Table of contents

0. Fundamental Concepts of Algebra3h 32m
1. Equations and Inequalities3h 27m
2. Graphs1h 43m
3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
19. Conic Sections2h 36m
20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

15. Polar Equations

Graphing Other Common Polar Equations

Precalculus

15. Polar Equations

Graphing Other Common Polar Equations

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Multiple Choice

Identify whether the given equation is that of a cardioid, limaçon, rose, or lemniscate.
$r=1-\sin\theta$

Cardioid

Limacon

Rose

Lemniscate

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Verified step by step guidance

Step 1: Understand the general forms of polar equations for different types of curves. A cardioid typically has the form r = a ± a sin(θ) or r = a ± a cos(θ). A limaçon has the form r = a ± b sin(θ) or r = a ± b cos(θ), where a ≠ b. A rose curve has the form r = a sin(nθ) or r = a cos(nθ), where n is an integer. A lemniscate has the form r² = a² sin(2θ) or r² = a² cos(2θ).

Step 2: Compare the given equation r = 1 - sin(θ) with the general forms. Notice that it resembles the form r = a ± a sin(θ) with a = 1.

Step 3: Recognize that when a = b in the limaçon form, the curve is specifically a cardioid. In this case, since the equation is r = 1 - sin(θ), it matches the cardioid form r = a - a sin(θ) with a = 1.

Step 4: Confirm that the equation does not fit the forms of rose curves or lemniscates, as it does not involve r² or a multiple angle like nθ.

Step 5: Conclude that the given equation r = 1 - sin(θ) is indeed a cardioid based on its form and comparison with the general equations for polar curves.