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Name:______________________________
NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND
OF REAL NUMBERS
Date:________________Period:_________
Mrs. Nguyen’s Initial:_________________
LESSON 6.4 – THE LAW OF SINES
1) SSS
2) SAS
3) ASA
4) AAS
Never: SSA and AAA
Triangles with no right angles.
Drawings:
Area of Oblique
Triangles (SAS
case)
The area A of a triangle with sides of
lengths a and b and with included angle 
1
is: A  ab sin 
2
Example:
Law of Sines
(in the case of ASA,
SAA, SSA)
In triangle ABC we have:
Example:
Review: Shortcuts
to prove triangles
congruent
Definition of
Oblique Triangles
Drawings:
sin A sin B sin C


a
b
c
Practice Problems: Find the missing sides and angles in each problem. Round to 2 decimal places.
1.
ABC , m A  54, m B  29, a  10
2.
AHS , m A  25, m H  111, a  110
3.
B
43
92
58
A
C
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 1
Practice Problems: Find the area of each triangle. Round to 2 decimal places.
4.
ABC , if b  10, c  6, and m A  65
6.
The triangle has sides of length 10 cm, 3 cm, with included angle 120  .
5.
ABC , if a  5, b  10, and m C  18
Practice Problems: Applying what you know.
7.
A water tower 30 m tall is located at the top of
a hill. From a distance of 120 m down the hill, it is
observed that the angle formed between the top and
the base of the tower is 8  . Find the angle of
inclination of the hill.
8.
A communications tower is located at the top
of a steep hill. The angle of inclination of the hill is
58  . A guy wire is to be attached to the top of the
tower & to the ground, 100 m downhill from the base
of the tower. The angle of elevation from the bottom
of the guy wire to the top of the tower is 70  . Find the
length of the cable required for the guy wire.
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 2
Ambiguous Case of
Law of Sines
If you’re given SSA, then there
can either be 0, 1, or 2 triangles
formed. This is the ambiguous
case.
Drawings:
Practice Problems: Solve for all possible triangles that satisfy the given conditions. Round all answers to 2
decimal places.
9.
ERW , m R  35, e  5, r  4
10.
DWC , m D  12, d  11, w  6
11.
MLT , m M  15, m  10, l  15
12.
A = 39, a = 10, b = 14
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 3
LESSON 6.5 – THE LAW OF COSINES
Law of Cosines
(in the case of SAS
or SSS)
In triangle ABC we have:
Example:
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
Practice Problems: Solve each triangle.
1.
In ABC , b  6, c  8, and m A  62
2.
In ABC , a  6, c  8, and m B  109
3.
In ABC , a  3, b  7, and c  5
4.
In BAT , b  7, a  9, and t  12
Heron’s Formula:
(SSS case)
The area A of triangle ABC is given by:
A  s ( s  a )( s  b)( s  c) where
1
(a  b  c) is the semi-perimeter of
2
the triangle; that is, s is half the perimeter.
s
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 4
Example:
Practice Problems: Find the area of the triangle whose sides have the given lengths.
5.
MAP, if m  5, a  8, and p  12
7.
CAT , c  29, a  45, and t  18
Heading and
Bearing…
6.
MEW , if m  5, e  7, and w  11
is a direction of navigation indicated by an acute angle measured from due north or due
south.
Practice Problems: Solve each triangle.
8.
A pilot sets out from an airport and heads in the direction N15  W, flying at 250 mph. After one hour, he
makes a course correction and heads in the direction of N45  W. Half an hour after that, he must make an
emergency landing.
(A) Find the distance between the airport & his final landing point.
(B) Find the bearing from the airport to his final landing point.
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 5
9.
Airport B is 300 mi from airport A at a bearing of N50  E. A pilot wishes to fly from A to B mistakenly
flies due east at 200 mph for 30 minutes, when he notices his error.
(A) How far is the pilot from his destination at the time he notices the error?
(B) What bearing should he head his plane in order to arrive at airport B?
10.
Two ships leave a harbor at the same time. One ship travels on a bearing of S12  W at 14 mph. The
other ship travels on a bearing of N75  E at 10mph. How far apart will the ships be after three hours?
11.
You are on a fishing boat that leaves its pier and heads east. After traveling for 25 miles, there is a report
warning of rough seas directly south. The captain turns the boat & follows a bearing of S40  W for 13.5 miles.
(A) At this time, how far are you from the boat’s pier?
(B) What bearing could the boat have originally taken to arrive at this spot?
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 6
LESSON 7.1 – THE UNIT CIRCLE
The Unit Circle
The unit circle is the circle of radius 1
centered at the origin. The equation of the
unit circle is:
x2  y 2  1
Note:
Every point on the unit circle can be linked
to the values of cos  and sin  .
If point P whose coordinates are (x, y) lies
on the unit circle for a given angle  , then
we know that x  cos  and y  sin 
Practice Problems: Find the missing coordinate of P, using the fact that P lies on the unit circle in the given
quadrant.
1.

