notes 6 & 7: trigonometric functions of angles and...

Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes –

43 92

58

A

B

C

NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND

OF REAL NUMBERS Name:______________________________ Date:________________Period:_________ Mrs. Nguyen’s Initial:_________________

LESSON 6.4 – THE LAW OF SINES Review: Shortcuts to prove triangles congruent

1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA

Drawings:

Definition of Oblique Triangles

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

is: 1

sin2

A ab

Example:

Law of Sines (in the case of ASA, SAA, SSA)

In triangle ABC we have: sin sin sinA B C

a b c

Example:

Practice Problems: Find the missing sides and angles in each problem. Round to 2 decimal places. 1. , 54, 29, 10ABC m A m B a 2. , 25, 111, 110AHS m A m H a

3.


Practice Problems: Find the area of each triangle. Round to 2 decimal places. 4. , if 10, 6, and 65ABC b c m A 5. , if 5, 10, and 18ABC a b m C

6. The triangle has sides of length 10 cm, 3 cm, with included angle 120 . 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.


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. , 35, 5, 4ERW m R e r 10. , 12, 11, 6DWC m D d w

11. , 15, 10, 15MLT m M m l 12. A = 39, a = 10, b = 14


LESSON 6.5 – THE LAW OF COSINES Law of Cosines (in the case of SAS or SSS)

In triangle ABC we have:

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc A

b a c ac B

c a b ab C

Example:

Practice Problems: Solve each triangle. 1. In , 6, 8, and 62ABC b c m A 2. In , 6, 8, and 109ABC a c m B

3. In , 3, 7, and 5ABC a b c 4. In , 7, 9, and 12BAT b a t

Heron’s Formula: (SSS case)

The area A of triangle ABC is given by:

( )( )( )A s s a s b s c where

1

( )2

s a b c is the semi-perimeter of

the triangle; that is, s is half the perimeter.

Example:


Practice Problems: Find the area of the triangle whose sides have the given lengths. 5. , if 5, 8, and 12MAP m a p 6. , if 5, 7, and 11MEW m e w

7. , 29, 45, 18CAT c a and t Heading and Bearing…

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 N15W, 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.


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 S12W at 14 mph. The other ship travels on a bearing of N75E 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 S40W 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?


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:

2 2 1x y

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 cos and sinx y

Practice Problems: Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant.

1. 7

,25

P

in QIV 2. 2

, 5

P

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. 2

3t

4. 5

4t

5. 7

6t

6.

11

6t

7. 3

t

8. 2

t


9. t 10.

3

2t

x

y

0

30

45

6090

120135

150

180

210

225240

270300

315

330

0

2

3

2

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

THE UNIT CIRCLE


Practice Problems: Find the reference angle for each of the given angles. 11. 170t

12. 410t

13. 5

7t

14. 11

9t

15. 8

7t

16. 5.8t

Let be an angle in standard position. Its reference angle is the acute angle ' formed by the terminal side of and the x-axis.

Review: Definition of Reference Angle

Quadrant II

'

' 180 deg

rad

ree

Quadrant III

'

' 180 deg

rad

ree

Quadrant IV

' 2

' 360 deg

rad

ree


LESSON 7.2 – TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS Definitions of Trigonometric Functions

Let t be a real number and let ,x y be the

point on the unit circle corresponding to t.

sin t y 1

csc , 0t yy

cos t x 1

sec , 0t xx

tan , 0y

t xx

, cot , 0x

t yy

Remember: SOH CAH TOA

sinopp

hyp csc

hyp

opp

cosadj

hyp sec

hyp

adj

tanopp

adj cot

adj

opp

sin 90 cos

sin cos2

cos 90 sin

cos sin2

tan 90 cot

tan cot2

cot 90 tan

cot tan2

Cofunctions

sec 90 csc

sec csc2

csc 90 sec

csc sec2

Reciprocal Identities

1sin

csc

1csc

sin

1

cossec

1

seccos

1

tancot

1

cottan

Quotient sintan

cos

cos

cotsin

Fundamental Trigonometric Identities

Pythagorean 2 2

2 2

2 2

sin cos 1

1 tan sec

1 cot csc


Practice Problems: Evaluate the six trig functions at each real number without using a calculator. Plot the ordered pair.

sin csc

cos sec

1. 5

6t

tan cot

sin csc

cos sec

2. 3

2t

tan cot

sin csc

cos sec

3. 3

2t

tan cot

Domain of the Trigonometric Functions

sin, cos: All real numbers

tan, sec: All real numbers other than2

n for any integer n.

cot, csc: All real numbers other than n for any integer n.

