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Find the value of tan for the angle 45 degrees on the unit circle
To find the value of $\tan 45^{\circ}$ on the unit circle, we use the definition of $\tan$:
$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$
For $\theta = 45^{\circ}$, we know that $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ and $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$.
Substituting these values in, we get:
$$\tan 45^{\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$
So, the value of $\tan 45^{\circ}$ is 1.
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:
$$\cos(-\pi/3) = \frac{1}{2}$$
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