a. The equation of a circle with center (0,0) can be written as x^2 + y^2 = r^2, where r is the radius of the circle. Since the point (-√5/3, -2/3) lies on the circle, we can substitute the coordinates into the equation:
(-√5/3)^2 + (-2/3)^2 = r^2
5/3 + 4/9 = r^2
15/9 + 4/9 = r^2
19/9 = r^2
Therefore, the equation of the circle is x^2 + y^2 = 19/9.
b. To find the measure of θ, we first need to determine the angle formed by the terminal arm and the positive x-axis. This can be done using trigonometry. Let's denote the angle as θ.
First, calculate the lengths of the sides of the right triangle formed by the point (-√5/3, -2/3) and the origin (0,0). The length of the adjacent side is -√5/3, and the length of the opposite side is -2/3.
Using the tangent function, tan(θ) = (opposite side)/(adjacent side)
tan(θ) = (-2/3)/(-√5/3)
tan(θ) = 2/√5
θ = arctan(2/√5)
Using a calculator to find the arctan of 2/√5, we get θ ≈ 61.9 degrees.
Therefore, the measure of θ in degrees to the nearest tenth of a degree is 61.9 degrees.
The point (−√5/3, −2/3) lies at the intersection of a terminal arm and a circle with the centre (0, 0). Answer the following questions.
a. What is the equation of this circle?
b. What is the measure of 𝜃 in degrees to the nearest tenth of a degree? Explain by using a diagram.
1 answer