INTRO TO
TRIGONOMETRY
Math 12
Miss Kersting
Room 227
Period 7
Name:____________________________
Topic
Day 1 – The Right Angle
Day 2 – Angles as Rotations
Day 3 – The Unit Circle
Day 4 – Special Angles
Day 5 – Reference Angles
Day 6 – Using the Calculator to find Function Values and an Angle / DMS
Day 7 – Reciprocal Functions
Day 8 – Converting Degrees to Radian Measure
Day 9 – s    r
Day 10 – Finding Remaining Trig Values When One is Known
Day 11 – Co-functions
Day 12 – Review
NOTE: Quizzes for this unit will be announced in class. Please record the dates when announced.
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Day 1 – The Right Triangle
Angle A
Angle B
Applying SOHCAHTOA
sin θ =
1.)
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
RST is a right triangle with m  s  90 , RS = 15, ST = 8, and RT = 17.
Give the value of each ratio as a fraction.
a.) sin R
d.) sin T
b.) cos R
e.) cos T
c.) tan R
f.) tan T
2.) In the right triangle PQR, <R is a right triangle, PQ = 25, QR = 24, and PR = 7.
Find the exact value of each trigonometric ratio.
a.) sin P
d.) sin Q
b.) cos P
e.) cos Q
c.) tan P
f.) tan Q
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Finding Sides and Angles using SOHCAHTOA
3.) In triangle QPR, <P is a right angle, QR=12,
and m<Q = 30. Find PR, to the nearest tenth.
4.) Right triangle BOY, m<O = 90, OY = 4.2,
and BY = 13.4, Find the measures of <B and <Y
to the nearest tenth of a degree.
5.) In triangle CTH, m<T = 90, m<C = 48, and CH = 50cm. Find the length of CT to the nearest
tenth of a centimeter.
6.) Amelia Ann is standing on the beach looking up at a lighthouse whose base is 650 feet from
where she is standing. If the angle of elevation from where Amelia Ann is standing to the top of the
lighthouse measures 52  , to the nearest foot, how tall is the lighthouse?
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Day 1 – The Right Triangle
HOMEWORK
1.) The lengths of the sides of ABC are 6, 8, 10, <C is the right angle, and the m<A is less than
m<B. Find:
a.) sin A
b.) cos A
c.) tan A
Directions: Solve each triangle for x, to the nearest tenth of a degree or to the nearest tenth of a
centimeter.
2.)
3.)
4.)
5.)
6.) A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the sine,
cosine, and tangent values of the angle that the ladder makes with the ground.
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7.) An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet long,
what is the sine of the angle that the ramp makes with the ground?
8.) The bed of a truck is 5 feet above the ground. The driver of the truck uses a ramp 13 feet long
to load and unload the truck. Find the sine, cosine, and tangent values of the angle that the ramp
makes with the ground.
9.) A 20-meter line used to keep a weather balloon in place. The sine of the angle that the line
3
makes with the ground is . How high is the balloon in the air?
4
10.) 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 building?
4
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Review:
11.) Use the discriminant to determine all values of k that would result in the equation
x 2  kx  16  0 having equal roots. HINT: when roots are equal, the discriminant equals 0
12.) Find the sum and product of the roots of the equation 3x 2  13x  32  0 .
Answers to Homework (Day 1):
1.) a.)
2.)
3.)
4.)
5.)
6
10
b.)
8
10
c.)
6
8
7.)
6.4
52.6
14.7
61.5
6.) sin A 
2.5
10
8.) sin A 
16
20
cos A 
12
20
tan A 
5
13
cos A 
12
13
tan A 
9.) 15
10.) 125
11.) 8
16
12
12.) sum =
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13
3
product =
32
3
5
12
Day 2 – Angles as Rotations
Do Now: (Question 1)
1.) Kathy is lying on a beach blanket 60 feet from the base of a 150-foot cliff. Determine the angle
of elevation, to the nearest tenth of a degree, at which Kathy sights the top of the cliff.
Classifying Angles by Quadrant
Positive Angles Measures
Quadrants
Negative Angle Measures
Initial side of an angle – the ray from which the rotation begins.
Terminal side of an angle – the ray at which the rotation ends.
Standard Position – an angle whose vertex is the origin and its initial side is the positive x-axis.
Directions: Determine the quadrant in which each angle lies.
2.) 145 
3.)  120 
4.) 410 
5.)  400 
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6.) 270 
Coterminal Angles | add or subtract 360°
Coterminal Angles – angles in standard position that share the same terminal side.
7.)
a.) Draw a 30  angle in standard position.
b.) Find a coterminal angle with the angle 30 
Directions: Find the angle of smallest positive measure coterminal with an angle of the given
measure.
8.) 580 
9.)  110 
10.)  800 
Directions: For each given angle, find a coterminal angle with a measure of  such that 0    360 .
11.)  85
12.) 980 
13.)  500 
Practice Problems
14.) Which angle is not
coterminal to 112 ?
(1) 248
(2) 68
(3) 472
(4) 832
15.) An angle whose measure is 16.) Which angle rotates
clockwise?
214 is coterminal with
all of the following except
(1) 104
(3) 128
(1) 574
(3) 146
(2) 87
(4) 372
(2) 34
(4) 506
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Day 2 – Angles as Rotations
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Determine the quadrant in which each angle lies.
1.) 215
2.)  110 
3.) 318
4.) 72 
5.) 422
6.)  300 
7.) 1050 
8.) 540 
Directions: For each given angle, find a coterminal angle with a measure of  such that 0    360 .
9.) 390 
10.) 412
11.)  10 
12.) 1,000 
14.)  1320 
13.) 980 
15.) a.) To insert a screw, should the screw be turned clockwise or counterclockwise?
b.) The thread spirals six and a half times around a certain screw.
How many degrees should the screw be turned to insert it completely?
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Review:
x
5
16.) The graph of the equation y    has an asymptote. On the grid below, sketch the graph of
2
x
5
y    and write the equation of the asymptote.
2
17.) Express 7 2 x5  3 18x5 in simplest radical form.
Answers to Homework (Day 2):
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
11.)
12.)
13.)
14.)
III
III
IV
I
I
I
IV
Quadrantal Angle
30
52
350
280
260
120
15.) a.) Clockwise
b.) 2340
16.)
x
y
-3
.064
-2
.16
1
.4
0
1
1
2.5
2
6.25
3
15.625
17.) 2 x2 2 x
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Day 3 – The Unit Circle
Do Now: (Questions 1&2)
1.) Which is coterminal with an angle of 45 ?
(1) 90  (2) 225 (3) 315 (4) 405
2.) Name the quadrant where 452 terminates.
The Unit Circle
The Unit Circle is the circle whose center is at the origin and whose radius is 1.
FIVE IMPORTANT PARTS OF THE UNIT CIRCLE:





