# some trigonometry questions

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?

2. What is x (adj)?

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.

https://brainmass.com/math/basic-calculus/some-trigonometry-questions-613685

#### 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.