Find the cosine of $-pi/3$ using the unit circle
Answer 1
Sophia Williams
To find the cosine of $-\pi/3$ using the unit circle, follow these steps:
1. Recognize that the angle $-\pi/3$ is a negative angle, which means it is measured clockwise from the positive x-axis.
2. The angle $-\pi/3$ is equivalent to $-60^\circ$.
3. On the unit circle, an angle of $-60^\circ$ corresponds to an angle of $300^\circ$ when measured counterclockwise from the positive x-axis.
4. The coordinates of the point on the unit circle at $300^\circ$ are $(\cos 300^\circ, \sin 300^\circ)$. These coordinates are $(1/2, -\sqrt{3}/2)$.
5. Therefore, the cosine of $-\pi/3$ is the x-coordinate of this point, which is $1/2$.
So, $\cos(-\pi/3) = \frac{1}{2}$.
Answer 2
Abigail Nelson
To find the value of $cos(-pi/3)$ using the unit circle:
1. The angle $-pi/3$ is equal to $-60^circ$.
2. Since the unit circle is symmetric, we can convert the negative angle to its positive equivalent by adding a full circle (or $2pi$ radians). This gives us:
$-pi/3 + 2pi = frac{-pi + 6pi}{3} = frac{5pi}{3}$
3. The angle $frac{5pi}{3}$ is coterminal with $-pi/3$ and lies in the fourth quadrant.
4. In the fourth quadrant, the cosine value is positive. Therefore, we look at the point where the terminal side intersects the unit circle, which corresponds to the reference angle of $pi/3$.
5. The coordinates for $pi/3$ on the unit circle are $left(frac{1}{2}, frac{sqrt{3}}{2}
ight)$. Since we are in the fourth quadrant, the y-coordinate is negative, giving us $left(frac{1}{2}, -frac{sqrt{3}}{2}
ight)$.
6. Thus, the cosine of $-pi/3$ is $frac{1}{2}$.
So, $cos(-pi/3) = frac{1}{2}$.
Answer 3
William King
To determine $cos(-pi/3)$ using the unit circle:
1. Identify $-pi/3$ as a $-60^circ$ angle.
2. This angle corresponds to $300^circ$ in the unit circle.
3. The coordinates at $300^circ$ are $left(frac{1}{2}, -frac{sqrt{3}}{2}
ight)$.
4. Therefore, the cosine of $-pi/3$ is $frac{1}{2}$.
So, $cos(-pi/3) = frac{1}{2}$.
Start Using PopAi Today