Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
Given a point on the unit circle at an angle θ, find the coordinates of the point
To find the coordinates of a point on the unit circle at an angle $ \theta $, use the definitions of sine and cosine.
The coordinates of the point are given by:
$$ (\cos(\theta), \sin(\theta)) $$
Find the cosine value of the angle formed by a point on the unit circle
To find the cosine value of the angle formed by a point on the unit circle in the complex plane, consider a point $ z = e^{i\theta} $, where $ \theta $ is the angle in radians.
The cosine value of the angle $ \theta $ is the real part of $ z $, which is $ \cos(\theta) $.
Therefore:
$$ \text{Re}(e^{i\theta}) = \cos(\theta) $$
Find the point of intersection of a line passing through the origin at an angle θ with the unit circle
To find the point of intersection of a line passing through the origin at an angle $ \theta $ with the unit circle, we start by writing the equation of the line. The equation of the line in the form $ y = mx $ is:
$$ y = \tan(\theta) x $$
Since the line intersects the unit circle, we substitute $ y = \tan(\theta) x $ into the equation of the unit circle $ x^2 + y^2 = 1 $:
$$ x^2 + (\tan(\theta) x)^2 = 1 $$
Simplifying, we have:
$$ x^2 + x^2 \tan^2(\theta) = 1 $$
$$ x^2(1 + \tan^2(\theta)) = 1 $$
We use the trigonometric identity $ 1 + \tan^2(\theta) = \sec^2(\theta) $:
$$ x^2 \sec^2(\theta) = 1 $$
$$ x^2 = \cos^2(\theta) $$
So, we get two possible values for $ x $:
$$ x = \cos(\theta) $$
$$ x = -\cos(\theta) $$
For each $ x $, we find the corresponding $ y $:
When $ x = \cos(\theta) $:
$$ y = \tan(\theta) \cos(\theta) = \sin(\theta) $$
When $ x = -\cos(\theta) $:
$$ y = \tan(\theta) (-\cos(\theta)) = -\sin(\theta) $$
So, the points of intersection are:
$$ (\cos(\theta), \sin(\theta)) $$
$$ (-\cos(\theta), -\sin(\theta)) $$
Find the value of trigonometric functions at pi/3 on the unit circle
To find the value of trigonometric functions at $\frac{\pi}{3}$ on the unit circle, we need to calculate $\sin\left(\frac{\pi}{3}\right)$ and $\cos\left(\frac{\pi}{3}\right)$.
Since $\frac{\pi}{3}$ corresponds to 60 degrees:
$$\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$$
$$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$$
Find the angle θ on the unit circle where the equation cos^2(θ) – sin^2(θ) = 1 – 2sin^2(θ) holds true
To solve for $ \theta $ on the unit circle in the equation $ \cos^2(\theta) – \sin^2(\theta) = 1 – 2\sin^2(\theta) $, start by using trigonometric identities:
\n
We know that $ \cos^2(\theta) = 1 – \sin^2(\theta) $, so the equation becomes:
\n
$$ 1 – \sin^2(\theta) – \sin^2(\theta) = 1 – 2\sin^2(\theta) $$
\n
Simplify both sides:
\n
$$ 1 – 2\sin^2(\theta) = 1 – 2\sin^2(\theta) $$
\n
The equation holds for any $ \theta $ where $ 1 – 2\sin^2(\theta) $ is defined, which simplifies to $ \theta = n\pi $, where $ n $ is an integer.
Find the exact values of sin(θ), cos(θ), and tan(θ) at θ = 3π/4
To find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ at $ \theta = \frac{3\pi}{4} $, we use the unit circle:
For $ \theta = \frac{3\pi}{4} $, the corresponding point on the unit circle is in the second quadrant where both $ \sin(\theta) $ and $ \cos(\theta) $ have specific values:
$ \sin(\frac{3\pi}{4}) $: The sine value is given by:
$$ \sin(\frac{3\pi}{4}) = \sin(\pi – \frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
$ \cos(\frac{3\pi}{4}) $: The cosine value is given by:
$$ \cos(\frac{3\pi}{4}) = \cos(\pi – \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2} $$
$ \tan(\frac{3\pi}{4}) $: The tangent value is given by:
$$ \tan(\frac{3\pi}{4}) = \frac{\sin(\frac{3\pi}{4})}{\cos(\frac{3\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $$
Find the exact coordinates of the point where the angle 7π/6 intersects the unit circle
To find the coordinates of the point where the angle $ \frac{7\pi}{6} $ intersects the unit circle, we first identify the reference angle. The reference angle for $ \frac{7\pi}{6} $ is $ \frac{\pi}{6} $.
The coordinates for the angle $ \frac{\pi}{6} $ on the unit circle are $ \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $.
Since $ \frac{7\pi}{6} $ is in the third quadrant, both x and y coordinates will be negative:
$$ \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $$
Therefore, the coordinates are $ \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $.
Draw a point on the unit circle at angle pi/4
The unit circle has a radius of 1. To draw a point at angle $ \frac{\pi}{4} $, use the coordinates:
$$ (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) $$
Since $ \cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $, the point is:
$$ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $$
Find the coordinates of the point where the terminal side of theta intersects the unit circle at theta = 5π/6
To find the coordinates of the point where the terminal side of $ \theta $ intersects the unit circle at $ \theta = \frac{5\pi}{6} $, we use the unit circle definition and the corresponding reference angle.
The reference angle for $ \theta = \frac{5\pi}{6} $ is $ \frac{\pi}{6} $. The coordinates on the unit circle for $ \frac{\pi}{6} $ are $ \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $.
Since $ \frac{5\pi}{6} $ is in the second quadrant, we adjust the signs of the coordinates:
$$ \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$
What is a unit circle in trigonometry, and how is it used to define the trigonometric functions?
A unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The equation of the unit circle is given by:
$$ x^2 + y^2 = 1 $$
In trigonometry, the unit circle is used to define the trigonometric functions sine and cosine. For a given angle $ \theta $, measured from the positive x-axis, the coordinates of the corresponding point on the unit circle are $(\cos(\theta), \sin(\theta))$. These definitions can be extended to all real numbers by considering the angle $ \theta $ to be the result of wrapping the real line around the unit circle.
Furthermore, the unit circle allows us to define the other trigonometric functions as follows:
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