Quadrants
Degree Measures
4 Ordered Pairs
Point (cos  , sin  )
Positive Function Values (ASTC)
3.) What line segment represents cos  ?
4.) What line segment represents sin  ?
5.) What line segment represents tan  ?
Directions: Use the unit circle given above to find each measure.
6.) cos 0º
7.) sin 0º
8.) tan 0º
9.) cos 90º
10.) sin -90º
11.) tan 90º
12.) cos 180º
13.) sin 180º
14.) tan 180º
15.) cos 270º
16.) sin -270º
17.) tan 270º
18.) cos 360º
19.) sin -360º
20.) tan 360º
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21.) P is a point on the unit circle with coordinates (0.6, 0.8) as shown below.
Find:
a.) cos 
b.) sin 
Directions: Name the quadrant in which the angle  terminates.
22.) sin   0 and cos  0
23.) sin   0 and cos  0
24.) tan   0 and sin   0
25.) cos  0 and tan   0
27.) tan   2.7 and sin   0
26.) tan x  1 and
2
cos x  
2
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Day 3 – The Unit Circle
HOMEWORK
 4 3
1.) Points R(1,0) and P   ,  are on the unit circle O. If mROP   , find:
 5 5
a.) sin 
b.) cos
c.) tan 
 5 12 
2.) If  is an angle in standard position and P  ,  is a point on the unit circle on the terminal
 13 13 
side of  , what is the value of tan  ?
Directions: Name the quadrant in which each angle terminates.
3
2
3.) sin B 
and cos B  0
4.) sin x 
and
5
2
cos x  
5.) sin   0 and tan   0
2
2
6.) If  is an angle in standard position on the
unit circle and its terminal side passes through
 3 1
,  , in what quadrant is  ?
the point 
2
2

