos

4

30-60-90
sin

4

)?
4

tan

4

What would be the values of values of the
coorniates for all angles whose reference


angle is 30o  or  ?
6

cos

6

sin

6

tan

6

7
60-30-90
What would be the values of values of the
coorniates for all angles whose reference


angle is 60o  or  ?
3

cos

3

sin


3
tan

3

Notes on the Unit Circle (Day 2)
1. Determine the point on the Unit Circle that  
7
lies on.
4
2. Determine which quadrant(s) that  lies in under the given conditions.
a. sin   0 cos  0
b. tan   0 cos  0
c. sin   0
d. sin   cos
Evaluate the sine, cosine, and tangent of the given angle,  . Use the unit circle.
3.  
7
6
4.  
7.  
11
4
8.  

*5.  
3

17
6
6.  
10.  
9.   2
2
3
2
11
6
Use the Unit circle to find the angle,  , that has the following trigonometric value.
11. sin  
3
2
12. cos   
2
2
13. sin   
1
2
14. tan  undefined
8
Notes on Arc Length and Area of a Sector
Arc Length
Find the length of the arc on a circle of radius, r, with a central angle,  .
3


1. r  4,  
2. r  3.5,  
3. r  10,   
12
4
6
4. r  100 ft ,   2.57r
5. r  10,   60
6. r  4,   340
Find the radian measure of the central angle of a circle with radius, r, that intercepts an arc length, s.
7. r = 20, s = 15
8. r= 33 inches, s = 6 inches
Area of a Sector:
Find the area of a sector with radius, r, and central angle,  .
1. r  5;  120
2. r  8.4; 
2
3
9
Applications:
1. Pittsburg, PA is located at 40.5  N while Miami is located at 25.5  N. Assuming the Earth is a perfect sphere,
how many miles apart are the two cities (Earth radius is 4,000 miles).
2. A bike track is in the shape of a circle with a radius of 40 ft. Bilbo travels around the track 8.5 times.
What is the distance that he traveled? If it took him 30 minutes, what was his average speed?
2. A sprinkler can spray water 75 feet and rotates through an angle of 135  .
Find the area of the region that the sprinkler covers.
More Practice:
1. Assume that the Earth travels in a circle around the sun (it is actually elliptical, but for the sake of this
problem, we will assume it’s a circle). How many miles does the Earth travel in half a year? (hint, how many
radians?)
Assume the radius from the Sun to the Earth is 93 million miles.
What is the average speed of the Earth throughout that time? (Recall 1yr = 365 days)
10
Review:
Find the Cosine, Sine, and Tangent Values for the following:
1.   2
2.  
2
3
3.  
7
4
Find the angle(s),  , that has the following trigonometric values.
1
5. tan  1
6. sin   
2
4.  
3
2
7. cos  0
Bonus: Find the angle(s),  , that has the following trigonometric values: 1  2 cos  0
Notes on Linear and Angular Velocity
Terms and formulas you need to know:
1. Arc Length
2. Area of a Sector
4. Revolution
5. Rotation
3. Converting Degrees to Radians
Angular and Linear Velocity:
6. Angular displacement
7. Angular Velocity
8. Linear Velocity
Examples:
1. A CD makes
4
a rotation about its axis. Find the angular displacement in radians
5
2. Find the Angular Velocity in radians per second of a wheel turning 25 revolutions per minute.
11
3. A sector has an arc length of 16 cm and a central angle measuring .95 radians. Find the radius of the circle and the area of the sector
4. The wheel of a truck is turning at 6 rps. The wheel is 4 ft in diameter. Find the angular velocity of the wheel in radians per second.
Find the linear velocity.
5. Two pulleys, one 6 inches in diameter and the 2 feet in diameter, are connected by a belt. The larger pulley revolves at the rate of 60
rpm. Find the linear velocity of the belt in feet per minutes. Calculate the angular velocity of the smaller pulley in radians per minute.
6. A pulley of radius 12 cm turns at 7 rps. What is the linear velocity of the belt driving the pulley in meters per second?
7. A trucker drives 55 mph. Her truck’s tires have a diameter of 26 inche4s. What is the angular velocity of the wheels in revolutions
per second?
Extra Practice:
1. An arc is 6.5 cm long and is intercepted by a central angle of 45 degrees. Find the radius of the circle.
2. How many revolutions will a car wheel of diameter 30 inches make as the car travels a distance of one mile?
12
3. Pittsburg and Miami are on the same meridian. Pittsburg has a latitude of 40.5 degrees north and Miami, 25.5 degrees north. Find
the distance between the two cities. The radius of the earth is 3960 miles.
4. Kim’s bicycle wheel has a 26 inch diameter. To the nearest revolution, how many times will the wheel turn if it is ridden for one
mile? Suppose the wheel turns at a constant rate of 2.5 revolutions per second. What is the linear speed of a point on the tire in feet per
second? What is the speed in miles per hour?
5. A merry-go-round makes 8 revolutions per minute.
a) What is the angular velocity of the merry-go-round in radians per minute?
b) How fast is a horse 12 feet from the center traveling?
c) How fast is a horse 4 feet from the center traveling?
6. A circular gear rotates at the rate of 200 rpm.
a) What is the angular speed of the gear in radians per minute?
b) What is the linear speed of a point on the gear 2 inches from the center in inches per minute?
7. A merry-go-round horse is traveling at 10 feet per second and the merry-go-round is making 6 revolutions per minute. How far is the
horse from the center of the merry-go-round?
13
Notes on Right Triangle Trigonometry
(6 Trig Functions)
Right Triangle Trigonometric Functions.
Trig. Functions
Reciprocal Function
cos  
sec 
sin  
csc 
tan  
cot  
Find the exact values of the 6 trigonometric functions of the angle  .
1.
2.
3.
Given the following trigonometric function of the acute angle  , find the other five trigonometric
functions.
3
3
4. sin  
5. cot  
4
2
6. sec 
5
4
8. tan  2
7. csc  3
Given the coordinate, determine the exact value of the six trig. functions.
  5 12 
9. 
, 
 13 13 
 15  8 
10.  ,

