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Question #1:
The transformation formulas between Cartesian coordinates (x,y) and polar coordinates (r,Î¸) are as follows:
x = rcos(Î¸) y = rsin(Î¸)
r = sqrt(x2 + y2) tan(Î¸) = y/x, where you have to determine which quadrant Î¸ is in.
Every Î¸ has a reference angle Î± as defined on pp.721-722, 747 in the book.
r is always positive. x and y can be either positive or negative.
For the following 5 cases, find the unknown quantities and draw the appropriate triangles, clearly labeling x, y, r, Î¸, and Î±. Round any numbers to 2 decimal places.
1. If r = 5 and Î¸ = 3Ï€/4, find and label x, y, and Î±.
2. If x = 3 and y = 4, find and label r, Î¸, and Î±.
3. If x = 3 and y = -4, find and label r, Î¸, and Î±.
4. If x = -3 and y = 4, find and label r, Î¸, and Î±.
5. If x = -3 and y = -4, find and label r, Î¸, and Î±.

Question #2:
Verify the following trigonometric identity by using trigonometric identities on the right hand side of the equation. Hint: remember the math student's pick up line. Hi, I'm sine squared. If you're cosine squared, then we can be one for a date tonight!
sin2(x) = [ 1 - cos(2x) ]/2
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Question #3:
Given the following triangle. Let hyp =12 and Î¸ = 60 deg.
1. What is Î²?
3. What is y (opp)?
4. What is the value of cos(Î²)?
5. What is the value of the reference angle Î± that corresponds to Î¸?

Question #4:
Solve: 2sin(2x) = 1
1. What are the 2 values that (2x) can have?
2. Draw the unit circle with the 2 triangles which correspond to your answers in (1). Clearly mark each of your (2x) angles in your figure.
3. What are the 4 values that x can have?

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Question #5:
Given the following triangle. Let hyp = 5 and Î¸ = 210 deg.
1. Draw the reference angle Î±.
2. What is the value of Î±?
3. Find Î².
4. Find x.
5. Find y.
6. What is the value of tan(Î¸)?

Question #6:
1. Let sin(Î±) = cos(Î±+Î²). Expand cos(Î±+Î²) with the appropriate trigonometric angle addition formula and find a value for Î².
2. Let cos(Î±) = sin(Î±+Î²). Expand sin(Î±+Î²) with the appropriate trigonometric angle addition formula and find a value for Î².
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Question #7:
Find the value of z = tan(Î¸), where Î¸ = arcos(4/5) or z = tan[ arcos(4/5)].
1. Draw a circle and then the reference triangle (i.e. Î¸ = Î±) with adj (or x) = 4, hyp (or r) = 5, and opp (or y) positive.
2. What is the value of Î±?
3. arccos is a double value function for [0,360). Draw the 2nd triangle (i.e. Î¸ not equal to Î±) with opp (or y) negative.
4. If Î¸ = arcos(4/5), then what are the 2 values of Î¸?
5. What are the 2 values of z?
Question #8:
Two sinusoidal functions of the same frequency, when added together or subtracted from each other, will produce a single sinusoidal function of that same frequency. The standard transformations using the angle addition formulas on p775 and p795 are:
y(t) = Acos(Ï‰t) + Bsin(Ï‰t)
= Dcos(Ï†)cos(Ï‰t) + Dsin(Ï†)sin(Ï‰t)
= Dcos(Ï‰t - Ï†)
where D = sqrt(A2 + B2) and tan(Î±) = B/A, A=Dcos(Ï†), B=Dsin(Ï†)
Ï† = Î± if A > 0
Ï† = Î± + Ï€ if A < 0
Ï† = Ï€/2 if A = 0 and B > 0
Ï† = -Ï€/2 if A = 0 and B < 0
Given: y(t) = 12cos(2Ï€t/12 + Ï€/4)
1. What is the angular frequency Ï‰? (be sure to state the units if t is time in sec)
2. What is D?
3. What is Ï†?
4. What is A?
5. What is B?
6. Does D = sqrt(A2 + B2)? (yes or no)
7. What is Î±?
8. What is the period T?
9. What is the amplitude?
10. y(t) is written in terms of radians. Re-write y(t) in terms of degrees.

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Solution Preview

Question #1:
The transformation formulas between Cartesian coordinates (x,y) and polar coordinates (r,Î¸) are as follows:
x = rcos(Î¸) y = rsin(Î¸)
r = sqrt(x2 + y2) tan(Î¸) = y/x, where you have to determine which quadrant Î¸ is in.
Every Î¸ has a reference angle Î± as defined on pp.721-722, 747 in the book.
r is always positive. x and y can be either positive or negative.
For the following 5 cases, find the unknown quantities and draw the appropriate triangles, clearly labeling x, y, r, Î¸, and Î±. Round any numbers to 2 decimal places.
1. If r = 5 and Î¸ = 3Ï€/4, find and label x, y, and Î±.
Solution:
x = rcos(Î¸) = 5cos(3Ï€/4) = 5( ) =
y = rsin(Î¸) = 5sin(3Ï€/4) = 5( ) =
tan(Î±) =|y|/|x| =
Î± = Ï€/4

2. If x = 3 and y = 4, find and label r, Î¸, and Î±.
Solution:
r = sqrt(x2 + y2) = sqrt(32 + 42) = sqrt(25) = 5
tan(Î¸) = y/x = 4/3
Î¸ = 53.13Â°
tan(Î±) =|y|/|x| = 4/3
Î± = 53.13Â°

3. If x = 3 and y = -4, find and label r, Î¸, and Î±.
Solution:
r = sqrt(x2 + y2) = sqrt(32 + (-4)2) = sqrt(25) = 5
tan(Î¸) = y/x = -4/3
Î¸ = 306.87Â° (since point is in fourth quadrant)
tan(Î±) =|y|/|x| =|- 4|/3 = 4/3
Î± = 53.13Â°

4. If x = -3 and y = 4, find and label r, Î¸, and Î±.
Solution:
r = sqrt(x2 + y2) = sqrt((-3)2 + 42) = sqrt(25) = 5
tan(Î¸) = y/x = 4/-3 = -4/3
Î¸ = 126.87Â°(since point is in second quadrant)
tan(Î±) =|y|/|x| =|- 4|/3 = 4/3
Î± = 53.13Â°

5. If x = -3 and y = -4, find and label r, Î¸, and Î±.
Solution:
r = sqrt(x2 + y2) = ...

Solution Summary

This posting includes step by step solutions of questions on reference angle, unit circle and trigonometric equations. Transformation formulas between Cartesian coordinates and polar coordinates are examined.

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