91Ó°ÊÓ

When working with trigonometric equations, knowing your identities is crucial. The double-angle identity for sine is a valuable tool. This identity states that

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)

This means that if you have an equation involving \( \sin 2\theta \), you can replace it with \( 2 \sin \theta \cos \theta \). This is helpful in simplifying expressions and solving equations. In our exercise, this substitution is a key step.
Understanding and memorizing these identities aids you in solving problems more efficiently. They allow you to transform trigonometric expressions into more manageable forms.

Rectangular coordinates, also known as Cartesian coordinates, are a method for representing points on a plane using two numbers or variables. Normally, these points are denoted as

• \( (x, y) \)

Here, \( x \) is the horizontal coordinate, and \( y \) is the vertical coordinate.
This system helps you easily locate points with a specific distance to each axis. In trigonometric problems, converting polar equations into rectangular forms is often necessary.
The conversion relies on the relationships

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)
• \( r^2 = x^2 + y^2 \)

Understanding these conversions is crucial to shift seamlessly between the coordinate systems.

Polar coordinates describe a point on a plane uniquely, using a distance from a reference point and an angle from a reference direction. A point in polar coordinates is written as

• \( (r, \theta) \)

Here, \( r \) is the distance from the pole (a central point, usually the origin), and \( \theta \) is the angle measured counterclockwise from the positive x-axis.
Polar coordinates are particularly useful in scenarios involving circular or spiral patterns, where angles and distances offer more intuitive descriptions than their rectangular counterparts.
When converting polar equations to rectangular form, it's key to utilize the relationships between the two systems. This includes substituting \( r\cos\theta \) for \( x \) and \( r\sin\theta \) for \( y \). These conversions assist in expressing polar equations within the familiar x-y plane.