Trig Packet Notes #1.notebook

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February 16, 2017

Mar 26­1:51 PM

Trigonometry Packet #1 Name: ________________

θ

opposite side

adjacent side

hypotenuseS O H C A H T O A

a

bc

Pythagorean Theorem: ________________

Right Triangle Definitions of Trig Functions

sinθ = ______ cscθ = ______

cosθ = ______ secθ = ______

tanθ = ______ cotθ = ______

Objectives: Students will be able to solve triangles using trig ratios and find trig ratios of a given angle.

Mar 26­6:39 PM

Examples Evaluate the six trig functions of the angle θ.

1.)

2.)

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5θ

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

55√2

θ

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

Trig Packet Notes #1.notebook

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February 16, 2017

Mar 26­6:41 PM

Example: Let θ be an acute angle of a right triangle. Find the values of the other five trig functions of θ. tanθ = 7 3

Example: Find x and y.

30o

y

x4

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

cotθ = ____

Mar 26­7:02 PM

a

15A C

B

c

28o

Example: Solve ΔABC. Note: This means to find all of the missing angles measures and side lengths.

Example: A tree casts a shadow as shown. What is the height of the tree?

25 ft31o

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­9:55 AM

standard position:

Examples: Draw an angle with the given measure in standard position.

1.) 240o 2.) 500o 3.) -50o

Objectives: Students will be able to work with angles in standard position, convert between radians and degrees and use the unit circle to solve problems.

Apr 7­10:18 AM

coterminal angles:

Examples: Find one positive angle and one negative angle that are coterminal with the given angles.

1.) 45o 2.) -380o

Angles can also be measured in __________.There are ____ radians in a full circle._____ radians = 360o , so ____ radians = 180o.

-To convert degrees to radians, multiply by π . 180

-To convert radians to degrees, multiply by 180 . π

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­10:30 AM

Examples:1.) Convert 125o to radians. 2.) Convert -π to degrees.

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Degree measure Radian measure0o 30o

π/460o

π/22π/3

135o

150o

180o

7π/65π/4

240o

270o

5π/3315o

11π/6360o

Apr 7­10:44 AM

Fill in the ratios using O = opposite, A = adjacent and H = hypotenuse.

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

General Definitions of Trig FunctionsLet θ be an angle in standard position, and let (x,y) be the point where the terminalside of θ intersects the circle x2 + y2 = r2. The six trig functions of θ are as follows:

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

r

(x,y)

θ

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­10:53 AM

Example: Let (-4,3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ.

The Unit Circle : the circle x2 + y2 = 1, which has center (0,0) and radius 1.

1

(x,y)

θ

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Apr 7­11:02 AM

Example Use the unit circle to evaluate the six trig functions of θ=270o.

Reference Angles Acute angles formed by the terminal side of θ and the x-axis.

Recall:30o = 45o =60o =

30o

60o

45o

45o

1

112 √2

√3

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Trig Packet Notes #1.notebook

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February 16, 2017

Feb 16­12:13 PM

Feb 16­11:59 AM

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­11:12 AM

Examples: Evaluate the six trig functions of θ. Simplify and rationalize.

1.) θ = π/3

2.) θ = 7π/6

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Feb 16­12:21 PM

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­11:15 AM

3.) θ=7π/4

4.) θ=2π/3

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Apr 7­11:24 AM

So far, we've learned how to evaluate trig functions of a given angle.Now, we'll study how to reverse the problem - find an angle thatcorresponds to a given value of a trig function.

Example sinθ = 1

Note: There are many θs that could satisfy the above equation. For this reason, we must make some restrictions.

Inverse Trig Functions:-Sine Inverse: -90o≤θ≤90o Cosine Inverse: 0 o≤θ≤180o

-Tangent Inverse: -90o≤θ≤90o

Objectives: Students will be able use inverse trig functions to solve for angles.

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­11:40 AM

Examples Evaluate the expression in both radians and degrees.

1.) cos-1√3 2

2.) sin-1-√2

2

Apr 7­11:43 AM

Examples Find the measure of angle θ.

1.)