7.) If tan A  0 and tan Asin A  0 , in what
quadrant does  A lie?
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8.) Name a line segment whose directed distance
is the value of:
a.) sin 
b.) cos 
c.) tan 
9.) In the accompanying diagram, PR is tangent to circle O
at R, QS  OR , and PR  OR . Which segment has the
same measure as tan  ?
Review:
10.) Rationalize the denominator:
1 3
1
x
x
11.) Express in simplest form:
1
1
x
1 3
Answers to Homework (Day 3):
1.) a.) 
2.) 
3
5
b.) 
4
5
c.)
6.) IV
7.) I
8.) a.) MN
9.) PR
10.) 1 3
11.) 1  x
3
4
12
5
3.) I
4.) II
5.) IV
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b.) KN
c.) LP
Day 4 – Special Angles
Do Now: (Questions 1-8) Multiple Choice
1.) If cos  is positive, and sin  is negative,
what could be the value of  ?
(1) 70º
(3) 195º
2 2
, in what quadrant(s) can
5
this angle terminate?
3
2
3
(4) sin 60   
2
(2) sin 240  
1
and tan  > 0, which must be
3
true?
(1) sin  < 0
(3) sin  = 0
(2) sin  > 0
(4) sin  undefined
6.) Which cannot be true?
(1) sin 30  
(2) 128º
(4) 313º
7.) If tan  > 0, which of the following is true?
(1)
(2)
(3)
(4)
4.) If cos  = 
(2) I and II
(4) II and IV
5.) If sin  < 0 and tan  < 0, what might be
the measure of  ?
(1) 81º
(3) 201º
3
2
3
(3) sin 120  
2
(1) sin 300  
(2) 100º
(4) 303º
3.) If tan  
(1) I only
(3) I and III
2.) Which could be true?
1
2
(3) tan 240    3
(2) tan 180   0
(4) cos 300  
8.) Which of the following is a false statement?
(1) tan  is undefined whenever cos  equals
zero
(2) tan  equals zero whenever sin  equals
zero
(3) sin  can equal cos  in quadrant I or
quadrant III of the unit circle
(4) sin  can equal cos  only in quadrant I of
the unit circle
sin  must be negative
cos  must be negative
 must be in Quadrant I
sin  may be positive or negative
Special Angles | Exact Values
sin
cos
tan
1
2
30º
1
2
3
2
3
3
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45º
2
2
2
2
60º
3
2
1
2
1
3
Directions: Without looking back at any notes fill in the 5 important parts of the Unit Circle and fill
in the Special Angles Chart.
Unit Circle
Special Angles Chart
Directions: Find the exact numerical value of each expression.
9.) sin 30   cos 60 
10.) tan 45   2 cos 60 

11.) tan 60 

2
 sin 180 

12.) cos 60  cos 0   cos180 
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
Day 4 – Special Angles
HOMEWORK
Directions: Find the exact numerical value for each expression.
1.) tan 45   sin 30 
2.) sin 45   cos 45 





  sin 30 
3.) tan 60  tan 30 
5.) cos 45
7.) cos 30 

2
2




4.) sin 60  cos 60 
6.) tan 30 
 2


2
8.) sin 30   tan 60 
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
2


 

9.) 2 sin 30 
10.) sin 30  cos 60   cos 30  sin 60 
11.) sin 90   cos 0 
12.) cos 180   sin 270 



13.) sin 0  tan 30   cos 0 

14.) cos180   2 tan 45
Review:
15.) Factor completely: 10 xy 2  23xy  5x
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
2

