Trig Packet Notes #1.notebook - Perry...
Transcript of Trig Packet Notes #1.notebook - Perry...
Trig Packet Notes #1.notebook
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February 15, 2017
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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
5θ
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
55√2
θ
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
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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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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.
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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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Examples:1.) Convert 125o to radians. 2.) Convert -π to degrees.
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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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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February 15, 2017
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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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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.
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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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
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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February 15, 2017
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Find the length of the arc.
Feb 258:55 AM
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February 15, 2017
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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
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 = ____
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4.) Solve ΔABC.
35o16
A
BC a
b
B = ____
b = ____
a = ____
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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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February 15, 2017
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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?
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
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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.
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 =
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February 15, 2017
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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 =
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?
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February 15, 2017
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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?
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.