Question #1:
You used Pythagorus' theorem to determine whether or not a triangle was a right triangle. The sides of the triangle are:
a = sqrt(416), b = sqrt(601), and c = sqrt(1009) so that a2 + b2 did not equal c2. Thus it is not a right triangle. Let the α, β, γ be the angles of the triangle across from sides a,b,c respectively.
Use the law of cosines to determine how far the angle γ across from c deviates from 90 degrees. (Round all answers to 2 decimal places.)
Use the law of sines to determine the other 2 angles α and βof the triangle.
Does (α + β + γ) = 180 degrees? (yes or no)
Find the area of the triangle using theorem 11.4, p903.
Find the area of the triangle using Heron's formula (theorem 11.6, p914).
Do your answers in (4) and (5) agree? (yes or no)
If your answer is no, which answer do you think is correct? The one from (4) or the one from (5)? Both of them are wrong?
Question #2:
A vector is like an arrow with a length and a direction. A vector P ? as measured from the origin has it tail at (0,0) and its tip at (x,y).
P ? = (x,y) = xi ^ + y j ^
P ?= |P|( cos (θP) i ^ + sin(θP)j ^ )
where |P| = sqrt(x2 + y2), cos(θP) = x/|P|, sin(θP) = y/|P|.
Let P ? = (3,4) and Q ? = (5,-12).
Plot (x,y) for P ? on graph paper
Draw the vector P ? on your graph paper.
What is |P|?
What is θP?
Draw θP on your graph.
What reference angle αP corresponds to θP?
Plot (x,y) for Q ?.
Draw the vectorQ ? on your graph paper
What is |Q|?
What is θQ?
Draw θQ on your graph.
What reference angle αQ corresponds to θQ?
The vector PQ is defined as(PQ) ? =Q ? - P ? = Q ?+ (-P ?), where (-P ?) = (-x,-y).
What are the componentsof(PQ) ? ?
Using the vector addition laws (p. 1014), draw (PQ) ? on your graph starting by placing its tail on the tip ofP ?, so as to obtain P ? + (PQ) ? = Q ?.
Does the tip of (PQ) ?actually end up at the tipof Q ?? (yes or no)
What is |PQ|?
What is θPQ?
What reference angle αPQ corresponds to θPQ?
Question #3:
Again using the data from the first problem of Quiz 1. A = (-8,19), B = (-13,-5), and C = (7,-9).
By definition, the vector(PQ) ? = Q ? - P ? (p.1012-1013).
What are the components of the vector (CA) ??
Plot (CA) ? on graph paper.
What is |CA|?
What is θCA?
What are the components of the vector (CB) ??
Plot (CB) ? on your graph.
What is |CB|?
What is θCB?
Using the Dot Product (pp1034-1036) to calculate angle ACB (angle β in question #1).
Does your value of β from the Dot Product agree with your value of β from using the Law of Sines in question #1? (yes or no)
If your answer is no, which value of β do you think is correct? The one you calculated using the Law of Sines, or the one you calculated using the Dot Product? Both values are wrong?
Does β = θCB - θCA? (yes or no)
Question #4:
Vectors are used in physics to estimate how much tension is in the cables holding a weight. Suppose a 100 lb weight is hung from the ceiling using 2 steel cables. We want to calculate the tension that will be needed so we can use a cable which is sufficiently strong to hold the weight.
The left hand figure is the physical setup. The right hand picture is the physicist's model.
The vectors we will be using are:
A ? = |A| (cos(θA)i ^ + sin(θA) j ^)
B ? = |B| (cos(θB)i ^ + sin(θB) j ^ )
W ? = |W| (cos (θW) i ^ + sin(θB)j ^)
What is θA?
What is θB?
What is θW?
What is |W|?
Newton's first law of motion says that for a point which is not moving, the vector sum of the forces equals 0. The physicist chose the origin to be the point which is not moving. So
0 = A ? + B ? + W ?
Using the above, you should get two separate equations, one for i ^and a second for j ^.There will be two unknowns, |A| and |B|. Thus, you can solve for the tensions in the cables.
What is |A|?
What is |B|?
What is the minimum strength of the cables in lbs that must be used in order to hold the weight?
Question #5:
Rewrite the following quantity as an algebraic expression of x and state the domain on which the equivalence is valid
sin (arctan(x)) = sin(θ), whereθ = arctan(x/1)
Draw a right triangle for θ with opp = x and adj =1 so that tan(θ) = x.
What is hyp?
What is sin(θ) for this triangle in terms of x and other constants?
What is the domain of x for f(x) = sin(θ)?
Question #6
Solve the following equation giving the exact solutions which lie in x = [0,2π).
tan(4x) = tan(x)
Use the tangent addition formula with 4x = 3x + x to get an equation in tan(3x) and tan(x).
What is the maximum possible number of solutions for tan(x)?
What is the maximum possible number of solutions for tan(3x)?
What are the actual solutions for x which are based on tan(x)?
What are the actual solutions for x which are based on tan(3x)?
Note: for each actual solution of (3x), there are two other solutions as well consisting of (3x +2π) and (3x + 4π). So for each x, (x +2π/3), and (x + 4π/3) are also solutions.
Question #7:
Again, everyone should memorize the following:
2π radians = 360 degrees = 1 cycle = 1 revolution
T (period in time/cycle) = 1/f (frequency in cycles/time), and ω = 2πf (angular frequency in rad/sec or rev/sec).
Pendulums obey the law of simple harmonic motion, i.e. they move as sine and cosine functions with respect to time.
S(t) = Acos(ωt -φ) , where A is the amplitude, ω is the angular frequency, and φ is the phase shift.
Given the following two equations of motion with t in sec:
S1(t) = 5cos(2πt/8)
S2(t) = 10 cos(2πt/16)
What is ω1, the angular frequency of pendulum #1? (Be sure to state the units for all quantities)
What is ω2, the angular frequency of pendulum #2?
What is f1, the frequency of pendulum #1?
What is f2, the frequency of pendulum #2?
What is T1, the period of pendulum #1?
What is T2,the period of pendulum #2?
What is A1, the amplitude of pendulum #1?
What is A2, the amplitude of pendulum #2?
The period of a pendulum is T = 2πsqrt(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. If L1 = 10 ft,
what is L2? (Note: you do not need to know g to do this part.)
Does the period of oscillation of a pendulum depend on its amplitude? (yes or no)
Plot S1(t) on graph paper for t = (0,16).
Plot S2(t) on graph paper for t = (0,16).
How many cycles N1did you plot for S1?
How many cycles N2did you plot for S2?
Does N1/N2 = f1/f2 ? (yes or no)
Plot S3(t) = 10cos(2πt/16 - π) on your graph for t = (0,16), let φ = π.
Is S3(t) a reflection of S2(t) across the x-axis? (yes or no)
Show mathematically that S3(t) = -S2(t) using the cosine subtraction formula cos(α-β) on S3(t). I.e. show that a phase shift of π or -π is the equivalent of a multiplication by -1.