16.)
A circle shown in the diagram below has a center of  5,3 and passes through the point  1,7
Write an equation that represents this circle.
Answers to Homework (Day 4):
1.)
9.) 1
10.) 1
11.) 2
12.) -2
13.) 1
14.) 1
15.) x  2 y  5 5 y  1
3
2
2.) 2
3.) 1
3
4.)
4
5.) 1/2
6.) 1/3
7.) 1
13  4 3
8.)
4
16.)
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 x  5    y  3
2
2
 32
Reference Angles
Day 5 – Reference Angles
Reference Angle – an acute angle formed by the terminal side of a given angle and the x-axis.
Quadrant I
1.)
Draw an angle of 30º
Quadrant II
2.)
Draw an angle of 150º
Quadrant III
3.)
Draw an angle of 210º
Quadrant IV
4.)
Draw an angle of 330º
What is the reference
angle?
What is the reference
angle?
What is the reference
angle?
What is the reference
angle?
Finding Reference Angles
Special Angles Chart
Steps for Expressing Functions as Positive Acute Angles and Finding Exact Values
1.) Determine the quadrant in which the angle lies. (You may have to first find a coterminal
angle.)
2.) Determine the sign (positive or negative) of the function in the quadrant.
3.) State the function.
4.) Find the reference angle.
5.) Exact Value only – find the value from the unit circle or special angles chart.
Directions: Express each of the following as a function of a positive acute angle.
5.) sin 140 
6.) cos 250 
7.) tan 300 
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Directions: Find the exact value of each of the following expressions.
cos 135 
8.) sin 300 
9.)
10.) tan 240 
11.)
sin( 30  )
12.) cos(150 )
13.)
tan 630 
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Day 5 – Reference Angles
HOMEWORK
Directions: Express each of the following as a function of a positive acute angle.
1.) tan 237 
2.) cos 690 

3.) sin  158

4.) tan 500 
Directions: Find the exact value of each of the following expressions.
5.) tan 600 
6.) cos(30  )
7.) sin 900 
8.)  sin 135
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Directions: Find the exact value of each of the following expressions.
9.) cos 90   tan 225
10.) sin 210   cos 120 
11.) tan 135  sin 330 
12.) cos135   cos 225 
13.) sin 300  sin( 240 )
14.)
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sin 60 cos150   tan  45 



15.) tan( 315 )  tan 135
16.)
tan 30   cos 30 
 2
 2
Review:
2
2

17.) Express  x  2  as a trinomial.
5

18.) Solve algebraically for: 252 x 3  125x  2
Answers to Homework (Day 5):
1.)
2.)
3.)
4.)
5.)
tan 57
cos 30
 sin 22
 tan 40
3
6.)
11.) 
12.)  2
1  3
13.)
2
1
14.) 
2
15.) 0
39
16.)
36
4 2 8
x  x4
17.)
25
5
18.) 0
3
2
7.) 0
8.) 
3
2
2
2
9.) 1
10.) -1
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Day 6 – Using the Calculator to find Function Values and an Angle / DMS
Do Now: (Questions 1-6) Multiple Choice
1.) For what value of x is the expression
2.) If f ( x)  sin 2 x  cos x , then f (180 ) 
1
undefined?
1  sin x
(1) 1
(2) 2
(3) -1
(4) 0
(1) 1
(2) 90º
(3) 180º
(4) 270º
3.) The expression cos 290º is equivalent to
(1) cos 70º
(3) -cos 20º
(2) cos 20º
(4) -cos 70º
5.) What single transformation moves a fourthquadrant angle to its equivalent first
quadrant reference angle?
(1)
(2)
(3)
(4)

