The rectangular and polar coordinates of a
point are (x, y) and (r, θ), where x = 4 and
θ = 38◦
.
Determine the value of r.
To determine the value of r, we can use the given information that x = 4 and θ = 38°.
In rectangular coordinates, x represents the horizontal distance from the origin, and y represents the vertical distance from the origin. Therefore, we need to find the value of y.
Using the Pythagorean theorem, we can calculate y as follows:
y = √(r^2 - x^2)
Substituting x = 4 into the equation:
y = √(r^2 - 4^2)
Since we don't know the value of r yet, we can't solve for y directly using this equation. However, we can use the information from the polar coordinates to find r.
In polar coordinates, r represents the distance from the origin to the point. We are given that θ = 38°, which is the angle formed between the positive x-axis and the line segment connecting the point to the origin.
Using trigonometry, we can use the cosine function to relate x, r, and θ:
cos(θ) = x / r
Substituting x = 4 and θ = 38°:
cos(38°) = 4 / r
To solve for r, we can rearrange the equation:
r = 4 / cos(38°)
Now, we can use a calculator to find the value of cos(38°) and calculate r.