Find the point (x, y) on the unit circle that corresponds to the real number n.
n = 5pi/3
(x, y) = ?
Is it (1/2, -sqrt3/2)?
Look up the unit circle online and it will help you. Go to google and you type calculus unit circle. The answer provided was correct. But make sure it's just (1/2,-sqrt(3)/2) don't have the 2 with the square root.
Well, isn't it funny that you've brought up circles! I love circles – they're so well-rounded! Now, let's talk about this unit circle. For n = 5π/3, the point (x, y) on the unit circle would be (-1/2, √3/2). So close, but not quite there! It seems you've taken a slight detour, but don't worry, we're all on a journey here!
To find the point (x, y) on the unit circle that corresponds to the real number n, you can use the trigonometric functions. The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in a Cartesian coordinate system.
To find the point corresponding to the real number n = 5π/3, we can use the following formulas:
x = cos(n)
y = sin(n)
Plugging in the value n = 5π/3:
x = cos(5π/3)
y = sin(5π/3)
To evaluate these trigonometric functions, we need to know the values for π/3. Using the values for π/3, we have:
cos(π/3) = 1/2
sin(π/3) = √3/2
Now, let's evaluate cos(5π/3) and sin(5π/3):
cos(5π/3) = cos(3π + 2π/3) = cos(2π/3) = -1/2
sin(5π/3) = sin(3π + 2π/3) = sin(2π/3) = √3/2
Therefore, the point (x, y) on the unit circle that corresponds to the real number n = 5π/3 is:
(x, y) = (-1/2, √3/2)
So, the point is not (1/2, -√3/2), but (-1/2, √3/2).
To find the point (x, y) on the unit circle that corresponds to a given angle, you can use the trigonometric functions sine and cosine.
In this case, the given angle is n = 5π/3.
To find x, you can use the cosine function. Recall that the cosine of an angle in the unit circle is equal to the x-coordinate of the corresponding point on the circle. So, you can calculate x as follows:
x = cos(n)
Substituting the given value of n:
x = cos(5π/3)
Now, to find y, you can use the sine function. The sine of an angle in the unit circle is equal to the y-coordinate of the corresponding point on the circle. So, you can calculate y as follows:
y = sin(n)
Substituting the given value of n:
y = sin(5π/3)
To evaluate these trigonometric functions, you need to remember the values of sine and cosine for certain angles on the unit circle.
For angle 5π/3, you can break it down as follows: 5π/3 = 2π + π/3.
At π/3 radians (60 degrees), on the unit circle, the x-coordinate is 1/2 and the y-coordinate is √3/2.
Since this is a full rotation plus an additional π/3, the coordinates are simply the negation of the coordinates at π/3.
So, the correct coordinates for (x, y) are:
(x, y) = (-1/2, -√3/2)
Therefore, the given point (1/2, -√3/2) is incorrect, and the correct answer is (-1/2, -√3/2).