4.) Find the exact value of tan 120 
(1) 3  1
(3) 3

2
 cos180  .
(2) 2
(4) 4
6.) Which statement is a false statement?
(1) tan  is undefined whenever cos  equals
zero
3
1
(2) If sin  =
, cos   .
2
2
(3) If cos  = 0, then sin   1 .
(4) sin  = cos  only in Quadrant I.
reflection in the y-axis
reflection in the origin
reflection in the x-axis
reflection in the line y = x
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Steps for using the Calculator to find Function Values
**YOUR CALCULATOR MUST BE IN DEGREE MODE**
1.) Enter the function (sin, cos, tan)
2.) Enter the degree measure.
3.) Round to the appropriate decimal value.
Directions: Find each function value to four decimal places.
7.) sin (-10º)
8.) tan 375º
9.) cos 255º
Steps for using the Calculator to find Angle Measures
**YOUR CALCULATOR MUST BE IN DEGREE MODE**
1.) Press the 2nd button.
2.) Enter the function ( sin 1 , cos 1 , tan 1 ) .
3.) Enter the value.
4.) Round to the appropriate value.
Directions: Find the smallest positive value of  to the nearest degree.
10.) sin  = 0.3455
11.) cos  = 0.4383
12.) tan  = 0.7000
ONE DEGREE = 60 MINUTES (1º = 60’)
Steps for using the Calculator to find Degrees, Minutes, and Seconds
Finding a value: 1.) Enter the function (sin, cos, tan)
2.) Enter the DMS using the 2nd button, then the “purple” APPS button.
3.) Round to the appropriate decimal value.
Find a degree measure: 1.) Press 2nd button.
2.) Enter the function ( sin 1 , cos 1 , tan 1 ) .
3.) Enter the value.
4.) 2nd APPS
5.) Option 4 (DMS)
6.) Round to the appropriate value.
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Directions: Find each function value to four decimal places.
13.) cos 18º 12’
14.) sin 57º 40’
15.) tan 61º 23’
Directions: Find the smallest positive value of  to the nearest minute.
16.) sin  = 0.2672
17.) cos  = 0.9692
18.) tan  = 0.0892
Directions: Express the measure of each acute angle  : a.) to the nearest degree b.) to the
nearest minute c.) to the nearest ten minutes.
19.) sin  = 0.5505
20.) tan  = 3
21.) A 20-foot ladder leans against a wall. The top of the ladder reaches 18.5 feet up the side of the
building. Find the measure of the angle the ladder makes with the ground.
a.) to the nearest degree
b.) to the nearest minute
c.) to the nearest ten minutes
p28
Day 6 – Using the Calculator to Find Function Values and an Angle / DMS
HOMEWORK
Directions: Find each function value to four decimal places.
1.) cos 100º
2.) tan 15º
3.) sin (-82º)
Directions: Find the smallest positive value of  to the nearest degree.
4.) tan  = 0.2126
5.) cos  = 0.7660
6.) sin  = 0.9990
Directions: Find each function value to four decimal places.
7.) tan 88º 30’
8.) sin 105º 50’
9.) cos 205º 12’
Directions: Find the smallest positive value of  to the nearest minute.
10.) cos  = 0.2672
11.) sin  = 0.9692
12.) tan  = 7.3478
Directions: Express the measure of each acute angle  : a.) to the nearest degree b.) to the
nearest minute c.) to the nearest ten minutes.
13.) sin  = 0.8811
14.) cos  = 0.7454
15.) A standard rectangular sheet of paper measures 8 ½ inches by 11 inches. A diagonal is drawn,
connecting opposite corners of the paper. Find, to the nearest minute, the measures of the two
acute angles formed by the diagonal.
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16.) Three roads intersect to enclose a small triangular park. A path that is 72 feet long extends
from the intersection of the two roads to the third road. The path is perpendicular to that road
at a point 65 feet from one of the intersections and 58 feet from the third. Find, to the nearest
ten minutes, the measures of the angles at which the roads intersect.
Review:
17.) Solve algebraically for x:
1
2
4

 2
x2 2 x x 4
Answers to Homework (Day 6):
1.
2.
3.
4.
5.
6.
7.
-.1736
.2679
-.9903
12°
40°
87°
38.1885
8.
9.
10.
11.
12.
13.
.9621
-.9048
74°30’
75°45’
82°15’
a.) 62°; b.) 61°47’;
c.) 61°50’
p30
14. a.) 42°; b.) 41°48’;
c.) 41°50’
15. 52°18’ and 37°42’
16. 42°0’ and 38°50’
2
17.
3
Day 7 – Reciprocal Functions
Do Now: (Questions 1 & 2)
1.) Express as a function of a positive acute
angle.
tan( 50  )
2.) Evaluate 2 sin 330  cos(60  ) .
3 Reciprocal Functions
Reciprocal Functions follow all the rules of the original three functions.
sin θ
cos θ
tan θ
Directions: Determine the quadrant in which each angle terminates.
3.) csc  0 and cot   0
4.) sec  5 and csc  0
5.) sec  0 and tan   1
Directions: Express as a function of a positive acute angle.
7.) sec 140º 10’
6.) cot( 130  )
p31
Directions: Find the exact value of each expression.
9.) cot 45  csc 45
8.) csc 270 
Practice Problems
Directions: Find the exact value of each expression.
10.) sec 300
11.) cot 225
12.) csc 420
13.) sec(210)
p32
Day 7 – Reciprocal Functions
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Determine the quadrant in which each angle terminates.
1.) sec A  0 and csc A  0
2.) sin x  0 and cot x  0
3.) csc  0 and cot   0
4.) sec B  0 and sin B  0
5.) cot   0 and sec   0
Directions: Find the exact value of each expression.
7.) csc 225 
8.) cot 270 
9.) cot 420 
10.) csc( 210  )
p33
6.) cos E  0 and csc E  0


