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Find the value of cos(π/4) on the unit circle
On the unit circle, the angle $ \frac{\pi}{4} $ corresponds to 45 degrees. The coordinates of this point are ( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} ). Therefore,
$$ \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
Determine the coordinates of a point on the unit circle given the angle θ = π/4
To find the coordinates of a point on the unit circle given the angle $\theta = \frac{\pi}{4}$, we use the definitions of sine and cosine:
$$ x = \cos(\theta) $$
$$ y = \sin(\theta) $$
For $\theta = \frac{\pi}{4}$:
$$ x = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
$$ y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
Thus, the coordinates of the point are:
$$ \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$
Find the values of sin, cos, and tan for an angle of π/4 on the unit circle
To find the values of $ \sin, \cos, $ and $ \tan $ for an angle of $ \frac{\pi}{4} $ on the unit circle, we start with:
$$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
$$ \tan\left(\frac{\pi}{4}\right) = 1 $$
Calculate the coordinates of a point on the unit circle at an angle of 5π/6
To find the coordinates of a point on the unit circle at an angle of $ \frac{5\pi}{6} $, we use the unit circle properties.
In the unit circle, the coordinates of a point at an angle $ \theta $ are given by $ ( \cos(\theta), \sin(\theta) ) $.
So for $ \theta = \frac{5\pi}{6} $:
$$ \cos(\frac{5\pi}{6}) = -\frac{ \sqrt{3} }{2} $$
$$ \sin(\frac{5\pi}{6}) = \frac{1}{2} $$
Therefore, the coordinates are:
$$ \left( -\frac{ \sqrt{3} }{2}, \frac{1}{2} \right) $$
Find the value of tan(theta) using the unit circle
To find the value of $\tan(\theta)$ using the unit circle, we need to know the coordinates of the point on the unit circle that corresponds to the angle $\theta$.
On the unit circle, the coordinates of a point can be given as $(\cos(\theta), \sin(\theta))$.
The tangent of the angle $\theta$ is given by:
$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
Hence, if we know $\cos(\theta)$ and $\sin(\theta)$, we can find $\tan(\theta)$ by dividing $\sin(\theta)$ by $\cos(\theta)$.
Determine the values of tan(θ) on the unit circle where tan(θ) = 1 or tan(θ) = -1
First, note that $ \tan(\theta) = 1 $ when $ \theta = \frac{\pi}{4} $ or $ \theta = \frac{5\pi}{4} $ on the unit circle. Also, $ \tan(\theta) = -1 $ when $ \theta = \frac{3\pi}{4} $ or $ \theta = \frac{7\pi}{4} $. Therefore, the angles where $ \tan(\theta) = 1 $ or $ \tan(\theta) = -1 $ are:
$$ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} $$
Find the value of arctan(sin(3π/4))
To find the value of $ \arctan(\sin(\frac{3\pi}{4})) $, we first need to find the value of $ \sin(\frac{3\pi}{4}) $.
$$ \sin(\frac{3\pi}{4}) = \sin(\pi – \frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
Now, we need to determine the value of $ \arctan(\frac{\sqrt{2}}{2}) $.
Since $ \arctan(x) $ is the inverse of $ \tan(x) $, we seek an angle $ \theta $ such that:
$$ \tan(\theta) = \frac{\sqrt{2}}{2} $$
One such angle is $ \theta = \frac{\pi}{4} $, but considering the range of $ \arctan $, the solution is:
$$ \arctan(\sin(\frac{3\pi}{4})) = \arctan(\frac{\sqrt{2}}{2}) = \frac{\pi}{4} $$
Find the coordinates of the point on the unit circle where the angle with the positive x-axis is pi/3
The unit circle is defined as a circle with radius 1 centered at the origin. The coordinates of any point on the unit circle can be given by $(\cos(\theta), \sin(\theta))$ where $\theta$ is the angle with the positive $x$-axis.
Given that $\theta = \frac{\pi}{3}$:
$$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$
$$\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$
Thus, the coordinates are:
$$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$
Find the coordinates of the point where the terminal side of an angle $\theta$ intersects the unit circle, given that $\theta = \frac{5\pi}{6}$
To determine the coordinates where the terminal side of $\theta = \frac{5\pi}{6}$ intersects the unit circle:
First, recall that on the unit circle, the coordinates are given by $(\cos(\theta), \sin(\theta))$.
Calculate the cosine and sine values:
$$ \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$
$$ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} $$
Thus, the coordinates are:
$$ \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$
Calculate the area of a sector of a circle with radius r and central angle θ (in radians)
The area of a sector of a circle with radius $ r $ and central angle $ \theta $ can be calculated using the formula:
$$ A = \frac{1}{2} r^2 \theta $$
For example, if $ r = 5 $ and $ \theta = \frac{\pi}{3} $:
$$ A = \frac{1}{2} \cdot 5^2 \cdot \frac{\pi}{3} = \frac{25 \pi}{6} $$
So, the area is $ \frac{25 \pi}{6} $ square units.
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