Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 1

NOTES 2.5, 6.1 – 6.3

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

LESSON 2.5 – MODELING VARIATION Direct Variation y mx b when 0b or

y mx or y kx y kx and 0k

- y varies directly as x - y is directly proportional to x - k is the constant of variation - k is the constant of proportionality

Express the statement as an equation. Use the given information to find the constant of proportionality. B is directly proportional to L. If L = 15, then B = 1350.

Direct variation as thn power

ny kx and 0k

- y varies directly as the thn power of x - y is directly proportional to the thn power

of x

B is directly proportional to the square of L. If L = 15, then B = 1350.

Inverse variation

ky or k xy

x

As ,x goes up y goes down or As ,y goes up x goes down

n varies inversely as f. If n = 3, then f = 5.

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 2

Joint variation z kxy z varies directly as x and y

M varies jointly as h and n. If h = 7 and n = 9, then M = 504.

Practice Problems: 1. The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. a. Write an equation that expresses this variation. b. Find the constant of proportionality if 100L of gas exerts a pressure 33.2 kPa at a temperature of 400 K. c. If the temperature is increased to 500K and the volume is decreased to 80L, what is the pressure of the gas?

2. The power P (measured in horse power, hp) needed to propel a boat is directly proportional to the cube of its speed s. An 80-hp engine is needed to propel a certain boat at 10 knots. Find the power needed to drive the boat at 15 knots.

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 3

LESSON 6.1 – ANGLE MEASURE Trigonometry Measurement of triangles

Angles Note: Labeled with Greek letters:

, , ,...

An angle is in standard position if… 1. its initial side is along the positive x-axis 2. its vertex is at the origin, and

Coterminal Angles Angles with the same initial and terminal sides.

360

2

and n

and n

Review 1. Central angle 2. Acute angle

3. Right angle 4. Obtuse angle 5. Arc measure

6. Arc length 7. Complementary angles 8. Supplementary angles

Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles. 1. 210

2. 45 3. 540

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 4

Radian Measure

One radian is the measure of a central

angle that intercepts an arc s equal in length to the radius of a circle.

C = 2r 6.28r

The radian measure of an angle of one full revolution is 2 . Since one full circle has 360 ,

Note: In a full revolution, the arc length s is equal to

2C r s . Also, there are just over six radius lengths in a full circle. Therefore, the central

angle is s

r where

is measured in radians.

360 2 rad 180 rad 90

2

rad 60

3

rad

Degrees to Radians

Multiply by 180

Example:

Radians to Degrees Multiply by

180

Example:

Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without using a calculator. 4. 150

5. 7

6

6. 240 7.

11

30

Practice Problems: Convert the following angles from degrees to radians and from radians to degrees using a calculator and round to 3 decimal places. 8. 87.4

9. 2 10. 0.54 11. 0.57

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 5

Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co-terminal with the given angle. Sketch the angles.

12. 13

6

13. 3

4

14.

2

3

Length of a Circular Arc

In a circle of radius r, the length s of an arc that subtends a central angle of radians is: s r Note: must be in radians.

Example:

Review Problem 15: Find the following arc lengths using geometry then use s r to validate your answers. Given 24CB in ,

0 60m A B , find the following arc lengths using 2 methods

60O

D

C B

A

24 in

mAB

mCA

mCDB

mADB

mADC

1. Circumference =

2. Length of AB =

3. Length of CA =

4. Length of CDB =

5. Length of ADB =

6. Length of ADC =

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 6

Practice Problems: Find the unknown value. 16. A central angle in a circle of radius 24 cm is subtended by an arc of length 6 cm. Find the measure of in radians.

17. Find the radius of the circle if an arc of length

8 in on the circle subtends a central angle of 4

.

18. A bicycle’s wheels are 14 inches in diameter. How far (in miles) will the bike travel if its wheels revolve 500 times without slipping?

19. How many revolutions will a Ferris wheel of diameter 60 feet make as the Ferris wheel travels a distance of a ½ mile?

20. An ant is sitting 5 cm from the center of a c.d.. If the c.d. turns 40 , how far has the ant moved in meters?

21. A bug is on a car’s windshield wiper and is 10 inches from the base of the windshield wiper. If the bug moves 34 inches, at what angle did the windshield wiper turn?

Angular Speed The angular velocity of a point on a rotating object is the

number of degrees (radians, revolutions, etc.) per unit time through which the point turns.

t

Linear Speed The linear velocity of a point on a rotating object is the distance per unit time that the point travels along its circular path.

sv

t

Note: The linear velocity depends on how far the object is from the axis of rotation, whereas the angular velocity is the same no matter where the object lies on the rotating object.

Relationship between Linear and Angular Speed

If a point moves along a circle of radius r with angular speed , then its linear speed v is given by: v r

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 7

Practice Problems: Solve the following problems. 22. A woman is riding a bike whose wheels are 26 inches in diameter. If the wheels rotate at 125 revolutions per minute (rpm), find the speed at which she is traveling, in miles per hour.

23. The rear wheels of a tractor are 4 feet in diameter, and turn at 20 rpm. (a) How fast is the tractor going (feet per second)? (b) The front wheels have a diameter of only 1.8 feet. What is the linear velocity of a point on their tire treads? (c) What is the angular velocity of the front wheels in rpm?

24. The pedals on a bike turn the front sprocket at 8 radians per second. The sprocket has a diameter of 20 cm. The back sprocket, connected to the wheel, has a diameter of 6 cm. (a) Find the linear velocity of the chain. (b) Find the angular velocity of the back sprocket.

