find the exact coordinates of the point on a circle of radius 5 at an angle of 5pi/3. thanks for any help! (:
I will assume you want your circle to centre (0,0), then its equation is
x^2 + y^2 = 25
2x + 2y dy/dx = 0
dy/dx = -x/y <---- the slope of the tangent
you want the tangent angle to be 5π/3 or 300°
tan 5π/3 = -√3
then -x/y = -√3
x = √3 y
subbing back into my circle equation
3y^2 + y^2 = 25
4y^2 = 25
y = ± 5/2
if y = 5/2, then y = 5√3/2 -------> (5√3/2 , 5/2)
if y = -5/2, then y = -5√3/2 -----> ( -5√3/2 , -5/2)
https://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+%3D+25+%2C+y+-+5%2F2+%3D+-%E2%88%9A3%28x+-+5%E2%88%9A3%2F2%29%2C+y+%2B+5%2F2+%3D+-%E2%88%9A3%28x+%2B+5%E2%88%9A3%2F2%29
if your circle has center at (0,0) then you have
x = r cosθ
y = r sinθ
Now just plug in your numbers
Well, buckle up because we're going on a math-filled circus ride! To find the coordinates of a point on a circle, we can use some circus tricks called trigonometry.
First, let's break down the question. We have a circle with a radius of 5. The angle given is 5π/3. To find the coordinates, we'll use the angle to determine the position on the circle.
Now, to get the coordinates, we use two main circus acts called cosine (cos) and sine (sin). These two frenemies will help us, so hang on tight!
The x-coordinate of the point is given by the formula: x = r * cos(θ)
where r is the radius and θ is the angle.
Substituting the values, we get:
x = 5 * cos(5π/3)
Now, let's grab our calculator and evaluate this:
x ≈ 5 * (-0.5)
x ≈ -2.5
So, the x-coordinate is approximately -2.5.
Similarly, the y-coordinate can be found using the formula: y = r * sin(θ).
Substituting the values, we get:
y = 5 * sin(5π/3)
Calculating this:
y ≈ 5 * (√3/2)
y ≈ 5√3/2
So, the y-coordinate is approximately 5√3/2.
Putting it all together, the exact coordinates of the point on the circle of radius 5 at an angle of 5π/3 are approximately (-2.5, 5√3/2).
And that, my friend, is the grand finale of our math circus performance! Hope you enjoyed the show!
To find the coordinates of a point on a circle, you can use the trigonometric functions sine and cosine.
The general formula for finding the coordinates (x, y) of a point on a circle with radius r and angle θ is:
x = r * cos(θ)
y = r * sin(θ)
In your case, the radius of the circle is given as 5 and the angle is given as 5π/3.
Let's substitute the values into the formulas:
x = 5 * cos(5π/3)
y = 5 * sin(5π/3)
Now, let's calculate the values using a scientific calculator or any calculator that has the trigonometric functions.
First, let's calculate cos(5π/3):
cos(5π/3) = -½
Now, let's calculate sin(5π/3):
sin(5π/3) = -√3/2
Finally, let's substitute the values back into the formulas:
x = 5 * (-½) = -2.5
y = 5 * (-√3/2) = -2.5√3
So, the exact coordinates of the point on a circle of radius 5 at an angle of 5π/3 are (-2.5, -2.5√3).
Note: In some cases, you might need to simplify or approximate the values of trigonometric functions depending on the requirements of the problem.