 17 17 
14
Use the given function values to find the indicated trigonometric functions.
11. sin(
2
3
2
1
and cos( )  
)
3
2
3
2
tan
2
=
3
cot
2
=
3
csc
2
=
3
Notes on Unit Circle (revisited)
Evaluate all six trigonometric functions for given angle.
1.  
2
3
2.  
5
4
Domain Sine and Cosine
3.  
3
2
11
6
4.  
Period of Sine and Cosine
A function, f, is periodic if there is a number, c, such
that f (t  c)  f (t ) for all t. The smallest number for
which f is periodic is called the period.
Evaluate the trigonometric functions using its period:
1. cos(7 )
2. sin
11
4
3. sin
 7
2
4. cos
 15
6
15
Review of Even Functions
sec( x) 
cos( x) 
Review of Odd Functions
sin( x) 
cot( x) 
tan( x) 
csc( x) 
Examples:
1. If csc t  5 , then:
sin(t )  ______
csc(t )  _____
sin(t )  ______
2
, then:
3
cos(t )  ______
cos(t )  ______
3. If cot(t )  3 , then:
cot(t )  ______
tan(t )  ______
sin(  t )  ______
sin(t   )  _____
2. If sec( t )  
4. If sin(t ) 
4
, then:
5
tan(t )  ______
Notes on Co-Functions and Calculator Trigonometry
Trigonometric Identities:
Reciprocal Identities
Quotient Identities
Use the identities to transform the left side into the right side.
1. sec cot   csc
2. sin  sec  tan
3. sin

6
cot

6

4.
cos 60

cos 30
16
Co-Functions
Co-Function Identities – Will be used in Calc BC 


sin  u   cos u
2



cos  u   sin u
2



tan  u   cot u
2



csc  u   sec u
2



sec  u   csc u
2

Remember

2


cot   u   tan u
2

 90
Examples: Draw a triangle, use Trigonometric Values, or use your co-functions
1. cos 30 
1
2
2. tan   3
a. tan 30  __________
b. sec 30  __________
c. sin(90  30)  __________
d. cos(90  30)  __________
a. cot   __________
b. sin   __________
c. sin(
3. csc  
3
2

2
  )  __________
a. sin   __________
b. sec  __________
c. cos   __________
d. tan(

2
  )  __________
Right triangle Problems: Solve for x,y, or r as indicated.
1.
2.
17
Calculator Trigonometry: Round to 4 decimal places (be careful of Mode)
sin

3
7
tan1.28
sec .77
cot 77.5
sin 57.8
csc 29.5
csc
5
cos 38
(sin 26)2  (cos 26)2 
The Doctor is standing 50 feet from the base of a tall building staring at a Dalek located on the 45 th floor. He estimates the
angle of elevation to be 65 .
a. What is the height of the building to the 45th floor?
b. What is the distance between the Doctor and the Dalek?
Notes on Trigonometric Functions of any Angle
How to get the Trig Function of any angle:
Let  be an angle in standard position with (x, y) being a point on the terminal side of  and r 
cos 
sec 
sin  
csc 
tan 
cot  
x2  y 2 .
18
Determine the exact value of the 6 trig functions given (4, -7). Draw a triangle.
State the quadrant(s) in which  lies.
1. tan  0
4.
sin  0 cos  0
2.
sin  0 tan  0
3.
cos  0
5.
sin   cos
6.
sin   0 cos   0
Find the values of the 6 trig functions with the given constraints. Draw a triangle.
cos  
7.
3
5
 lies in Quadrant IV
8.
csc  2
cos   0
Evaluate the Sine, Cosine, and Tangent without a Calculator
9.
10.  840
300
Find
12.
 , for 0     , for the following problems.
sin  
3
 =_____
2
13.
11.
10
3
Put answer in radians.
cos  1

= ______
14.
tan  undefined

=______
19
Notes on Finding the Angle Using Inverses
Use a calculator to evaluate the following. Round to 4 decimal places.
1. cos 125
2. csc168
3. cot 150
5. sec
3
5
6. cos
 2
7
Verify using the trigonometric identities:
9. tan x  cot x  1
7. tan
4. sin 345
13
8
8. sin 3.42 r
10. sin x  cot x  cos x
Find the two solutions for the given equations.
3
 3
1. cos  
2. sin  
2
2
3. csc   2
4. cot   3
*6.
*5. 1  2 sin x  0
3  2 cos x  0
20