25. Dan and Ella are riding on a Ferris wheel. Dan observes that it takes 20 seconds to make a complete revolution. Their seat is 25 feet from the axle of the wheel.

(a) What is their angular velocity? (b) What is their linear velocity?

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 8

xx 2

x

90 45

45

bc

a

90 45

45

LESSON 6.2 – TRIGONOMETRY OF RIGHT TRIANGLES The Trigonometric Ratios

Let has an acute angle of a right triangle. The six trigonometric functions of the angle are defined below.

sinopp

hyp csc

hyp

opp

cosadj

hyp sec

hyp

adj

tanopp

adj cot

adj

opp

SOH CAH TOA

Review: Special Right Triangles

2xx 3

x

30

6090

Practice Review Problems: Evaluate the following

1. 4a b c

2. a 6 2b c

3. a b 10c

4. 7 3a b c

5. a b hat c

6. a b 7c iPod

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 9

7. 4a b c

8. a 6 2b c

9. a b 10c

10. 7 3a b c

11. a b hat c

cb

a

30

6090

12. a b 7c iPod

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

sin

17csc

4

cos sec

1.

tan cot

sin csc

cos sec 6

2.

tan

cot

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 10

Practice Problems: Evaluate without using a calculator. Draw and label triangles. 3. tan 60

4. csc45

5. tan30

6. sec30

Practice Problems: Evaluate the given expression without using a calculator. Leave your answer in simplest radical form. 7. sin 60 cos30 8. tan 45 cot(60)

9. tan 60sec60 10. 15sin 30cos 45

Practice Problems: Solve the right triangle.

c

b=19.4

a

34.2C

B

A

a b c

m A m B m C

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 11

100

75

6

72.3

12. a b c

m A m B m C

13. a b c

m A m B m C

Angles of Elevation and Depression

The angle of elevation is the angle from a horizontal line UP to an object. The angle of depression is the angle from a horizontal line DOWN to an object.

Practice Problems: Set up an equation for each word problem and solve. 14. Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes you dizzy so you decide to do the whole job at ground level. From a point 47.3 meters from the base, you find that you must look up at an angle of 53 degrees to see the top of the tower. How high is the tower?

15. When landing, a jet will average a 3 angle of descent. What is the altitude, to the nearest foot, of a jet on final descent as it passes over an airport radar 6 miles from the start of the runway?

16. At a point 300 feet from the base of a building, the angle of elevation to bottom of a smokestack is 40 , and the angle of elevation to the top is 55 . Find the height of the smokestack alone.

17. The distance between a plane and a building on the ground is 350 feet. The angle of depression from the plane to the building is 20 . Find the horizontal distance from the plane to the building.

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 12

LESSON 6.3 – TRIGONOMETRIC FUNCTIONS OF ANGLES Definitions of Trigonometric Functions of any Angle

Let be an angle in standard position with ,x y

a point on the terminal side of and 2 2 0r x y .

siny

r csc , 0

ry

y

cosx

r sec , 0

rx

x

tan , 0y

xx

cot , 0x

yy

sin csc

cos sec

Practice Problem 1: Let 4, 3

be a point on the terminal side of . Find:

tan cot

sin csc

cos sec

Practice Problem 2: Let 1 3

3 , 72 4

be a point on the

terminal side of . Find:

tan cot

Signs of the Trigonometric Functions

Use “All Student Take Calculus” to figure out in which quadrant each trig function has a positive value.

cos/sec positive

Calculus

tan/cot positive

Take

sin/csc positive

Students All

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 13

sin :

cos :tan :

csc :

sec :

cot :

sin :

cos :tan :

csc :

sec :

cot :

sin :

cos :tan :

csc :

sec :

cot : cot :

sec :

csc :

tan :cos :

sin :

3

2

2

0

( + , - )

( + , + )

( - , - )

( - , + )

IVIII

II I

sin csc

cos sec

Practice Problem 3: Find the value of the six trigonometric

functions. Given: 15

tan8

;

sin 0 tan cot

sin csc

cos sec

Practice Problem 4: Find the value of the six trigonometric functions. Given: cot is

undefined; 3

2 2

tan

cot

Practice Problem 5: Evaluate the following:

a. sin 0

b. sin

c. sin2

d.

3sin

2

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 14

Practice Problem 6: Find the reference angle ' for the following. Graph the angle a. 309 '

b. 145 '

c. 7

'4

d. 11

'3

Evaluating Trigonometric Functions of any Angle

To find the value of a trig function of any . 1. Determine the function value for the associated ' . 2. Depending on the quadrant in which lies, affix the appropriate sign to the

function value.

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

Definition of Reference Angle

Quadrant II

'

' 180 deg

rad

ree

Quadrant III

'

' 180 deg

rad

ree

Quadrant IV

' 2

' 360 deg

rad

ree

Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 15

FUNDAMENTAL TRIGONOMETRIC INDENTITIES

Reciprocal Identities

1sin

csc

1

cscsin

1cos

sec

1

seccos

1tan

cot

1

cottan

Quotient Identities

sintan

cos

cos

cotsin

Pythagorean Identities

2 2sin cos 1 2 21 tan sec

2 21 cot csc

Practice Problem 7: Evaluate the trig functions

a. 4

cos3

b. tan 210

c. 11

csc4

d. cot2

Practice Problem 8: Find the indicated trig function

a. 5

cos8

and Quad III

sec

b. 3

sin5

and Quad IV

cos

Practice Problem 9: Find two exact solutions of the equation. in degrees 0 360 and radians

0 2 .

a. cot 1

b. tan 2.3545 . Round to 2 decimal places.