Precalculus Trigonometry Review: Polar Coordinates, Vectors, Conic Sections

Q1. Match the point \( \left(4, \frac{4\pi}{3}\right) \) in polar coordinates with either A, B, C, or D on the graph.

Background

Topic: Polar Coordinates

This question tests your ability to interpret and plot points given in polar coordinates on a graph, and to match them with labeled points.

Key Terms and Formulas:

• Polar coordinates: \( (r, \theta) \), where \( r \) is the radius (distance from origin) and \( \theta \) is the angle measured from the positive x-axis.

• To plot: Move \( r \) units from the origin along the direction of angle \( \theta \).

Step-by-Step Guidance

1. Identify the value of \( r = 4 \) and \( \theta = \frac{4\pi}{3} \).

2. Recall that \( \frac{4\pi}{3} \) radians is greater than \( \pi \) (180°), so it is in the third quadrant.

3. On the graph, locate the ray corresponding to \( \frac{4\pi}{3} \) and count 4 units from the origin along that direction.

4. Compare the location with the labeled points A, B, C, and D to determine which matches.

Try solving on your own before revealing the answer!

Final Answer: D

The point \( \left(4, \frac{4\pi}{3}\right) \) is located 4 units from the origin along the ray at \( \frac{4\pi}{3} \) radians, which matches point D on the graph.

Q2. Match the point \( \left(5, \frac{5\pi}{3}\right) \) in polar coordinates with either A, B, C, or D on the graph.

Background

Topic: Polar Coordinates

This question tests your ability to plot a point given in polar coordinates and match it to a labeled point on a graph.

Key Terms and Formulas:

• Polar coordinates: \( (r, \theta) \)

• \( \frac{5\pi}{3} \) radians is just before completing a full circle (close to 300°).

Step-by-Step Guidance

1. Identify \( r = 5 \) and \( \theta = \frac{5\pi}{3} \).

2. Determine which quadrant \( \frac{5\pi}{3} \) lies in (it is in the fourth quadrant).

3. Locate the ray for \( \frac{5\pi}{3} \) and move 5 units from the origin along that direction.

4. Compare with the labeled points to find the match.

Try solving on your own before revealing the answer!

Final Answer: B

The point \( \left(5, \frac{5\pi}{3}\right) \) is 5 units from the origin along the ray at \( \frac{5\pi}{3} \) radians, which matches point B on the graph.

Q5. Plot the point \( \left(7, \frac{7\pi}{4}\right) \) in polar coordinates, and find other polar coordinates (r, θ) for the point for which the following are true: (a) r > 0, -2π < θ < 0; (b) r < 0, 0 < θ < 2π; (c) r > 0, 2π < θ < 4π.

Background

Topic: Polar Coordinates and Multiple Representations

This question tests your understanding of how a single point in polar coordinates can be represented in multiple ways, depending on the sign of r and the range of θ.

Key Terms and Formulas:

• Polar coordinates: \( (r, \theta) \)

• \( r > 0 \): θ is the standard angle.

• \( r < 0 \): θ is opposite direction (add π to θ).

• \( r > 0 \), θ outside [0, 2π]: add or subtract multiples of 2π.

Step-by-Step Guidance

1. Start with \( r = 7 \), \( \theta = \frac{7\pi}{4} \).

2. For (a): Find an equivalent angle in the range \( -2\pi < \theta < 0 \) by subtracting \( 2\pi \).

3. For (b): To use \( r < 0 \), add \( \pi \) to \( \theta \) and change the sign of r.

4. For (c): To use \( r > 0 \), add \( 2\pi \) to \( \theta \) to get an angle in \( 2\pi < \theta < 4\pi \).

5. Write each new coordinate pair in the form \( (r, \theta) \).

Try solving on your own before revealing the answer!

Final Answer:

(a) \( (7, -\frac{\pi}{4}) \) (b) \( (-7, \frac{3\pi}{4}) \) (c) \( (7, \frac{15\pi}{4}) \)

Each representation uses the appropriate sign and range for r and θ.

Q7. The polar coordinates of a point are (-5, 0). Find the rectangular coordinates of this point.

Background

Topic: Polar to Rectangular Conversion

This question tests your ability to convert from polar coordinates to rectangular (Cartesian) coordinates.

