Unit Circle

Find the coordinates of the point on the unit circle corresponding to 210 degrees

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Points on the unit circle can be represented as (cos θ, sin θ), where θ is the angle in degrees.

To find the coordinates of the point on the unit circle corresponding to $210^{\circ}$:

1. Convert the angle to radians: $210^{\circ} = \frac{210 \pi}{180} = \frac{7 \pi}{6}$.

2. Use the unit circle values:

$$\cos \left(\frac{7 \pi}{6} \right) = -\frac{\sqrt{3}}{2}$$

$$\sin \left(\frac{7 \pi}{6} \right) = -\frac{1}{2}$$

Thus, the coordinates are $\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)$.

Calculate cos(-π / 3) using the unit circle

Using the unit circle, we know that the angle $-\pi / 3$ corresponds to moving $\pi / 3$ radians clockwise from the positive x-axis.

Since $\cos$ is the x-coordinate of the point on the unit circle, and moving $\pi / 3$ radians clockwise is the same as moving $2\pi – \pi / 3 = 5\pi / 3$ radians counterclockwise from the positive x-axis, we need to find the cosine of $5\pi / 3$.

On the unit circle, the coordinates of the angle $5\pi / 3$ are $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$. Therefore, the value of $\cos(5\pi / 3)$, corresponding to $\cos(-\pi / 3)$, is $\frac{1}{2}$.

$$\cos(-\pi / 3) = \frac{1}{2}$$

On the unit circle, find the values of angles in radians for which the secant function is undefined

The secant function $\sec(\theta)$ is undefined when the cosine function $\cos(\theta)$ is zero. On the unit circle, $\cos(\theta)$ is zero at the points where the x-coordinate is zero, which happens at $\theta = \frac{\pi}{2} + k\pi$ for any integer $k$.

Therefore, the angles in radians for which the secant function is undefined are:

$$\theta = \frac{\pi}{2} + k\pi$$

where $k \in \mathbb{Z}$ (any integer).

Find the value of \( \cot(\theta) \) when \( \theta = \frac{\pi}{4} \) on the unit circle

Given:

\( \theta = \frac{\pi}{4} \)

On the unit circle, the coordinates for \( \theta = \frac{\pi}{4} \) are:

\( (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \)

\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

Substituting the values:

\( \cot(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \)

Therefore, \( \cot(\frac{\pi}{4}) = 1 \).

Find the tangent of 45 degrees using the unit circle

To find the tangent of 45 degrees using the unit circle, we first locate the point corresponding to 45 degrees on the circle. The coordinates of this point are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Recall that the tangent function is defined as the ratio of the y-coordinate to the x-coordinate:

$$\tan(45^\circ) = \frac{\text{y-coordinate}}{\text{x-coordinate}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

Therefore, $$\tan(45^\circ) = 1$$.

Find the coordinates of points on the unit circle corresponding to a given angle in a flipped configuration

Given a unit circle, we need to find the coordinates of points corresponding to the angle $\theta = \frac{5\pi}{4}$, but with the configuration flipped over the x-axis.

In the standard unit circle, the point corresponding to $\theta = \frac{5\pi}{4}$ is:

$$\left(\cos \frac{5\pi}{4}, \sin \frac{5\pi}{4}\right)$$

Using the trigonometric values:

$$\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$$

Since the configuration is flipped over the x-axis, we change the sign of the y-coordinate:

$$\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$

Thus, the coordinates are:

$$\boxed{\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)}$$

Find the sine and cosine of the angle \(\theta\) when \(\theta = \frac{\pi}{6}\)

To find the sine and cosine of the angle $\theta$ when $\theta = \frac{\pi}{6}$, we can use the unit circle.

The angle $\frac{\pi}{6}$ radians corresponds to 30 degrees.

On the unit circle, the coordinates of the point at angle $\frac{\pi}{6}$ are:

$$\left(\cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right)\right)$$

From trigonometric values:

$$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$

$$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$$

Therefore, the sine and cosine of the angle $\theta$ when $\theta = \frac{\pi}{6}$ are given by:

$$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$$

$$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$

Given a unit circle centered at the origin, but flipped in a non-standard way such that the positive x-axis points downwards and the positive y-axis points to the left, find the coordinates of the point corresponding to an angle of 5π/6 radians

To solve this problem, we first need to understand the transformation of the coordinate system.

In the standard unit circle, an angle of $\frac{5\pi}{6}$ radians would correspond to the point $(-\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6}))$.

Therefore, in the standard unit circle, the coordinates would be:

$$(-\frac{\sqrt{3}}{2}, \frac{1}{2})$$

Now, since the unit circle is flipped such that the positive x-axis points downwards and the positive y-axis points to the left, we need to adjust these coordinates accordingly:

1. The x-coordinate will become the negative of the original y-coordinate.

2. The y-coordinate will become the negative of the original x-coordinate.

Thus, the transformed coordinates are:

$$( -\frac{1}{2}, -\left(-\frac{\sqrt{3}}{2}\right) )$$

which simplifies to:

$$\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$$

Find the value of cos(θ) on the unit circle for a given θ and determine the exact coordinates of the corresponding point

Let’s consider the angle $ \theta = \frac{7\pi}{6}$.

First, we determine the reference angle. Since $\frac{7\pi}{6}$ is in the third quadrant, we find the reference angle by subtracting $\pi$:

$$ \theta_{ref} = \frac{7\pi}{6} – \pi = \frac{7\pi}{6} – \frac{6\pi}{6} = \frac{\pi}{6} $$

The cosine of the reference angle $\frac{\pi}{6}$ is $\frac{\sqrt{3}}{2}$, but since we are in the third quadrant, the cosine value is negative:

$$ \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$

The exact coordinates of the point on the unit circle corresponding to $\theta = \frac{7\pi}{6}$ are:

$$ \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) $$

Find the coordinates of the point on the unit circle where the angle with the positive x-axis is 5π/6 radians

Given the angle $\theta = \frac{5\pi}{6}$ radians, we need to find the coordinates of the point on the unit circle.

On the unit circle, the coordinates of a point at an angle $\theta$ are $$(\cos(\theta), \sin(\theta))$$.

Therefore,

$$x = \cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$$

$$y = \sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$$

So, the coordinates are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.