Math

Find the cosecant of an angle in the unit circle

To find the $\csc(\theta)$ of an angle $\theta$ in the unit circle, first determine the sine of the angle. The cosecant is the reciprocal of the sine.

For example, consider $\theta = \frac{\pi}{6}$.

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

Therefore,

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

Given the point P(x, y) on the unit circle where x = -1/sqrt(2) and y = -1/sqrt(2), find the angle θ in radians

First, recognize that the coordinates given are $x = -\frac{1}{\sqrt{2}}$ and $y = -\frac{1}{\sqrt{2}}$. These values correspond to specific angles on the unit circle. We need to determine where both sine and cosine are negative and equal in magnitude.

Looking at the unit circle, we see that $\theta = \frac{5\pi}{4}$ radians has these properties.

Therefore, the angle $\theta$ in radians is \(\theta = \frac{5\pi}{4}\).

Find the cosine of the angle at the unit circle

To find the cosine of $$\theta = \frac{\pi}{3}$$ on the unit circle, we look at the x-coordinate of the point where the terminal side of the angle intersects the unit circle. For $$\theta = \frac{\pi}{3}$$, the point is $$ (\frac{1}{2}, \frac{\sqrt{3}}{2}) $$.

Therefore,

$$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$

Find the coordinates of a point on the unit circle given an angle in radians

Given an angle $\theta = \frac{\pi}{3}$ radians, find the coordinates of the point on the unit circle.

The unit circle is defined by the equation $x^2 + y^2 = 1$. For an angle $\theta$, the coordinates $(x, y)$ can be found using:

$$x = \cos(\theta)$$

$$y = \sin(\theta)$$

Substituting $\theta = \frac{\pi}{3}$:

$$x = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$

$$y = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$

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

Find all angles θ in the interval [0, 2π) for which cotangent is equal to 1/√3 on the unit circle

To solve for $\theta$ in $\cot(\theta) = \frac{1}{\sqrt{3}}$, we start by recalling the definition of cotangent:

$$\cot(\theta) = \frac{1}{\tan(\theta)}$$

Given $\cot(\theta) = \frac{1}{\sqrt{3}}$, we have:

$$\frac{1}{\tan(\theta)} = \frac{1}{\sqrt{3}}$$

By taking reciprocals, we find:

$$\tan(\theta) = \sqrt{3}$$

The angles that satisfy $\tan(\theta) = \sqrt{3}$ on the unit circle are:

$$\theta = \frac{\pi}{3} + k\pi$$

where $k$ is any integer. Considering the interval $[0, 2\pi)$, we get:

$$\theta = \frac{\pi}{3}, \frac{4\pi}{3}$$

So, the angles are:

$$\boxed{\frac{\pi}{3}, \frac{4\pi}{3}}$$

Find the value of tan(θ) on the unit circle where θ=150°

To find $\tan(150^\circ)$, we first note that $150^\circ$ can be written as $180^\circ – 30^\circ$.

The reference angle here is $30^\circ$.

Since $\tan\theta$ is negative in the second quadrant:

$$\tan(150^\circ) = -\tan(30^\circ)$$

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}\ or \ \frac{\sqrt{3}}{3}$.

Therefore,

$$\tan(150^\circ) = -\frac{1}{\sqrt{3}}\ or \ -\frac{\sqrt{3}}{3}$$

Find the coordinates on the unit circle for an angle of \(\frac{\pi}{3}\) radians

To find the coordinates of an angle of $\frac{\pi}{3}$ radians on the unit circle, we use the cosine and sine values of the angle.

The cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$.

The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$.

Therefore, the coordinates are:

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

Find the values of cos(θ) and sin(θ) for θ = 5π/4

To find the values of $\cos(\theta)$ and $\sin(\theta)$ for $\theta = \frac{5\pi}{4}$, we start by locating the angle on the unit circle. The angle $\frac{5\pi}{4}$ is in the third quadrant.

In the third quadrant, both sine and cosine values are negative. The reference angle for $\frac{5\pi}{4}$ is $\pi/4$, for which the cosine and sine values are both $\frac{\sqrt{2}}{2}$.

Therefore:

$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Find the cosine of an angle using the unit circle in the complex plane

Given an angle \( \theta \) in the complex plane, the unit circle can be used to find the cosine of the angle. The cosine of the angle \( \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Let’s consider \( \theta = \frac{\pi}{4} \), find \( \cos(\theta) \).

On the unit circle, the coordinates of the point at angle \( \frac{\pi}{4} \) are \( \left( \cos \left( \frac{\pi}{4} \right), \sin \left( \frac{\pi}{4} \right) \right) \).

Since \( \frac{\pi}{4} \) is a 45-degree angle, the coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Thus, the cosine of \( \frac{\pi}{4} \) is:

$$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

Convert the point on the unit circle given in Cartesian coordinates (sqrt(3)/2, 1/2) to its corresponding angle in degrees and radians, and verify the solution by converting the angle back to Cartesian coordinates

We are given the point $(\sqrt{3}/2, 1/2)$ on the unit circle. To find the corresponding angle, we use the following trigonometric relationships:

$$x = \cos(\theta)$$

$$y = \sin(\theta)$$

Thus, we have:

$$\cos(\theta) = \sqrt{3}/2$$

$$\sin(\theta) = 1/2$$

For the angle $\theta$ that satisfies these equations, we recognize that these are standard values. The angle $\theta$ is $30^{\circ}$ or $\pi/6$ radians.

To verify, we will convert $30^{\circ}$ back to Cartesian coordinates:

$$\cos(30^{\circ}) = \sqrt{3}/2, \sin(30^{\circ}) = 1/2$$

Thus, the point $(\cos(30^{\circ}), \sin(30^{\circ})) = (\sqrt{3}/2, 1/2)$ matches the given point. Therefore, $\theta = 30^{\circ}$ or $\pi/6$ radians.