11.) sec150 cos150


Review
13.) Solve algebraically for x: log x 3
x3  x  2
2
x
Answers to Homework (Day 7):
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
IV
II
IV
I
II
III
 2
0

12.) tan 300  cot 300 
9.)
3
3
10.) 2
11.) 1
12.) 1
1

13.) 2,  
5

p34

Day 8 – Converting Degrees to Radians
Do Now: (Questions 1 & 2)
1.) If  is an angle in standard position and
its terminal side passes through the point
2.)
 1
3  on the unit circle, what is the
  ,

 2
2 

Evaluate 2sec 330  cot(60 ) .

degree measure of  ?
Converting Between Degrees and Radians
Definition: A radian is the measure of an angle that, when drawn as a central angle of a circle,
intercepts an arc whose length is equal to the length of a radius of the circle.
 radians = 180 degrees
To convert from degrees to radians, multiply by

180
.
180
To convert from radians to degrees, multiply by
.


Directions: Convert each radian measure to degrees.

3.) 2
4.)
3
2

5.)

7
3
6.)

p35
5
6
Directions: Convert each degree measure into radians.
7.) 45
8.) 60
9.)
135
10.) 120
Practice Problems
11.) What is the measure of an angle formed by
the hands of a clock at 4pm?
a) In degrees
b) In radians
13.) An angle of
quadrant?
3
radians lies in which
4
12.) What is the measure of an angle formed by
the movement of the minute hand of a
clock when it moves from 0 minutes to 10
minutes?
a) In degrees
b) In radians
14.) Express as the function of a positive acute
5
angle: sin
4


15.) Find the exact value: csc
2
3
2
2
and cos x  
, then x =
2
2
3
(3)
4
7
(4)

4
16.) If sin x 

4
5
(2)
4
(1)





p36
Day 8 – Converting Degrees to Radians
HOMEWORK
**Complete any Practice Problems from class work that have not been completed**
Directions: Change each angle from degree measure to radian measure.
1.
270
2.
–50
3.
330
Directions: Change each angle from radian measure to degree measure.
2

4.
5.
6.

3
2



Directions: Express as the function of a positive acute angle.
5
7
7.
8.
sin
tan
4
6


Directions: Find the exact value of each expression.
cos2
9.
10.


p37
cot
11
6
5
6
Review:
11.) For a given set of rectangles, the length is inversely proportional to the width. In one of these
rectangles, the length is 18 and the width is 9. For this set of rectangles, calculate the width if
the length is 12.
Answers to Homework (Day 8):
1.)
2.)
3.)
4.)
5.)
3
2
5
18
11
6
120
90
6.)
150
7.)
 sin 45 or  sin
8.)
tan30 or tan
9.) 1
10.)  3
11.) 13.5
p38

6

4
Do Now: (Questions 1 & 2)
1.) Find, to the nearest minute, the angle whose
measure is 3.45 radians.
Day 9 – s    r
2.) Find the exact sum of sin

