Download CW# 2

UNIT 7: TRIGONOMETRY BASICS
CW #1: Right Triangle Trig and Special Right Triangles
Find the measure of x in each of the following. Leave irrational answers in radical form:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet
long, what is the measure of the angle that the ramp makes with the ground?
14.
From a point on the ground that is 100 feet from the base of a building, the tangent of the
5
angle of elevation of the top of the building is . To the nearest foot, how tall is the
4
building?
15.
Find the measure of the diagonal of a rectangle whose sides measure 14 and 48.
16.
The length of a rectangle is 7 meters more than its width. A diagonal of the rectangle
measures 17 meters. Use algebra (not guess and check!) to find the length of the rectangle.
17.
In a triangle, the lengths of the sides are 7, 8, and 12. Is this a right triangle? Show your
reasoning.
CW# 2: Creating a unit circle “calculator”: Coordinate Geometry, Rotations, Unit
Circle, Degree Measure, more Vocabulary
DEFINE Unit Circle:
I. Determine the quadrant in which an angle of the given measure lies:
_______________ 1.
140
________________ 2.
210
_______________ 3.
97
________________ 4.
315
_______________ 5.
80
________________ 6.
240
_______________ 7.
168
________________ 8.
200
II. Find the angle of smallest positive measure coterminal with an angle of the given measure:
_______________ 9.
400
________________ 10.
710
_______________ 11.
580
________________ 12.
790
_______________ 13.
30
________________ 14.
370
_______________ 15.
400
________________ 16.
800
III. Draw a sketch of an angle of the given measure, indicating the direction of the rotation by an
arrow.
17.
18.
100
210
19.
140
IV. Determine the quadrant in which an angle of the given measure lies:
_______________ 1.
475
________________ 2.
428
CW #3: Creating a unit circle “calculator”: Radian Measure & Conversions
1.
Define Unit Circle:
2.
Define RADIAN:
_______________3.
One radian is approximately how many degrees?
_______________4.
How many radians are in the circumference of a circle?
_______________5.
How many radians are in ½ the circumference of a circle?
_______________6.
How many radians are in ¼ the circumference of a circle?
_______________7.
What are the coordinates of the point at 0 and 360 degrees on the unit circle?
_______________8.
What are the coordinates of the point at 90 degrees on the unit circle?
_______________9.
What are the coordinates of the point at 180 degrees on the unit circle?
_______________10.
What are the coordinates of the point at 270 degrees on the unit circle?
11.  is the measure of a central angle that intercepts an arc of length s in a circle whose radius
has length r .
_______________ a.
If s  12 and r  4, find  .
_______________ c.
If r  4 and   2.5, find s .
_______________ b.
If s  12 and   6, find r .
12.
On a clock, the length of the pendulum is 30 centimeters. A swing of the pendulum
determines an angle of 0.8 radian. Find, in centimeters, the distance traveled by the tip of
the pendulum during this swing.
13.
Convert the following to radian measure:
_______________ a. 180
_______________b. 90
_______________ d. 360
_______________e. 60
_______________ c. 45
_______________
f. 30
_______________ g. 270
14.
_______________h. 150
_______________ i. 240
Convert the following to degree measure:
___________ a. 1 radian
____________b. 2 radians
_______________ d. 
_______________e.
_______________ g.
_______________ j.

10
3
4
_______________h.
_______________k.

3

2
7
2
___________ c. 5 radians
_______________ f.

9
_______________ i.
5
6
_______________ l.
4
3
CW #4: Creating a unit circle “calculator”: Unit Circle Trigonometry
1.
Define UNIT CIRCLE:
2.
Define RADIAN:
3.
What are the COORDINATES of the point at 90 degrees on the unit circle?
4.
What are the COORDINATES of the point at 30 degrees on the unit circle?
5.
What are the COORDINATES of the point at 45 degrees on the unit circle?
6.
Give the EXACT value of each of the following (NO DECIMALS … NO CALCULATORS!)
a. sin 90
b. cos 45
c. tan 90
d. cos 30
e. sin 60
f. cos180
g. tan 270
h. tan 30
i. sin
j. sin 
k. cos
m. sin
p. sin

4
3
4
n. cos
q. tan

2

3
11
6

6
l. tan 
o. cos
3
2
r. cos
4
3
7.
What is the SLOPE of every vertical line?
8.
What is the SLOPE of every horizontal line?
9.
Write the formula for tangent in terms of sine and cosine:
10.
Write the formula for tangent in terms of x and y:
11.
Use the idea that tangent = slope to find the EXACT value of each of the following:
a.
tan
d.
tan
g.
tan 0

6
11
6
b.
tan 
e.
tan
h.
tan 315
4
3
c.
tan
f.
tan
i.
tan
3
4

2
2
3
CW #5: Creating a unit circle “calculator”: Mastering Unit Circle Trigonometry
1.
Fill in the following unit circle completely and accurately. (Give each point all of its’ names)
Find the exact value of each of the following:

__________ 1.
sin 45
__________ 2.
cos
__________ 3.
tan
3
2
__________ 4.
sin
__________ 5.
cos
__________ 6.
tan 
__________ 7.
tan
__________ 8.
cos
__________ 9.
sin
__________ 11. tan

