TRIGONOMETRIC FUNCTIONS OF ANGLES Unit 5. ANGLE MEASURE 7.1.

TRIGONOMETRIC FUNCTIONS OF ANGLES

Unit 5

ANGLE MEASURE

7.1

What is an angle?

Angle: 2 rays with a common vertex

How many angles do I have here?

Positive Angle

Negative Angle

More about angles:

An angle is in “standard” position when its initial side is the positive x-axis and its vertex is at the origin. Positive angles are formed in a counter-clockwise direction and negative angles are formed in clockwise direction.

What is an angle measured in?

Radians One radian is the measure of the central

angle (Theta-Θ) whose arc (s) is equal to the length of the radius (r)

What is the circumference of a circle with a radius of 1 unit?

2So 1 revolution around the coordinate plane =

?2Therefore 1 revolution around the coordinate

plane = 2 radians

Radians

0

3/2

2-

-3/2

/2

-/2

Sketch the angle (find coterminal angles)

Coterminal angles are angles that have the same terminal side:

2/3-/47/57/3

What is an angle measured in – Part II

DegreesWhat is the difference between 3 and 3?1 = 1/360 of a circleWhy?

Sketch the angle and find coterminal angles: Degrees

150 282 -60-150 450

Conversion

Degrees to Radians Multiply by /180Radians to DegreesMultiply by 180/

Convert the following

150 282 -60-150 450

2/3-/47/57/3

Arc Length –

s = r where s is the arc length, r is the

radius and is the angle in RadiansIf r = 4 in find arc length if = 240 If r = 8 in and s = 15 in, find the angle

Area of a Sector 22

1rA

• What do you think is true about Θ here?

• Find the area of a sector with central angle 60∘ if the radius of the circle is 3m.

• The area of a sector of a circle with a central angle of 2 rad is 16 m2. What is the radius of the circle?

Homework 7.1

Pg 453-454: 2 – 50 even, 50 – 62 even, 45

RIGHT TRIANGLE TRIGONOMETRY

7.2a

Right Triangle Trigonometry

opp hyp

adj

Right Triangle Trig:

opp

adjcot

adj

opptan

hyp

adjcos

adj

hypsec

opp

hypcsc

hyp

oppsin

SOH CAH TOA

Right Triangle Trigonometry

Evaluate all 6 trig functions for .

3

4

Sketch the triangle:

cot = 5cos = 3/7

Special Right Triangle:

/6/4

/4/3

3tan,

6cos,30sin :Evaluate

Solving Triangles – solve for unknowns

50

15

Solving Triangles – solve for unknowns

32

10

Solving Triangles – solve for unknowns

25

12

Homework 7.2a

Pg 462 – 463: 1 – 4, 11 – 16, 22 – 25, 27, 28

RIGHT TRIANGLE TRIGONOMETRY – STORY PROBLEMS

7.2b

A kite is held at a 75 angle using 300 ft of string, how high off the ground is the kite if the person holds the kite 5 ft off the ground?

75

300x

30075sin

x

5 ft

A ladder is placed so that it reaches a point 8 feet from the ground on a wall, if the ladder makes a 15 with the wall, how long is the ladder?

15

8

x x

815cos

A 40 foot ladder leans against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle formed by the ladder and the building?

Farmer John looks down into a valley and sees his favorite cow Eugene grazing on grass. He notices that a balloon is floating directly above Eugene (his favorite cow). Farmer John determines that the angle of elevation to the balloon is 72 and the angle of depression to Eugene is 30, find how far above Eugene the balloon is if the hill is 40 ft from the valley below (and Farmer John’s eyes are 5 feet off the ground)

Horizontal line72

30

?45

x

yx

45tan300 x

y0tan72

From a point on the ground 500 ft from the base of a building, it is observed that the angle of elevation to the top of the building is 24 degrees, and the angle of elevation to the top of the flagpole atop the building is 27 degrees. Find the height of the building and the length of the flagpole.

Homework 7.2b

Pg 463 464: 34 – 46 even

7.4

LAW OF S INESHOW DO I SOLVE NON-RIGHT TRIANGLES?