P

,
7 
 in QIV
25 
2.
 2
P ,
 5

 in Q II

Practice Problems: Find (a) the reference angle for each value of t, and (b) find the terminal point P(x, y) on
the unit circle determined by the given value of t.
3.
t
2
3
4.
t
5
4
5.
t
7
6
6.
t
11
6
7.
t
8.
t

3
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 7

2
9.
t 
3
2
t
10.
THE UNIT CIRCLE
y
(
(
(
,
,
,
)
(
)
,
)

)
(
,
)
2
(
,
)
(
90
135
120
,
)

0
180
(
,
300
270 
(
3
)
(
(
,
2
,
)
315
240
)
,
0
330
225 
,
(
30
210
(
)
45
150
(
,
60
(
)
(
,
)
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 8
,
)
,
)
)
x
Review: Definition
of Reference Angle
Let  be an angle in standard position. Its reference angle is the acute angle  ' formed
by the terminal side of  and the x-axis.
Quadrant II
Quadrant III
 '      rad
 '      rad
 '   2    rad
 '  180    deg ree
 '    180  deg ree
 '   360    deg ree
Practice Problems: Find the reference angle for each of the given angles.
11.
t  170
13.
t
15.
t
5
7
8
7
Quadrant IV
12.
t  410
14.
t
16.
t  5.8
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 9
11
9
LESSON 7.2 – TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
Definitions of
Trigonometric
Functions
point on the unit circle corresponding to t.
1
sin t  y
csc t  , y  0
y
cos t  x
tan t 
Cofunctions
Fundamental
Trigonometric
Identities
 x, y  be the
Let t be a real number and let
y
, x  0,
x
sec t 
1
, x0
x
cot t 
Remember:
SOH CAH TOA
sin  
opp
hyp
csc  
hyp
opp
cos  
adj
hyp
sec  
hyp
adj
tan  
opp
adj
cot  
adj
opp
x
, y0
y
sin  90     cos 
cos  90     sin 


sin      cos 
2



cos      sin 
2

tan  90     cot 
cot  90     tan 


tan      cot 
2



cot      tan 
2

sec  90     csc 
csc  90     sec 


sec      csc 
2



csc      sec 
2

Reciprocal
Identities
sin  
1
csc 
csc  
1
sin 
cos  
1
sec 
sec  
1
cos 
1
cot 
sin 
tan  
cos 
cot  
tan  
Quotient
Pythagorean
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 10
1
tan 
cos 
cot  
sin 
Practice Problems: Evaluate the six trig functions at each real number without using a calculator. Plot the
ordered pair.
1.
2.
3.
5
t
6
t
3
2
3
t
2
sin

csc

cos

sec

tan

cot

sin

csc

cos

sec

tan

cot

sin

csc

cos

sec

tan

cot

Domain of the
Trigonometric
Functions
sin, cos: All real numbers
Definition of
Periodic Function
A function f is periodic if there exists a positive real number such that

 n for any integer n.
2
cot, csc: All real numbers other than n for any integer n.
tan, sec: All real numbers other than
f t  c   f t 
for all t in the domain of f. The smallest number c for which f is
periodic is called the period of f.
Practice Problems: Evaluate the trigonometric function using its period as an aid.
4.
cos5 
5.
sin
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 11
9

4
6.
sin  3  
7.
 8
cos  
 3
The cosine and secant functions are even.
Even and Odd
Trigonometric
Functions
cos(t )  cos t
sec(t )  sec t
The sine, cosecant, tangent, and cotangent
functions are odd.
sin(t )   sin t
tan(t )   tan t



Remember:
Even  f (t )  f (t )
Odd  f (t )   f (t )
csc(t )   csc t
cot(t )   cot t
Practice Problems: Use the value of the trig function to evaluate the indicated functions.
8.
sin(t ) 
9.
cos t 
3
8
4
5
sin t 
csct 
cos   t  
cos  t    
Practice Problems: Use a calculator to evaluate. Round to 4 decimal places.
10.
sin

4

11.
csc1.3 
12.
cos  2.5  
13.
cot1 
Practice Problems: Use a calculator to evaluate. Round to 4 decimal places.
14.
cos80 
15.
cot 66.5 
16.
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 12
sec

7

Practice Problem 17: Let  be
an acute angle such that
cos  0.6 . Find:
sin  
csc 
cos 
sec 
tan  
cot  
Practice Problem 18: Given
sin  
tan  
13
and
2
13
sec 
, find
3
cos 
sec  90    
cot  
tan  90    
cos  
csc  
csc 
Practice Problem 19: Given
tan   5 , find


Practice Problems: Evaluate the value of  in degrees  0    90  and radians  0     without using
2

a calculator.
20.
csc  2
22.
cot   1
21.
tan   1
23.
sin  
3
2
Practice Problems: Use a calculator to evaluate the value of  in degrees  0    90  and radians


 0     . Round to the nearest degrees and 3 decimal places for radians.
2

24.
sec  2.4578
25.
sin   0.4565
26.
cot   2.3545
27.
sin   0.3746
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 13