Unit Circle

Calculate the exact value of sin(5π/6) and verify it on the unit circle

To find the exact value of $\sin(\frac{5π}{6})$, we first determine the corresponding angle in degrees. Converting radians to degrees:

$$\frac{5π}{6} \times \frac{180^\circ}{π} = 150^\circ$$

Now, considering the unit circle, the angle $150^\circ$ lies in the second quadrant where the sine value is positive. The reference angle for $150^\circ$ is:

$$180^\circ – 150^\circ = 30^\circ$$

We know from the unit circle that:

$$\sin(30^\circ) = \frac{1}{2}$$

Therefore,

$$\sin(150^\circ) = \sin(\frac{5π}{6}) = \frac{1}{2}$$

Find the sine of π/6 on the unit circle

To find the sine of $\frac{\pi}{6}$ on the unit circle, we need to know the coordinates of the point on the unit circle corresponding to this angle. The unit circle has a radius of 1, and an angle of $\frac{\pi}{6}$ corresponds to 30 degrees in the first quadrant.

The coordinates of this point on the unit circle are $\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$. The y-coordinate of this point gives us the sine value.

Therefore,

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

Find the sine and cosine values for an angle of 45 degrees on the unit circle

Using the unit circle, we can determine the sine and cosine values of $45^\circ$.

$45^\circ$ (or $\frac{\pi}{4}$ radians) is a commonly known angle.

The coordinates of the point on the unit circle corresponding to $45^\circ$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Therefore, the sine value is $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ and the cosine value is $\cos(45^\circ) = \frac{\sqrt{2}}{2}$.

Find the Cartesian coordinates of a point on the unit circle at a given angle

First, recall that for any point on the unit circle, its coordinates can be represented as \((x, y) = (\cos \theta, \sin \theta)\).

Given an angle \(\theta = \frac{3\pi}{4}\), we can calculate the coordinates as follows:

$$ x = \cos \left( \frac{3\pi}{4} \right) = \cos \left(135^\circ \right) = -\frac{\sqrt{2}}{2} $$

$$ y = \sin \left( \frac{3\pi}{4} \right) = \sin \left(135^\circ \right) = \frac{\sqrt{2}}{2} $$

Therefore, the Cartesian coordinates of the point are \( \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Convert 135 degrees to radians and find the sine and cosine values

To convert 135 degrees to radians, we use the formula:

$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$

So,

$$135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}$$

Next, we find the sine and cosine values for $\frac{3\pi}{4}$:

$$\sin\left(\frac{3\pi}{4}\right) = \sin\left(\pi – \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$

$$\cos\left(\frac{3\pi}{4}\right) = \cos\left(\pi – \frac{\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Therefore,

$$\text{Radians} = \frac{3\pi}{4}, \sin = \frac{\sqrt{2}}{2}, \cos = -\frac{\sqrt{2}}{2}$$

Determine the value of tan for given angles on the unit circle

$$\text{Given an angle of } \theta = \frac{5\pi}{4}$$

We know that:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

On the unit circle, for \(\theta = \frac{5\pi}{4}, \sin \theta = -\frac{\sqrt{2}}{2} \) and \(\cos \theta = -\frac{\sqrt{2}}{2}\)

Therefore,

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

Simplifying, we get:

$$\tan \left(\frac{5\pi}{4}\right) = 1$$

Find the value of tan(θ) on the unit circle when θ = π/4

First, we need to determine the coordinates of the point on the unit circle corresponding to $\theta = \frac{\pi}{4}$.

On the unit circle, the coordinates for the angle $\frac{\pi}{4}$ are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

The tangent of an angle $\theta$ is given by the ratio of the y-coordinate to the x-coordinate of the corresponding point on the unit circle:

$$\tan\left(\frac{\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

So, the value of $\tan(\theta)$ for $\theta = \frac{\pi}{4}$ is 1.

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

To find $\cos(-\pi/3)$, we first need to understand its position on the unit circle. The angle $-\pi/3$ is equivalent to rotating $\pi/3$ radians in the clockwise direction.

On the unit circle, $\pi/3$ radians is located in the first quadrant, and its coordinates are $(1/2, \sqrt{3}/2)$. Since we are rotating clockwise, we need to reflect over the x-axis, thus the coordinates become $(1/2, -\sqrt{3}/2)$.

Therefore, $\cos(-\pi/3) = \cos(\pi/3) = 1/2$.

So, $$\cos(-\pi/3) = 1/2$$

Calculate the value of tan(4π/3) using the unit circle

To calculate $\tan\left(\frac{4\pi}{3}\right)$, we start by locating the angle $\frac{4\pi}{3}$ on the unit circle.

The angle $\frac{4\pi}{3}$ radians is equivalent to $240^\circ$.

This angle lies in the third quadrant where both sine and cosine are negative.

Using the unit circle, we find the coordinates of the point at $240^\circ$: $(\cos 240^\circ, \sin 240^\circ) = \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$.

The tangent of an angle is given by the ratio of the sine to the cosine:

$$\tan\left(\frac{4\pi}{3}\right) = \frac{\sin 240^\circ}{\cos 240^\circ} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}.$$

Determine the Quadrant of a Trigonometric Function

Consider the angle $\theta = 210^\circ$. To determine the quadrant where this angle lies, we will use the unit circle.

The angle $210^\circ$ is measured from the positive x-axis in the counter-clockwise direction. Since $210^\circ > 180^\circ$ and $210^\circ < 270^\circ$, it lies in the third quadrant.

In the third quadrant, both sine and cosine values are negative. Therefore, the angle $\theta = 210^\circ$ lies in Quadrant III.