Laws of Sines – angle and an opposite side

Given

C A

B

c

b

a

c

C

b

B

a

A sinsinsin

C

c

B

b

A

a

sinsinsin

Solve:

C A

B

c

80.4

a

20

25

Solve:

C A

B

32

b

a

100

50

But wait – there is a problem when you know 2 sides and an opposite angle

What is the sin30?½What is the sin150?½When you do sin-1(1/2) how do you know if its

supposed to be 30 or 150?

Scenario I:

C A

B

c

9.9

7 45

Scenario II:

C A

B

186

b

248

43

Scenario III:

C A

B

32

70

122

42

7.4a Homework

Pg 483-484: 1, 4, 6, 10, 16, 21, 22

One, Two or Zero?:

18,17,63 baA

20,32,112 baA

14,10,58 baA

LAW OF S INES – STORY PROBLEMS

7.4b

Story Problem #1

The course for a boat race starts at a point A and proceeds in the direction of S52W to point B, then in a direction of S40E to point C, and finally back to A (due North). If A and C are 8 km apart, what is the total distance for the race?

Story Problem #2

The pitchers mound on a softball field is 46 feet from home plate and the distance from the pitchers mound to first base is 42.6 feet. How far is first base from home plate?

Story Problem #3

A hot air balloon is flying above High Point. To the left side of the balloon, the balloonist measures the angle of depression to a soccer field to be 20⁰ and to the right he sees a football field and finds the angle of depression to be 62.5⁰. If the distance between the two fields is know to be 4 miles, what is the direct distance from the balloon to the soccer field?

How high is the balloon?

Story Problem #4

An architect is designing an overhang above a sliding glass door. During the heat of the summer, the architect wants the overhang to prevent the rays of the sun from striking the glass at noon. The overhang is to have an angle of depression equal to 55⁰ and starts 13 feet above the ground. If the angle of elevation of the sun during this time is 63⁰, how long should the architect make the overhang?

Homework 7.4b

Pg 484 – 485: 24 – 30 even

7.5

LAW OF COSINES

Laws of Cosines – angle and the two included sides

Given

C A

B

c

b

a

Abccba cos2222

Baccab cos2222

Cabbac cos2222

Solve:

C A

B

212

388

a

82

• Hint always find the smaller angle first when you use the law of sines after using the law of cosines – why?

Solve:

C A

B

c

105

18

47

Solve:

C A

B

8

12

5

• Why should we solve for the biggest angle first? Which is the biggest angle?

Story Problem #1

A ship travels 60 miles due east, then adjust its course 15 northward. After traveling 80 miles how far is the ship from where it departed?

Story Problem #2

A car travels along a straight road, heading east for 1 hour, then traveling for 30 minutes on another road that leads northeast. If the car has maintained a constant speed of 40 mph in what direction did the car turn?

Homework 7.5

Pg 491-492: 1-3, 17-21, 24, 25, 27, 32

Navigation Problem/Worksheet

A boat “A” sights a lighthouse “B” in the direction N65⁰E and the spire of a church “C” in the direction S75⁰E. According to a map, the church and lighthouse are 7 miles apart in a direction of N30⁰W. Find the bearing the boat should continue at in order to pass the lighthouse at a safe distance of 4 miles.

Navigation Worksheet

[Type the document title] 1

Name:

1. Suppose the lighthouse “B” in the example is sighted at S30⁰W by a ship “P” due north of the church “C”. Find the bearing “P” should keep to pass “B” at 4 miles distance.

2. In the fog, the lighthouse keeper determines by radar that a boat 18 miles away is heading to the shore. The direction of the boat from the lighthouse is S80⁰W. What bearing should the lighthouse keeper radio the boat to take to come ashore 4 miles south of the lighthouse?

3. To avoid a rocky area along a shoreline, a ship “A” travels 7 km to “B”, bearing 22.25⁰, then 8 km to “C”, bearing 60.5⁰, then 6 km to “D”, bearing 109.25 . Find the distance from “A” to “D”.⁰

Unit 5 Test Review