به انگلیسی

Finding points by distance and angle

Mathematics · Geometry · ages 17-18

Name ______________________ Date ____________

The polar point is (-9, pi/2), with the angle in radians. What is its Cartesian y-coordinate?
Answer: ______________
Convert the Cartesian point (-4, -4) to polar form with r > 0 and 0 <= theta < 360 degrees. What is theta?
- 45 degrees
- 135 degrees
- 225 degrees
- 315 degrees
A point has polar coordinates (2, 90 degrees). What are its Cartesian coordinates?
- (0, 2)
- (2, 0)
- (0, -2)
For r = 9, let theta run from 0 to 2pi radians. Calculate positions at 0, pi/2, pi, 3pi/2 and 2pi. Which ordered Cartesian trace and resulting curve are correct?
- (9, 0), (0, -9), (-9, 0), (0, 9), (9, 0): a circle traced clockwise.
- (9, 0), (9, 9), (9, 18), (9, 27), (9, 36): a vertical straight line.
- (9, 0), (0, 9), (-9, 0), (0, -9), (9, 0): circle, anticlockwise.
Compare an origin-centred circle of radius 6 and the vertical line x = 8. Which choice gives the shorter coordinate form for each curve and a reason?
- Cartesian for both: x² + y² = 36 fixes the circle's radius directly, and x = 8 fixes the line's x-coordinate.
- Polar for circle: r = 6 fixes radius. Cartesian for line: x = 8 fixes x without trigonometry.
- Polar for both: r = 6 fixes the circle's radius, and r = 8 fixes x for every point on the vertical line.
Which Cartesian equation describes the polar equation r sin theta = -3?
- y = -3
- y = 3
- x = -3
- x = 3
On the spiral r = 3 theta, theta >= 0 is in radians. How much does r increase during one full turn?
- 3pi units
- 9pi units
- 6pi units
- 12pi units
For r = 4 theta/pi, 0 <= theta <= 2pi in radians, use theta = 0, pi/2, pi, 3pi/2 and 2pi. Which ordered Cartesian trace and resulting curve are correct?
- (0, 0), (0, -2), (-4, 0), (0, 6), (8, 0): an outward spiral traced clockwise.
- (0, 0), (0, 2), (4, 0), (0, -6), (-8, 0): an outward spiral with reversed horizontal positions.
- (0, 0), (0, 2), (-4, 0), (0, -6), (8, 0): an outward spiral traced anticlockwise.
The Cartesian point is (3, 4). What is its nonnegative polar radius r?
Answer: ______________
Using r >= 0, what polar equation describes the same circle as x² + y² = 25?
- r equals 25
- r equals 5
- theta equals 5

Answer key

For grown-ups. Fold this page away before handing over the rest.

Finding points by distance and angle W1-mt_R5QMUJvA1v-s1

-9 · y = r sin theta = -9 sin(pi/2). Since sin(pi/2) = 1, y = -9. Negative r puts this point below the origin.
225 degrees · Both coordinates are negative, placing the point in quadrant III. Equal absolute coordinates give a 45-degree reference angle, so theta = 180 + 45 = 225 degrees.
(0, 2) · x equals 2 cos 90, which is 0, and y equals 2 sin 90, which is 2.
(9, 0), (0, 9), (-9, 0), (0, -9), (9, 0): circle, anticlockwise. · The radius stays 9. The angles point right, up, left, down and right, giving the listed axis points in anticlockwise order. Joining the curve through them completes a circle centred at the origin.
Polar for circle: r = 6 fixes radius. Cartesian for line: x = 8 fixes x without trigonometry. · The circle's polar form r = 6 states its fixed radius more briefly than x² + y² = 36. The line's Cartesian form x = 8 states its fixed horizontal coordinate more briefly than r cos theta = 8.
y = -3 · Use y = r sin theta. Replacing r sin theta with y gives y = -3, a horizontal line below the x-axis.
6pi units · A full turn adds 2pi radians to theta. Since r = 3 theta, the radial increase is 3 × 2pi = 6pi units.
(0, 0), (0, 2), (-4, 0), (0, -6), (8, 0): an outward spiral traced anticlockwise. · Substitution gives radii 0, 2, 4, 6 and 8. Place them on the right, up, left, down and right rays. Joining in angle order winds anticlockwise outward, ending 8 units right of the origin.
5 · r squared equals 9 plus 16, which is 25, so r equals 5.
r equals 5 · Substitute r² for x² + y² to get r² = 25. Since r >= 0, choose r = 5. Let theta make a full turn to trace the circle.