In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)

Verified step by step guidance

1. Identify the polar coordinates given: $(r,\theta )=(7.4,2.5)$.
2. Recall the formulas to convert polar coordinates to rectangular coordinates: $x=r\cdot \cos (\theta )$ and $y=r\cdot \sin (\theta )$.
3. Substitute the given values into the formulas: $x=7.4\cdot \cos (2.5)$ and $y=7.4\cdot \sin (2.5)$.
4. Calculate $\cos (2.5)$ and $\sin (2.5)$ using a calculator or trigonometric tables.
5. Multiply the results by 7.4 to find the rectangular coordinates $(x,y)$.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a two-dimensional space using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The first value indicates the radius (distance from the origin), while the second value indicates the angle in radians or degrees. Understanding this system is crucial for converting to rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express a point in a two-dimensional space using two values: the x-coordinate (horizontal position) and the y-coordinate (vertical position). The conversion from polar to rectangular coordinates involves using the formulas x = r * cos(θ) and y = r * sin(θ), where r is the radius and θ is the angle.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in relating angles to the ratios of sides in right triangles. In the context of polar to rectangular conversion, these functions are used to determine the x and y coordinates based on the angle and radius. Mastery of these functions is essential for accurately performing the conversion.