3
 cos

2
.
S = Ө·r
Definition: A radian is the measure of the angle that, when drawn as a central angle of a circle,
intercepts an arc whose length is equal to the length of the radius of that circle. We write this as
the equation: s    r , where:
s = length of arc
Ө = central angle (Radian measure)
r = radius
Directions: Answer each question.
3.) In a circle with radius 5 cm, find the central angle intercepted by an arc whose length is 20cm.
4.) In a circle a central angle of 30º intercepts an arc whose length is 20 cm. Find the radius of the
circle.
p39
5.) A 30 degree central angle intercepts an arc of a circle with a diameter of 10cm. What is the
length of the arc, to the nearest tenth of a cm?
6.) On a clock, the length of the pendulum is 30 centimeters. A swing of the pendulum determines
an angle of 0.8 radians. Find, in centimeters, the distance traveled by the tip of the pendulum
during this swing.
7.) A dog has a 20-foot leash attached to the corner where a garage and a fence meet, as shown in
the accompanying diagram. When the dog pulls the leash tight and walks from the fence to the
garage, the arc the leash makes is 55.8 feet.
What is the measure of angle Ө between the garage and the fence, in radians?
(1) 0.36
(2) 2.79
(3) 3.14
(4) 160
p40
1.) Find the length of the radius of a circle in
which a central angle of 4.5 radians
intercepts an arc of 9 meters.
3.) A sector has a radius of 12 cm and an angle
of 65º. To the nearest tenth of a centimeter,
find its arc length.
Day 9 – s    r
HOMEWORK
2.) A pendulum makes an angle of 3 radians as
its tip travels 18 feet. What is the length of
the pendulum?
4.) A wheel has a diameter of 6 feet. As the
wheel turns, a rope connected to a five
kilogram weight winds onto the wheel,
causing the weight to move. If the wheel
turns 135º, to the nearest foot, how far does
the weight move?
Review
5.) Find, to the nearest tenth of a degree, the angle whose measure is 3.5 radians.
p41
6.) The graph below represents the function y  f ( x) .
State the domain and range of this function.
Answers to Homework (Day 9):
1.) 2 m
2.) 6 ft
3.) 13.6 cm
4.) 7 ft
5.) 200.5°
6.) Domain: 8  x  5 or [–8, 5]
Range: 1  y  3 or [–1, 3]
p42
Day 10 - Finding the Remaining Trig Function Values when One Function is Known
Do Now: (Questions 1 & 2)
1.) A circle has a radius of 4 inches. In inches,
2.) In a circle whose radius is 4 centimeters,
what is the length of the arc intercepted by a
what is the length of an arc intercepted by a
central angle of 2 radians?
central angle of 2.5 radians?
(1) 2
(2) 2
(3) 8
(4) 8
Finding the Remaining Trig Function Values when One Function is Known
1.
2.
3.
4.
Determine the quadrant, and draw a right triangle in the appropriate quadrant.
**Remember, a point on the unit circle is always (cos θ, sin θ).
Label the triangle using the given trigonometric function value.
Use the Pythagorean Theorem to determine the missing third side of the triangle.
**Remember to check positive or negative based on ASTC.
Use the properties of SOH-CAH-TOA and reciprocal functions to determine the values for the
remaining trigonometric functions.
3.) If  is an angle in standard position and its terminal side passes through the point
 1
3
  ,
 on the unit circle,
 2

2


a.) In simplest radical form, find all six trigonometric function values.
sin Ө =
csc Ө =
cos Ө =
sec Ө =
tan Ө =
cot Ө =
b.) Find the degree measure of angle  .
p43
4.) If  is an angle in standard position and its terminal side passes through the point (3, 4).
Find all six trigonometric function values.
5.) If Ө is in quadrant II and sin  
5
, find the remaining five trigonometric function values.
13
p44
6
, and Ө is an angle that terminates is quadrant IV, find the values of the other five
10
trigonometric functions.
6.) If cos  
7.) If sec  3 and tan   0 , find the remaining five trigonometric function values.
p45
Day 10 - Finding the Remaining Trig Function Values when One Function is Known
HOMEWORK
5
1.) If csc    and Ө is in the third quadrant,
4
then what is the value of cos Ө?
1
and Ө is an acute angle, what is
5
the value of tan Ө cos Ө?
2.) If sin  
3.) If Ө is in quadrant IV and cot   6 , find the remaining five trigonometric function values.
p46
4.) If sec  0 and csc  
3
, find the remaining five trigonometric function values.
2
Review:
5.) Express
250a 7b10
5a3b7
in simplest radical form.
6.) What is the number of degrees in an angle
11
whose measure in radians is
?
12
Answers to Homework (Day 10):
1.)

2.)
1
5
3
5
 35
sin  
35
sin  
2
3
6 35
35
1
tan  
6
cos  
 5
3
cos  
3.)
2 5
5
3
csc  
2
3 5
sec  
5
tan  
4.)
csc    35
35
6
cot   6
sec  
cot  
p47
 5
2
5.)
6.)
5a 2b 2b
165°
Day 11 – Co-functions
Do Now: (Questions 1 & 2)
1.) If  is an angle in standard position and
its terminal side passes through the point
 3, 2 , find the exact value of csc .
2.)
In right triangle ABC, C is the right angle
3
and sin A 
.
2
What is the value of csc B ?
(1)
2
3
(2) 2
(3)
1
2
(4)
3 3
2
Co-functions
Co-functions have complementary angles.
3 sets of co-functions:
sin  = cos (90 – )
sec  = csc (90 – )
tan  = cot (90 – )
Directions: Solve each problem by following the applicable directions.
3.) If sin 6A  cos9A , then solve for the
4.) If sin( A  30)  cos60 , find the number of
measure of angle A.
degrees in the measure of angle A.