2

6

3

4
__________ 13. sin 2
__________ 15. cos
3

6

4
__________ 10. cos 
__________ 12. sin
__________ 14. tan

2

3

6
Find the EXACT value of each expression:
16.
tan(315 )  tan(135 )
17.
sin(300 )  sin(240 )
CW #6: Reference Angles
MUST DO:
Express the given function as a function of a positive acute angle.
1.
sin100
2.
cos190
3.
sin 248
4.
sin 98
5.
tan 345
6.
tan 620
7.
sin(20 )
8.
cos 290
9.
sin(158 )
10.
tan 300
11.
cos(200 )
12.
tan(80 )
13.
sin 340
14.
cos150
15.
tan 237
CW #7: Reciprocal Trig Functions
Find the EXACT numerical value of each expression:
1. csc150
2.
sec 240
3. cot 315
4.
csc120
5. sec2
6.
cot
3
2
8.
sec
10.
csc
12.
cot 2
7. csc
9. sin

tan
3
11. cos 2

4

6
 sec 2

4

6
5
4

2
 sec 

3
 csc

2
Write each expression in terms of sin  , cos  , or both. Simplify whenever possible.
1. tan 
2.
cot 
3.
sec
4. csc
5.
tan  csc
6.
tan 
sec 
cot 
csc 
8.
sin 
csc 
9.
sec 
csc 
7.
Name the quadrant in which angle B must lie:
10.
cot B  0 and sin B  0
11.
sec B  0 and tan B  0
CW #8: Parent Trig Functions
MUST DO:
Accurately sketch each of the following parent trig functions in the interval 0    2 without a
calculator:
1.
f ( x)  sin x
2.
f ( x)  cos x
3.
f ( x)  tan x
Use your knowledge of transformation of parent functions (unit 1) to graph each of the following
without a calculator.:
4.
f ( x)  sin x  2
5.
f ( x)   cos x
Use your knowledge of transformation of parent functions (unit 1) to graph each of the following
without a calculator:
6.
f ( x)  sin( x  2 )
7.
f ( x)  2 cos x
CW #9: Amplitude, Frequency, Period, Phase Shift & Midline of Trig Graphs
I. State the amplitude, frequency, period, midline, and phase shift (if applicable) of each of the
following functions:
1.
3.
5.
f ( x)  2 cos x
2.
f ( x)  cos 2 x
Amplitude = __________
Amplitude = __________
Frequency = __________
Frequency = __________
Period = _____________
Period = _____________
Midline = _______________
Midline = _______________
Phase Shift = _______________
Phase Shift = _______________
1
f ( x)  sin 3x
2
4.
f ( x)  2sin 4 x  3
Amplitude = __________
Amplitude = __________
Frequency = __________
Frequency = __________
Period = _____________
Period = _____________
Midline = _______________
Midline = _______________
Phase Shift = _______________
Phase Shift = _______________
1


f ( x)  sin  x  
2
2

6.
f ( x)  sin  x    1
Amplitude = __________
Amplitude = __________
Frequency = __________
Frequency = __________
Period = _____________
Period = _____________
Midline = _______________
Midline = _______________
Phase Shift = _______________
Phase Shift = _______________
For #7-13, graph the function in the interval   x  2 without a calculator.
7.
f ( x)  2sin x
8.
f ( x)  cos 2 x
9.
1
f ( x)  2 cos x
2
10.
f ( x)   sin 2 x  2
11.


f ( x)  sin  x  
4

12.
f ( x)  3cos( x   )
13.


f ( x)  5cos  2 x    1
2

(Change the scale on the y-axis to make it fit)
CW #10: Modeling with Trigonometry
In #1-4, write a function for the sinusoid.
1.
2.
3.
4.
5.
The lowest frequency of sounds that can be heard by humans is 20 hertz. The maximum
pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write a
sine model that gives the pressure P as a function of time t (in seconds).
6.
The table shows the numbers of employees N (in thousands) at a sporting goods company
each year for 11 years. The time t is measured in years, with t=1 representing the first
year. (a) Use sinusoidal regression to find a model that gives N as a function of T. (b) Use
your model to predict the number of employees at the company in the 12th year.
t
N
1
20.8
2
22.7
3
24.6
4
23.2
5
20
6
17.5
7
16.7
8
17.8
9
21
10
22
11
24.1
CW #11: Using Trig Identities
1.
An identity is a statement that is always ________________.
In #2-5, Find the values of the other five trigonometric functions of  .
2.
5
3
csc    ,    
3
2
3.
1

sin   , 0   
3
2
4.
3 
tan    ,    
7 2
4.
5
3
cos    ,    
6
2
In #6-9, simplify the expression
6.
sin x cot x
7.
cos  1  tan 2  
8.
cos 2 
cot 2 
9.


sin     sec 
2

In #10-13, verify each identity.
10.
12.
sin x csc x  1


cos      1
2
 1
1  sin( )
11.
tan  csc cos  1
13.
1  cos x
sin x

 2 csc x
sin x
1  cos x
CW #12: Using Sum & Difference Formulas
1.
Write the expression cos130 cos 40  sin130 sin 40 as the cosine of an angle.
In #2-7, find the exact value of the expression.
2.
tan  15

4.
sin  165
6.
 11 
cos 

 12 

3.
tan 195

5.
cos 105

7.
 17 
tan 

 12 
In #8-11, simplify the expression.
8.
tan  x   
9.