Definition of Periodic Function

A function f is periodic if there exists a positive real number such that

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. 9

sin4


6. sin 3

7. 8

cos3

Even and Odd Trigonometric Functions

The cosine and secant functions are even.

cos( ) cost t sec( ) sect t

The sine, cosecant, tangent, and cotangent functions are odd.

sin( ) sint t csc( ) csct t

tan( ) tant t cot( ) cott t

Remember:

( ) ( )

( ) ( )

Even f t f t

Odd f t f t

Practice Problems: Use the value of the trig function to evaluate the indicated functions.

8. 3

sin( )8

t sin t csct

9. 4

cos5

t cos t

cos t

Practice Problems: Use a calculator to evaluate. Round to 4 decimal places.

10. sin4

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. sec7


sin csc

cos sec

Practice Problem 17: Let be an acute angle such that cos 0.6 . Find:

tan cot

sin tan

Practice Problem 18: Given

13csc

2 and

13sec

3 , find

cos

sec 90

cot tan 90

Practice Problem 19: Given tan 5 , find

cos csc

Practice Problems: Evaluate the value of in degrees 0 90 and radians 02

without using

a calculator. 20. csc 2

21. tan 1

22. cot 1 23.

3sin

2

Practice Problems: Use a calculator to evaluate the value of in degrees 0 90 and radians

02

. Round to the nearest degrees and 3 decimal places for radians.

24. sec 2.4578

25. sin 0.4565

26. cot 2.3545 27. sin 0.3746


LESSON 7.3 – TRIGONOMETRIC GRAPHS Graph of Sine Function

siny x

Domain: Range: Period: x-intercepts: Relative Minima: Relative Maxima:

x y

2

0

2

3

2

2

Graph of Cosine Function

cosy x

Domain: Range: Period: x-intercepts: Relative Minima: Relative Maxima:

x y

2

0

2

3

2

2


Transformations

sin

cos

y d a bx c

y d a bx c

Vertical stretch: 1a

a Scaling factor

Vertical shrink: 1a

a Reflection over the x-axis

2b period

b

c Horizontal translation (left or right)

d Vertical translation (up or down)

Definition of Amplitude of Sine and Cosine Curves

The amplitude of siny a x and cosy a x represents half the distance

between the minimum and the maximum value of the function and is given by

Amplitude a .

Practice Problems: Write an equation for each dashed curve. 1. y _______________

2. y ________________

Practice Problem 3: Write equations for both curves

solidy _______________

dashedy ______________


Period of Sine and Cosine Functions

Let b be a positive real number. The period of siny a bx and cosy a bx is

2

b

.

Practice Problems: Write and equation for each dashed curve. 4. y _______________

5. y ________________

Graphs of Sine and Cosine Functions

The graph of siny a bx c and cosy a bx c have the following

characteristics. (Assume 0b ).

Amplitude a 2

Periodb

The left and right endpoints of a one-cycle interval can be determined by solving the

equations: 0

2

bx c

bx c

.

Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.

6. 2sin 4y x

Amplitude: Period: Left Endpoint (LEP): Right Endpoint (REP): Vertical shift: Reflection:


7. 3cos 22 2

xy


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 0.001sin880y t , where t is time in seconds.

a. What is the period of the function?

b. The frequency f is given by 1

fp

. What is the frequency of the note?

Practice Problems: Write an equation for each curve. 9. y _______________

10. y ________________


Practice Problems: Sketch the graphs. Include two full periods. Label everything carefully.

11. 1

sin 22 3

xy


12. 3 4cos 510

y x


13. 4 sin 33

y x


notes 6 & 7: trigonometric functions of angles and...

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