Key Terms and Formulas:

• Polar coordinates: \( (r, \theta) \)

• Rectangular coordinates: \( (x, y) \)

• Conversion formulas: $x = r \cos \theta$ $y = r \sin \theta$

Step-by-Step Guidance

1. Identify \( r = -5 \) and \( \theta = 0 \).

2. Calculate \( x = r \cos \theta \) and \( y = r \sin \theta \).

3. Recall that \( \cos 0 = 1 \) and \( \sin 0 = 0 \).

4. Plug in the values to find x and y.

Try solving on your own before revealing the answer!

Final Answer: (-5, 0)

Using the conversion formulas, the rectangular coordinates are (-5, 0).

Q8. The polar coordinates of a point are \( \left(2, \frac{6\pi}{5}\right) \). Find the rectangular coordinates of this point.

Background

Topic: Polar to Rectangular Conversion

This question tests your ability to convert polar coordinates to rectangular coordinates using trigonometric functions.

Key Terms and Formulas:

• Polar coordinates: \( (r, \theta) \)

• Rectangular coordinates: \( (x, y) \)

• Conversion formulas: $x = r \cos \theta$ $y = r \sin \theta$

Step-by-Step Guidance

1. Identify \( r = 2 \) and \( \theta = \frac{6\pi}{5} \).

2. Calculate \( x = 2 \cos \left(\frac{6\pi}{5}\right) \) and \( y = 2 \sin \left(\frac{6\pi}{5}\right) \).

3. Evaluate the cosine and sine values for \( \frac{6\pi}{5} \).

4. Express the answer as an ordered pair.

Try solving on your own before revealing the answer!

Final Answer:

The rectangular coordinates are \( \left(2 \cos \left(\frac{6\pi}{5}\right), 2 \sin \left(\frac{6\pi}{5}\right)\right) \approx (-1.618, -1.175) \).

These values are calculated using the conversion formulas.

Q9. The rectangular coordinates of a point are (13, 0). Find the polar coordinates.

Background

Topic: Rectangular to Polar Conversion

This question tests your ability to convert from rectangular coordinates to polar coordinates.

Key Terms and Formulas:

• Rectangular coordinates: \( (x, y) \)

• Polar coordinates: \( (r, \theta) \)

• Conversion formulas: $r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Step-by-Step Guidance

1. Identify \( x = 13 \) and \( y = 0 \).

2. Calculate \( r = \sqrt{13^2 + 0^2} \).

3. Calculate \( \theta = \arctan\left(\frac{0}{13}\right) \).

4. Express the polar coordinates as an ordered pair.

Try solving on your own before revealing the answer!

Final Answer: (13, 0)

The polar coordinates are (13, 0), since the point lies on the positive x-axis.

Q10. The rectangular coordinates of a point are (-10, 0). Find the polar coordinates.

Background

Topic: Rectangular to Polar Conversion

This question tests your ability to convert from rectangular coordinates to polar coordinates, especially when x is negative.

Key Terms and Formulas:

• Rectangular coordinates: \( (x, y) \)

• Polar coordinates: \( (r, \theta) \)

• Conversion formulas: $r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Step-by-Step Guidance

1. Identify \( x = -10 \) and \( y = 0 \).

2. Calculate \( r = \sqrt{(-10)^2 + 0^2} \).

3. Calculate \( \theta = \arctan\left(\frac{0}{-10}\right) \).

4. Since x is negative, \( \theta \) is \( \pi \) radians.

5. Express the polar coordinates as an ordered pair.

Try solving on your own before revealing the answer!

Final Answer: (10, \pi)

The polar coordinates are (10, \pi), since the point lies on the negative x-axis.

Q11. The rectangular coordinates of a point are (5, 5√3). Find polar coordinates for the point.

Background

Topic: Rectangular to Polar Conversion

This question tests your ability to convert from rectangular coordinates to polar coordinates using trigonometric relationships.

Key Terms and Formulas:

• Rectangular coordinates: \( (x, y) \)

• Polar coordinates: \( (r, \theta) \)

• Conversion formulas: $r = \sqrt{x^2 + y^2}$ $\theta = \arctan\left(\frac{y}{x}\right)$

Step-by-Step Guidance

1. Identify \( x = 5 \), \( y = 5\sqrt{3} \).

2. Calculate \( r = \sqrt{5^2 + (5\sqrt{3})^2} \).

3. Calculate \( \theta = \arctan\left(\frac{5\sqrt{3}}{5}\right) \).

4. Simplify the expressions for r and θ.

Try solving on your own before revealing the answer!

Final Answer: (10, \frac{\pi}{3})

The polar coordinates are (10, \frac{\pi}{3}), since \( r = 10 \) and \( \theta = \frac{\pi}{3} \).