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 , x0 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 , y0 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
LESSON 7.3 – TRIGONOMETRIC GRAPHS
Graph of Sine Function
x
y sin x
y
2
0
Domain: Range: Period: x-intercepts:
2
3 2 2
Relative Minima: Relative Maxima: Graph of Cosine Function
x
y cos x
2
0
Domain: Range: Period: x-intercepts: Relative Minima: Relative Maxima:
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 14
2
3 2 2
y
Transformations
y d a sin bx c y d a cos bx c Vertical stretch:
a
Scaling factor
Vertical shrink:
a
a 1
a 1
Reflection over the x-axis
b
2 period b
c
Horizontal translation (left or right)
d
Vertical translation (up or down)
Definition of The amplitude of y a sin x and y a cos x represents half the distance Amplitude of Sine and between the minimum and the maximum value of the function and is given by Cosine Curves Amplitude a . Practice Problems: Write an equation for each dashed curve.
1.
y _______________
Practice Problem 3: Write equations for both curves
2.
y ________________
ysolid _______________ ydashed ______________
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 15
Period of Sine and Cosine Functions
Let b be a positive real number. The period of
2 b
y a sin bx and y a cos bx
is
.
Practice Problems: Write and equation for each dashed curve.
4.
y _______________
5.
y ________________
Graphs of Sine and Cosine Functions
y a sin bx c and y a cos bx c characteristics. (Assume b 0 ). 2 Amplitude a Period b The graph of
have the following
The left and right endpoints of a one-cycle interval can be determined by solving the equations:
bx c 0 . bx c 2
Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.
6.
y 2sin 4 x
Amplitude: Period: Left Endpoint (LEP):
Right Endpoint (REP):
Vertical shift: Reflection:
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 16
7.
x y 3cos 2 2 2
Amplitude: Period: Left Endpoint (LEP):
Right Endpoint (REP):
Vertical shift:
Reflection:
Practice Problem 8: When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approximated by y 0.001sin 880 t , where t is time in seconds.
a. What is the period of the function?
b. The frequency f is given by
f
1 . p
What is the frequency of the note?
Practice Problems: Write an equation for each curve.
9.
y _______________
10.
y ________________
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 17
Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.
11.
1 x y sin 2 2 3
Amplitude: Period: Left Endpoint (LEP):
Right Endpoint (REP):
Vertical shift: Reflection:
12.
y 3 4 cos
10
x 5
Amplitude: Period: Left Endpoint (LEP):
Right Endpoint (REP):
Vertical shift: Reflection:
13.
y 4 sin
3
x 3
Amplitude: Period: Left Endpoint (LEP):
Right Endpoint (REP):
Vertical shift: Reflection:
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 18
LESSON 7.4 – MORE TRIGONOMETRIC GRAPHS Graphs of y csc x and y sec x
y csc x
y sec x
In order to graph y csc x and y sec x , use the graphs of the other sine & cosine as base models.
Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.
1.
y csc 2 x
3.
y 2 csc 4 x
2.
y 2sec 3x
4.
y sec x 3
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 19
5.
2 y 2 csc x 4 3 4
y 1 2sec
6.
Graph of Tangent Function
2
( x 4)
x
y tan x
2
0
2
Domain:
Range:
3 2 2
Period: x-intercepts: Vertical asymptotes: Two standard consecutive vertical asymptotes:
bx c
2
bx c
2
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 20
y
Practice Problem 7: Write an equation for each dashed curve.
y _______________
Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.
8.
y 3tan x
Amplitude: Period:
Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection:
9.
x y 3tan 2 2 2
Amplitude:
Period:
Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection: Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 21
Graph of Tangent Function
x
y cot x
2
0
2
3 2 2
Domain: Range: Period: x-intercepts: Vertical asymptotes: Two standard consecutive vertical asymptotes:
bx c 0
bx c
Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.
10.
y cot 2 x Amplitude: Period:
Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection:
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 22
y
11.
y 2 cot 4 x Amplitude:
Period:
Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection: 12.
y 2 cot 3 x 6 2
Amplitude: Period: Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection:
13.
y 3 5cot
4
( x 2)
Amplitude:
Period:
Left vertical asymptote:
Right vertical asymptote:
Vertical shift: Reflection:
Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 23