Question

The rectangular coordinates of a point are (5.00, y) and the polar coordinates of

this point are (r, 67.4°).
a. What is the value of the polar coordinate r in this case?
b. In what quadrant are both the sine and tangent negative?

Answer 1

Step-by-step explanation:

Polar coordinates formula

Summary. To convert from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) : x = r × cos( θ ) y = r × sin( θ )

Answer 2

To find the polar coordinate 'r', use the formula x = r × cos(θ) with x = 5.00 and θ = 67.4°. Sine and tangent are both negative in the fourth quadrant of the Cartesian coordinate system.

Step-by-step explanation:

Finding Polar Coordinate 'r' and Quadrants for Sine and Tangent Functions

The student has provided the rectangular coordinates of a point (5.00, y) and the corresponding polar coordinates (r, 67.4°).

a. Value of the polar coordinate r: To find the value of r, we use the relationship between polar coordinates and rectangular coordinates.

The formula for converting the x-coordinate in rectangular form to polar form is x = r × cos(θ). We know that the x-coordinate is 5.00 and the angle θ is 67.4°. Thus, r can be calculated as:

r = x / cos(θ)
r = 5.00 / cos(67.4°)

By calculating this value, we can find the polar coordinate r. The cosine of 67.4° can be found using a calculator with trigonometric functions. Once the cosine value is known, divide 5.00 by this value to obtain r.

b. Quadrant with negative sine and tangent: The sine and tangent functions are both negative in the fourth quadrant of the Cartesian coordinate system.

This is because sine correlates with the y-coordinate, and tangent is the ratio of y to x. Since x is positive and y is negative in the fourth quadrant, both sine and tangent will yield negative values.