2.) A monster truck drives off a ramp in order to jump onto a rowof cars. The ramp has a height of 8 feet and a horizontal length of20 feet. What is the angle θ of the ramp?

θ

49

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­11:55 AM

Some More Application Problems

1.) The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles has an angle of elevation of 30o. The length of the escalator is 152 feet. What is the height of the escalator?

2.) A fire truck has a 100 ft. ladder whose base is 10 feet above the ground. A firefighter extends a ladder toward a burning building to reach a window 90 ft. above the ground. Draw a diagram. At what angle should the firefighter set the ladder?

Feb 25­8:44 AM

Find the length of the arc.

Trig Packet Notes #1.notebook

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February 16, 2017

Feb 25­8:55 AM

Feb 25­8:56 AM

Trig Packet Notes #1.notebook

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February 16, 2017

Feb 25­8:56 AM

Mar 26­7:19 PM

Trig Homework #1 Name: ______________

1.) Find all 6 trig functions for 30o, 45o and 60o and fill in the table below. Make sure to rationalize all values.

30o

60o

45o

45o

θ sinθ cosθ tanθ cscθ secθ cotθ

30o

45o

60o

1

112 √2

√3

Trig Packet Notes #1.notebook

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February 16, 2017

Mar 26­7:38 PM

2.) Evaluate the six trig functions of θ.

3.) Let θ be an acute angle of a right triangle. Find the values of the other 5 trig functions of θ. cotθ = 6

11

1715

θ

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

Mar 26­7:43 PM

4.) Solve ΔABC.

35o16

A

BC a

b

B = ____

b = ____

a = ____

Trig Packet Notes #1.notebook

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February 16, 2017

Mar 26­7:46 PM

5.) Find the length, x, of the prop holding open the piano.

6.) A parasailer is attached to a boatwith a rope 300 feet long. The angle ofelevation from the boat to the parasaileris 48o. Estimate the parasailer's height above the boat.

25o

150 cm

x

48o

300 ft

Feb 25­8:41 AM

7.) On an analog clock, the minute hand has moved 128o from the hour. What number will it pass next?

8.) It is 2:46 p.m. What is the measure of the angle that the minute hand swept through since 2:00 p.m?

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­12:44 PM

Draw an angle with the given measure in standard position.

9.) 110o 10.) 450o 11.) -3π/2 (Hint: change to degrees f

Find one positive angle and one negative angle that are coterminal with the given angles.

12.) -87o 13.) 120o

Apr 7­12:50 PM

14.) Let (-3,-4) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ.

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

15.) Let (-6,9) be a point on the terminal side of an angle θ. Find all the trig ratios. Simplify and rationalize all values.

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­12:53 PM

Evaluate the six trig functions of θ. Simplify and rationalize.

16.) θ = π

17.) θ = 4π/3

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Apr 7­12:53 PM

Evaluate the six trig functions of θ. Simplify and rationalize.

18.) θ = -7π/6

19.) θ = -π/4

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Trig Packet Notes #1.notebook

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February 16, 2017

Apr 7­12:56 PM

Evaluate the expressions in both radians and degrees.20.) cos-1(1/2) 21.) tan-1(-1) 22.) sin-1(√2/2)

23.) A crane has a 200 ft. arm with a lower end that is 5 ft.off the ground. The arm has to reach to the top of the buildingthat is 160 ft. high. At what angle θ should the arm be set?

Feb 25­8:50 AM

24.) A geostationary satellite is positioned 35,800 km above the Earth's surface. It takes 24 hours to complete one orbit. The radius of the Earth is about 6,400 km. What distance does the satellite travel in 1 hour? 3 hours?

After how many hours has the satellite travelled 200,000 km?

Trig Packet Notes #1.notebook

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February 16, 2017

Feb 25­8:52 AM

25.) Suppose a windshield wiper arm has a length of 22 inches and rotates through an angle of 110o. What distance does the tip of the wiper travel as it moves once across the windshield?

26.) A CD with a diameter of 12 cm spins in a CD player. Calculate how much farther a point on the outside edge of the CD travels in one revolution than a point 1 cm closer to the center of the CD.