PopAi provides you with resources such as math solver, math tools, etc.
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:
$$ \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} $$
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:
$$\left( \cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right) \right)$$
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 $\sin(\frac{5π}{6})$, 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,
$$ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} $$
Given a unit circle centered at the origin, find the coordinates of a point P on the circle such that the angle θ between the line segment OP and the positive x-axis is an irrational multiple of π
To solve this problem, we need to find the coordinates of point $P$ on the unit circle given that the angle $\theta$ is an irrational multiple of $\pi$. Let’s denote this angle as $\theta = k\pi$ where $k$ is an irrational number.
Using the parametric equations of the unit circle, we have:
$$x = \cos(\theta)$$
$$y = \sin(\theta)$$
Since $\theta$ is an irrational multiple of $\pi$, we can choose $\theta = \sqrt{2}\pi$. Then, the coordinates $(x,y)$ of point $P$ are:
$$x = \cos(\sqrt{2}\pi)$$
$$y = \sin(\sqrt{2}\pi)$$
Thus, the coordinates of $P$ are:
$$P = (\cos(\sqrt{2}\pi), \sin(\sqrt{2}\pi))$$
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.
Start Using PopAi Today
Suggested Content
More >
Answer 1 I've flown hundreds of times, and nothing prepared me for that landing. There was a sudden jerk, and the plane tilted violently. People were screaming, kids were crying. I honestly thought we were going to crash. Afterward, Delta was pretty...
Answer 1 First, recall that in the unit circle, an angle of 45 degrees corresponds to $\frac{\pi}{4}$ radians. From trigonometric identities: $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ Therefore, the values...
Answer 1 To find the values of $\sin$, $\cos$, and $\tan$ at $45^\circ$ on the unit circle, we start by noting that $45^\circ$ is the same as $\frac{\pi}{4}$ radians. The coordinates of the point on the unit circle at $\frac{\pi}{4}$ radians are...
Answer 1 Given a point on the unit circle in the complex plane, represented by the complex number $z = e^{i\theta}$, determine the value of $\cos(\theta)$. Since $z = e^{i\theta}$, we know that: $z = \cos(\theta) + i\sin(\theta)$ Thus, the real part...
Answer 1 Given that $\theta$ is in the fourth quadrant and the point on the unit circle is $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$, we can find the exact values of $\sin\theta$, $\cos\theta$, and $\tan\theta$. First, we recognize that $(\cos\theta,...
Answer 1 We know that at $45^\circ$, the coordinates on the unit circle are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.Therefore,$\sin 45^\circ = \frac{\sqrt{2}}{2}$$\cos 45^\circ = \frac{\sqrt{2}}{2}$To find $\tan 45^\circ$, we use the...