Trigonometry Packet #1 Name: - Perry Local...

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1 Mar 261:51 PM Trigonometry Packet #1 Name: ________________ θ opposite side adjacent side hypotenuse S O H C A H T O A a b c 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.) 13 5 θ sinθ = ____ cscθ = ____ cosθ = ____ secθ = ____ tanθ = ____ cotθ = ____ 5 52 θ sinθ = ____ cscθ = ____ cosθ = ____ secθ = ____ tanθ = ____ cotθ = ____

Transcript of Trigonometry Packet #1 Name: - Perry Local...

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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.)

13

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

55√2

θ

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

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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

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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

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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

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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 =

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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 =

Apr 7­11:15 AM

3.) θ=7π/4

4.) θ=2π/3

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

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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 θ's 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.

Apr 7­11:40 AM

Examples Evaluate the expression in both radians and degrees.

1.) cos-1√3 2

2.) sin-1-√2 2

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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

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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?

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Mar 26­7:19 PM

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

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

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1715

θ

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

sinθ = ____ cscθ = ____

cosθ = ____ secθ = ____

tanθ = ____ cotθ = ____

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Mar 26­7:43 PM

4.) Solve ΔABC.

35o16

A

BC a

b

B = ____

b = ____

a = ____

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

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Apr 7­12:44 PM

Homework #2 Name: ______________

Draw an angle with the given measure in standard position.

1.) 110o 2.) 450o 3.) -3π/2 (Hint: change to degrees f

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

4.) -87o 5.) 120o

Apr 7­12:50 PM

6.) 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 =

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

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Apr 7­12:53 PM

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

8.) θ = π

9.) θ = 4π/3

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

sinθ = cscθ =

cosθ = secθ =

tanθ = cotθ =

Apr 7­12:56 PM

Evaluate the expressions in both radians and degrees.10.) cos-1(1/2) 11.) tan-1(-1)

12.) 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?