Unit Circle
Triangulation
Scale Changes
Spring Motion/Sound
Advanced Graphing
100
Cos(45°)
√2/2
Why: At 45° on the unit circle, the point is:
(√2/2, √2/2). Cosine is the first value (x).
100
Define sin, cos, and tan in terms of a triangle.
Sin= opposite/hypotenuse
Cos= adjacent/hypotenuse
Tan= opposite/adjacent
Why: Use the phrase SOH-CAH-TOA to remember this. (In a right triangle the adjacent side is one of the sides on the right angle, the opposite is the other side of the right angle, and the hypotenuse is the side that is always the longest.
100
In the basic equation: y=A sin B (x-C) + D, what do all the letters stand for in a scale change? (Not including x and y)
A=Amplitude: vertical stretch/shrink
B=Period: horizontal stretch/shrink
C=Phase shift: horizontal slide
D=Vertical displacement: vertical slide
100
In the equation of y=17 sin 2 (x-5) + 6, find the period, amplitude, phase shift, and vertical shift.
Amplitude=17, Phase shift= right 5, Vertical shift= up 6, and Period=π
Why: The amplitude is just about knowing where to look for it. The phase shift is always opposite so since it is -5 in the equation, it goes right 5 instead of left. The vertical shift always stays the same so since is is +6 it goes up 6. The period is found by taking 2π/B and the B value is 2, so that makes the period π.
100
Graph the following:
y=1+cosx
200
Cot(-480°)
√3/3
Why: First you take -480° and add 360° to get -120°. Then you find that -120° is the same as 240°. (360-120). Cotangent=cosine/sin. The point at 240° is (-1/2, -√3/2). Then you divide -1/2 by -√3/2 (or -1/2 x -2/√3) which gives you 1/√3. That can then be simplified to the answer.
200
In triangle ABC, a= 6 inches, angle B= 31°, and angle C=74°. Find the measure of side c. (Round to the nearest tenth)
15.2 inches
Why: First you find the 3rd angle by taking 180°-31°-74° to get angle A=75°. Then since this triangle is an ASA, you use law of sines. You set it up like this: sin75/6 = sin74/c. Then you cross multiply: (sin74 x 6)/sin75 = 15.24
200
Write an equation for a cos graph that is moved left 8 units, 9 units up, has an amplitude of 3 and change the period by 4 times.
y=3 cos 1/4 (x+8)+9
Why: Since is is moved 8 units left (left is negative) it would be switched to say postive. Since is is moved 9 units up, this doesn´t change and you +9 to the end. When you change the period by 4 times, you get 8 pi (2 pi being the original and multiplying it by 4). Then you set up 8pi=2pi/B and you B value is what goes in the equation.
200
Which of these equations has the loudest sound and which has the highest pitch?
a(t)=5sin(200πt)
b(t)=10sin(100πt)
c(t)=15sin(50πt)
Loudest sound: c(t)
Highest pitch: a(t)
Why: A bigger amplitude means a louder sound and c(t) has the biggest amplitude (15). A higher frequency means a higher pitch, so when you divide each of the frequencies by 2, a(t) has the highest pitch because the frequency is 100.
200
Graph the following
y=3sinx
300
Convert 832° to radians
208π/45
Why: Degrees to radians is: π/180 x degrees. So you do π/180 x 832 to get 832π/180. The biggest number that you can divide both numbers by is 4, when you do that you get 208π/45.
300
In triangle ABC, side a=8, side b=15, and side c=10. Find the measure of the largest angle. (Round to the nearest tenth)
112.4 degrees
Why: You know the largest angle is going to be angle B because it has the longest side. And you want to use law of cos becuase it is a SSS triangle. So 15²=8²+10²-2(8)(10)(cosB) Then you multiply it out and subtract from 15². You do 225-164 to get 61=-160cosB. Then divide and take the inverse of the answer to find the angle.
300
Explain the difference in the graphs following:
y=sinx
y=7 sin 4/3 (x-2) + 5
The first is an original sine wave. The second equation is the same as the first one but slid 5 units up and 2 units to the right. It is also vertically stretched by 7 units and horizontally stretched by 3π/2 units.
