25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(2, 7π/4)

Verified step by step guidance

1. Recall the formulas to convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\): \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)
2. Identify the given polar coordinates: \(r = 2\) \(\theta = \frac{7\pi}{4}\)
3. Substitute the values into the conversion formulas: \(x = 2 \cos\left(\frac{7\pi}{4}\right)\) \(y = 2 \sin\left(\frac{7\pi}{4}\right)\)
4. Evaluate the trigonometric functions \(\cos\left(\frac{7\pi}{4}\right)\) and \(\sin\left(\frac{7\pi}{4}\right)\) using the unit circle or known values for these angles.
5. Multiply the results by \(r = 2\) to find the Cartesian coordinates \((x, y)\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from the origin (radius r) and an angle θ measured from the positive x-axis. They are expressed as (r, θ), where r ≥ 0 and θ is typically in radians.

Conversion Formulas from Polar to Cartesian Coordinates

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). These relate the radius and angle to the horizontal and vertical distances from the origin.

Trigonometric Values for Common Angles

Understanding the sine and cosine values of common angles, such as 7π/4, is essential for conversion. For example, cos(7π/4) = √2/2 and sin(7π/4) = -√2/2, which help compute exact Cartesian coordinates.