Homework

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

First, let’s understand the position of $\frac{4\pi }{3}$ on the unit circle. The angle $\frac{4\pi }{3}$ radians is in the third quadrant.

In the third quadrant, the reference angle is $\frac{\pi }{3}$. The tangent is positive in the third quadrant.

We know that $\tan \left(\frac{\pi }{3}\right)=\sqrt{3}$. Therefore:

$$\tan \left(\frac{4\pi }{3}\right)=\tan (\pi +\frac{\pi }{3})=\tan \left(\frac{\pi }{3}\right)=\sqrt{3}$$

Find the sine and cosine of the angle 30 degrees using the unit circle

First, we need to convert ${30}^{\circ }$ to radians:

$${30}^{\circ }=30\times \frac{\pi }{180}=\frac{\pi }{6}$$

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

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

Using known values, we have:

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

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

Therefore, the sine of ${30}^{\circ }$ is $\frac{1}{2}$ and the cosine of ${30}^{\circ }$ is $\frac{\sqrt{3}}{2}$.

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

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

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

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,

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

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

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.