Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
Find the values of sin, cos, and tan for 30 degrees on the unit circle
For $ 30^\circ $ on the unit circle:
$$ \sin(30^\circ) = \frac{1}{2} $$
$$ \cos(30^\circ) = \frac{\sqrt{3}}{2} $$
$$ \tan(30^\circ) = \frac{1}{\sqrt{3}} \text{ or } \frac{\sqrt{3}}{3} $$
Find the value of tan(135°) using the unit circle
To find the value of $ \tan(135^\circ) $ using the unit circle, we need to recall that $ \tan\theta $ is the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
The angle $ 135^\circ $ is in the second quadrant, where the tangent is negative. It corresponds to the reference angle $ 45^\circ $.
For $ 45^\circ $, the coordinates on the unit circle are:
$$ ( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} ) $$
In the second quadrant, the x-coordinate is negative, so the point is:
$$ (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} ) $$
Thus,
$$ \tan(135^\circ) = \frac{\frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = -1 $$
Find the coordinates of the vertices of a triangle inscribed in a unit circle given angles
Given the angles $ \theta_1, \theta_2, \theta_3 $ of the vertices of the triangle, the coordinates of the vertices on the unit circle are:
Vertex 1: $ ( \cos(\theta_1), \sin(\theta_1) ) $
Vertex 2: $ ( \cos(\theta_2), \sin(\theta_2) ) $
Vertex 3: $ ( \cos(\theta_3), \sin(\theta_3) ) $
Let
Solve for the angle θ in the unit circle where sin(θ)cos(θ) = 1/4 and 0 ≤ θ < 2π
Given:
$$ \sin(\theta)\cos(\theta) = \frac{1}{4} $$
Using the double-angle identity:
$$ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $$
We have:
$$ \sin(2\theta) = 2 \times \frac{1}{4} = \frac{1}{2} $$
Thus:
$$ 2\theta = \sin^{-1}(\frac{1}{2}) $$
Giving:
$$ 2\theta = \frac{\pi}{6} \text{ or } \frac{5\pi}{6} $$
Hence:
$$ \theta = \frac{\pi}{12} \text{ or } \frac{5\pi}{12} $$
Checking the interval $ 0 \leq \theta < 2\pi $:
The possible solutions are:
$$ \theta = \frac{\pi}{12}, \frac{5\pi}{12} \text{ or } \frac{13\pi}{12}, \frac{17\pi}{12} $$
Find the value of sin(2x) and cos(2x) on the unit circle
To find the value of $\sin(2x)$ and $\cos(2x)$ on the unit circle, we can utilize the double-angle formulas:
$$ \sin(2x) = 2\sin(x)\cos(x) $$
$$ \cos(2x) = \cos^2(x) – \sin^2(x) $$
Given a point on the unit circle (a, b) where $a = \cos(x)$ and $b = \sin(x)$, we can substitute:
$$ \sin(2x) = 2ab $$
$$ \cos(2x) = a^2 – b^2 $$
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 $\cos(\theta)$.
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