SUMMARY
The polar curves r=5sin(θ) and r=5cos(θ) intersect at the points where 5sin(θ) = 5cos(θ). By dividing both sides by 5cos(θ), the equation simplifies to tan(θ) = 1. This results in θ = π/4 and θ = 5π/4 within the specified range of 0 ≤ θ < 2π. The corresponding polar coordinates for these intersections are (5√2/2, π/4) and (5√2/2, 5π/4).
PREREQUISITES
- Understanding of polar coordinates and their representation.
- Knowledge of trigonometric functions, specifically sine and cosine.
- Familiarity with the concept of tangent and its properties.
- Ability to solve equations involving trigonometric identities.
NEXT STEPS
- Explore the properties of polar coordinates and their applications in geometry.
- Study the unit circle and its relationship with sine, cosine, and tangent functions.
- Learn how to graph polar equations and identify their intersections.
- Investigate the implications of polar curves in calculus, particularly in area calculations.
USEFUL FOR
Mathematics students, educators, and anyone interested in polar coordinates and trigonometric functions.