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Find the sine of an angle whose terminal side passes through the point (sqrt(3)/2, -1/2) on the unit circle
To find the sine of the angle, we need to identify the y-coordinate of the given point $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ on the unit circle.
The y-coordinate is:
$$ -\frac{1}{2} $$
Therefore, the sine of the angle is:
$$ \sin(\theta) = -\frac{1}{2} $$
Determine the quadrant for the given angle
The angle $ \theta = 45^\circ $ is in the first quadrant.
Find the secant line to the unit circle that is equidistant from the x-axis
To find the secant line to the unit circle that is equidistant from the $x$-axis, we use the equation of the unit circle
$$ x^2 + y^2 = 1 $$
and the general equation of a line
$$ y = mx + b $$
Since the secant line is equidistant from the $x$-axis, the $y$-intercept $b$ must satisfy the condition that the distances from $b$ to the points of intersection with the circle are equal. So, we solve:
Substitute $y = mx + b$ into the circle
Determine the exact values of sine and cosine for the angle π/4 using the unit circle
To find the exact values of sine and cosine for the angle $ \frac{\pi}{4} $, we use the unit circle.
For $ \theta = \frac{\pi}{4} $, the coordinates on the unit circle are:
$$ ( \cos( \frac{\pi}{4} ), \sin( \frac{\pi}{4} )) $$
Since $ \frac{\pi}{4} $ is an angle in the first quadrant where sine and cosine values are positive, we use the 45-degree reference angle values. We have:
$$ \cos( \frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
$$ \sin( \frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
Thus, the exact values are:
$$ \cos( \frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
$$ \sin( \frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
Identify the coordinates of specific angles on the unit circle
To find the coordinates of specific angles on the unit circle, remember that the unit circle has a radius of 1.
For the angle $\theta = \frac{\pi}{4}$, the coordinates are:
$$(\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$$
Find the area of a sector in a unit circle with a central angle of θ radians
The formula to find the area of a sector in a unit circle is:
$$ A = \frac{1}{2} \theta $$
where $ \theta $ is the central angle in radians.
For example, if $ \theta = \frac{\pi}{4} $:
$$ A = \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8} $$
Find the image of the unit circle under the transformation f(z)=z^2
The unit circle in the complex plane is given by $ |z| = 1 $, meaning any point $ z $ on the unit circle can be written as $ z = e^{i\theta} $ for some real number $ \theta $.
Under the transformation $ f(z) = z^2 $, the image of $ z $ is:
$$ f(z) = (e^{i\theta})^2 = e^{i(2\theta)} $$
Since $ e^{i(2\theta)} $ is still a point on the unit circle, the image of the unit circle under $ f(z) = z^2 $ is the unit circle itself.
Identify the quadrant in which the angle lies
To identify the quadrant in which the angle $ \theta $ lies, follow these steps:
1. If $ 0 \leq \theta < \frac{\pi}{2} $, then the angle is in the first quadrant.
2. If $ \frac{\pi}{2} \leq \theta < \pi $, then the angle is in the second quadrant.
3. If $ \pi \leq \theta < \frac{3\pi}{2} $, then the angle is in the third quadrant.
4. If $ \frac{3\pi}{2} \leq \theta < 2\pi $, then the angle is in the fourth quadrant.
Find the value of cos(π/3)
The value of $ \cos\left( \frac{\pi}{3} \right) $ can be found using the unit circle. The angle $ \frac{\pi}{3} $ corresponds to 60 degrees. On the unit circle, the coordinates for the angle 60 degrees are:
$$ \left( \cos\left( \frac{\pi}{3} \right), \sin\left( \frac{\pi}{3} \right) \right) = \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $$
Therefore, the value of $ \cos\left( \frac{\pi}{3} \right) $ is:
$$ \frac{1}{2} $$
Find the coordinates of a point on the unit circle at angle π/4
To find the coordinates of a point on the unit circle at angle $\frac{\pi}{4}$, we use the trigonometric functions:
$$ x = \cos\left(\frac{\pi}{4}\right) $$
$$ y = \sin\left(\frac{\pi}{4}\right) $$
Since:
$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
$$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
The coordinates are:
$$ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $$
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