5.)
If cos  x  30  sin x , find the measure of
angle x.
6.)
p48
If cot  x 10   tan  4 x   , find a value of
angle x.
7.)
If tan x  cot  2 x  6 , find
9.)
Find the value of acute angle A if
sin A
1
cos 50
8.)
.
If sin 2 A  cos3A , find
10.) If cos  2 x 1   sin  3x  6  , find the value
of x.
Directions: Find the function of a positive acute angle less than 45.
11.) cos72
12.) tan120


13.) sec250

14.) csc 440

15.) tan272

.
16.)

p49
sin 60
cos60
Practice Problems
Directions: Express as a function of a positive acute angle less than 45º or
17.) cot
5
3

.
4
18.) csc(-75º)
Directions: Solve each equation for Ө.
19.) sin 10º = cos Ө
20.) sec 60º = csc Ө
21.) cos Ө = sin (2Ө +15)
22.) cot (3Ө -14) = tan (56 - Ө)
23.) If tan xº = cot (2x – 15), then x =
24.) Express sec 102º as the function of a
positive acute angle less than 45º.
(1)
(2)
(3)
(4)
15
25
35
45
25.) If Ө is the measure of an acute angle and
tan Ө = cot 2Ө, then tan Ө =
(1)
(1)
(2)
(3)
(4)
-sec 78º
–csc 12º
–sec 12º
csc 12º
26.) For what value of Ө does
cos (3Ө + 25) = sin (37 - Ө)?
(1)
(2)
(3)
(4)
1
2
3
3
(3) 3
(4) 30
(2)
p50
28
23
14
7
Day 11 – Co-functions and Review
HOMEWORK

Directions: Express as a function of a positive acute angle less than 45º or .
4
1.) sin 65º
13
2.) cos
30
Directions: Solve each equation for Ө.
3.) cot (3Ө - 6) = tan (Ө + 8)
4.)
sin (3Ө) = cos (4Ө + 13)
5.)
6.)
In right triangle ABC, C is the right angle.
BC = 5 and AB = 13. What is the value of
sec A?
If sin A < 0 and cos A < 0, in which
quadrant does < A terminate?
(1) I
(2) II
(3) III
(4) IV
(1)
7.)
5
13
(2)
13
5
(3)
12
13
(4)
13
12
A kite is flying 40 feet in the air. The kite string is 75 feet long and has been staked to the
ground. To the nearest minute, what is the measure of the angle of elevation of the kite?
p51
Directions: Find a coterminal angle with a measure of  such that 0     360 
8.) 505º
9.) -302º
Directions: For each function value, if 0    360 , find, to the nearest degree, two values of  .
10.) sin  = 0.3747
11.) tan  = -0.5775
Directions: Find each function value to four decimal places.
12.) cos 50 
13.) sin 110 
14.) From the top of a building that is 56 feet high, the angle of depression to the base of an
adjacent building is 72º. Find, to the nearest foot, the distance between buildings.
p52
Directions: Find the exact value of each expression.
 19 
16.)
15.) csc  

 6 
 csc150  cos150 
17.) How long, to the nearest tenth of an inch, is the arc traced by a minute hand that measures 6
inches when it moves from 1:05pm to 1:30pm?
p53
18.) Solve for x: cos(4 x  10)  sin  5 x  10 
Answers to Homework (Day 11):
1.)
cos 25°
2.)
sin12 or sin
3.)
4.)
5.)
6.)
7.)
8.)
9.)
22
11
(3)
(4)
3214'
145°
58°

15
10.)
11.)
12.)
13.)
14.)
15.)
16.)
17.)
18.)
QI: 22°, QII: 158°
QII: 150°, QIV: 330°
0.6428
0.9397
172 ft
2
 3
15.7 in
10
p54