Math

Find the coordinates of the point on the unit circle where the angle is 5π/4 radians

To find the coordinates of the point on the unit circle where the angle is $ \frac{5\pi}{4} $ radians, we can use the definitions of sine and cosine for the unit circle.

The angle $ \frac{5\pi}{4} $ is in the third quadrant, where both sine and cosine are negative.

For the unit circle, the coordinates are given by $(\cos \theta, \sin \theta)$.

Thus, we find:

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

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

Therefore, the coordinates are:

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

Find the coordinates of the point on the unit circle at an angle of 5π/4 radians

To find the coordinates of the point on the unit circle at an angle of $\frac{5\pi}{4}$ radians, we can use the cosine and sine functions:

$$ x = \cos \left( \frac{5\pi}{4} \right) $$

$$ y = \sin \left( \frac{5\pi}{4} \right) $$

First, let’s calculate the cosine value:

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

Next, let’s calculate the sine value:

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

Therefore, the coordinates of the point are:

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

Find the sine and cosine values for the angle 5π/6 on the unit circle, and determine the corresponding point on the unit circle

First, we need to determine the reference angle for $\frac{5\pi}{6}$. The reference angle is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

In the second quadrant, sine is positive and cosine is negative.

The sine value for $\frac{\pi}{6}$ is $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

The cosine value for $\frac{\pi}{6}$ is $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

Therefore, $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$ and $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.

The corresponding point on the unit circle is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Find the values of sine and cosine for an angle on the unit circle

Given an angle $\theta$, find the values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle equation. Suppose $\theta = \frac{\pi}{4}$.

The unit circle equation is given by:

$$x^2 + y^2 = 1$$

For $\theta = \frac{\pi}{4}$, the corresponding point on the unit circle is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Therefore,

$$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.$$

What is the cosine and sine of an angle of π/3 on the unit circle?

The angle $\frac{\pi}{3}$ corresponds to 60 degrees on the unit circle.

The coordinates of this angle on the unit circle are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

Therefore, the cosine of the angle $\frac{\pi}{3}$ is $\frac{1}{2}$, and the sine is $\frac{\sqrt{3}}{2}$.

Find the coordinates on the unit circle for an angle of 5π/6

To find the coordinates of the point on the unit circle corresponding to an angle of $\frac{5\pi}{6}$, we can use the sine and cosine functions.

The angle $\frac{5\pi}{6}$ is in the second quadrant.

The reference angle is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

We know that for $\frac{\pi}{6}$:

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

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

Since $\frac{5\pi}{6}$ is in the second quadrant, the x-coordinate (cosine) will be negative and the y-coordinate (sine) will be positive.

Therefore, the coordinates are:

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

Find the sine and cosine of the angle θ = 45° using the unit circle

To find the sine and cosine of the angle $\theta = 45°$, we use the unit circle where the radius is 1.

The coordinates of the point where the terminal side of a 45° angle intersects the unit circle are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Therefore, $$\cos(45°) = \frac{\sqrt{2}}{2}$$ and $$\sin(45°) = \frac{\sqrt{2}}{2}$$.

Determine the Values of Trigonometric Functions at π/3

Consider the angle $\frac{\pi}{3}$ on the unit circle. To find the values of sin, cos, and tan at this angle, we use the known values:

The sine of $\frac{\pi}{3}$ is:

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

The cosine of $\frac{\pi}{3}$ is:

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

Using the quotient identity for tangent:

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

Determine the cosine value of -π/3 using the unit circle

First, recall that the unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. In the unit circle, the angle $\theta = -\pi/3$ is measured in the clockwise direction.

To find the cosine of $-\pi/3$, we can use the symmetry of the unit circle. The angle $-\pi/3$ is the same as $5\pi/3$ in the standard position (i.e., measured counterclockwise from the positive x-axis).

Cosine corresponds to the x-coordinate of the point on the unit circle. Thus, we need to find the x-coordinate of the point corresponding to $5\pi/3$.

At $5\pi/3$, the point on the unit circle is $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$. Therefore, the cosine of $-\pi/3$ is:

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

Find the value of tan(-π/6) using the unit circle

We start by recognizing that the angle $-\frac{\pi}{6}$ is equivalent to rotating $\frac{\pi}{6}$ radians in the clockwise direction.

On the unit circle, the point corresponding to $\frac{\pi}{6}$ radians is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. When we rotate in the clockwise direction to $-\frac{\pi}{6}$, the coordinates of the point become $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$.

The tangent of an angle is given by the ratio of the y-coordinate to the x-coordinate:

$$\tan(-\frac{\pi}{6}) = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}.$$

So, the value of $\tan(-\frac{\pi}{6})$ is $-\frac{\sqrt{3}}{3}$.