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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
5θ
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
55√2
θ
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
2
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
3
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
4
Apr 710:30 AM
Examples:1.) Convert 125o to radians. 2.) Convert -π to degrees.
12
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
5
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 =
6
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 =
7
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
8
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
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?
9
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
11
1715
θ
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
sinθ = ____ cscθ = ____
cosθ = ____ secθ = ____
tanθ = ____ cotθ = ____
10
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
11
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.
12
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?