Why: 5 is the vertical slide and since it is positive it goes 5 up. 2 is the horizontal slide and since it is negative, it goes 2 right. 7 is the amplitude so it is vertically stretched by 7. 4/3 is the period so you set up 2π/B=4/3 which gives you a B value of 3π/2.
300
In the wave voltage equation:
I=6 sin (29160°t)
Find the peak value, frequency, and period.
Peak value=6
Frequency=81
Period= 1/81
Why: The peak value is always the amplitude. Then since it says 29160°, you divide that by 360° which gives you 81 for the frequency. The period is the reciprocal of the frequency so it is 1/81
300
Graph the following:
y=x² +cosx
400
Find all solutions (to the nearest tenth) of:
csc(x)= 1.873 in the domain 0<x<2π
32.3°, 144.7°
Why: First you take the inverse of csc(1.873) and you get 32.3°, that is the first answer. You know that csc is 1/sin so in order to get another answer you must reflect over y-axis to keep the value positive. Then you do 180°-32.3° and you get the second answer
400
In triangle ABC, angle A=51°, side b=13 cm and side c=16 cm. Find the length of side a. (Round to the nearest tenth)
12.8
Why: Since this is a SAS triangle, you know to use law of cos. You set it up like this: a²=13²+16²-2(13)(16)(cos51°). Then you multiply out the entire left side and you end up with 163.2. Since a is squared, you take the square root of your answer and that is side a.
400
Graph the following:
y=3 cos 2 (x+π/2) -1
400
A spring is pulled down 3 cm below the equilibrium and let go. It comes back to 3 cm below the equilibrium at 4 seconds. Find the period, amplitude, phase shift, and vertical shift of the motion of the spring when it is represented with sin.
Period: 4 seconds Amplitude: 3 cm
Phase shift: left 1 second Vertical shift: None
Why: Since the spring is pulled down 3 cm you know that the amplitude is going to be 3. The equilibrium is 4 seconds so that means it takes 4 seconds to make a complete revolution. Then, the halfway point where it goes up starts at 1 second rather than 0 so the ps is 1. There is no vertical shift because it does not move up or down.
400
Graph the following:
y=.5 to the x power (sinx)
500
Find all solutions (to the nearest thousandth) of:
cot(x)=-0.852 in the domain 0<x<2π
2.276, 5.418
Why: First you do the inverse of cot(-0.852) to get 2.276. This is in the 1st quadrant so you know the other point must be in the 4th (one positive and one negative value). So you take 2.276+π to get 5.418
500
In triangle ABC, angle A=30°, side a=10, and side b=16. Find the measure of angle B. (Hint: This is a SSA triangle so there could be more than one answer).
53° or 127°
Why: First, you want to solve for angle B, it is a SSA triangle so use law of sin. Sin(30°)/10 = sin(B)/16. Then after cross multiplying you get .8, which you need to take the inverse of. Then you get the first angle to be 53°. Then you need to consider that there could be more than 1 answer. So you take 180°-your first answer(53°) and you get 127°. If you add 127° to 30° (The other angle) it is less than 360° so you know that it works.
500
Write an equation in sin for the following graph:
y=4sin(x-π)+2
Why: By taking the highest value minus the lowest value and dividing by 2 you can find the amplitude. Then since it is a sin graph, you can see that it is slid over π units right, which means you put it as a negative in the equation. It is also moved 2 units up from the original sin graph so you put +2 on the end.
500
A ferris wheel has a diameter of 40 feet and makes a complete rotation every 20 seconds. The bottom of the wheel is 5 feet off the ground. Write an equation to represent this motion. (Use sin)
y=20 sin π/10 (x-5) + 25
Why: You know the amplitude is 20 because it is the (high-low)/2, the high is 45 (40 feet but 5 ft off the ground) and low is 5, so 40/2=20. Then the period is π/10 because it is 20 seconds and 2π/20 equals π/10. Since it is a sin graph, the center point of the line where it goes up would be at (5, 25) so it goes right 5 and up 25 because the original sin wave starts at (0,0).
500
Graph the following:
y=√x +sinx
In the domain: x>0
