In calculus, it can be shown that e^(iθ) = cos θ + i sin θ. In Exercises 87–90, use this result to plot each complex number. e^(πi/4)
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
9. Polar Equations
Polar Coordinate System
Multiple Choice
The graph of the polar curve is shown above for . What is the area of the shaded region?
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Verified step by step guidance1
Identify the given polar curve: \(r = \pi\), which is a circle with radius \(\pi\) centered at the origin in polar coordinates.
Recall the formula for the area enclosed by a polar curve from \(\theta = a\) to \(\theta = b\): \(A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta\).
Substitute the given values into the formula: since \(r = \pi\) is constant, the integral becomes \(A = \frac{1}{2} \int_{0}^{3\pi} (\pi)^2 \, d\theta\).
Simplify the integral by factoring out constants: \(A = \frac{1}{2} \pi^2 \int_{0}^{3\pi} d\theta\).
Evaluate the integral \(\int_{0}^{3\pi} d\theta = 3\pi\), then multiply by the constants to express the area as \(A = \frac{1}{2} \pi^2 \times 3\pi\).
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