Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
Find the derivative of cos(x^2) with respect to x
To find the derivative of $ \cos(x^2) $ with respect to $ x $, we use the chain rule:
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$$ \frac{d}{dx} \cos(u) = -\sin(u) \cdot \frac{du}{dx} $$
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Here, let $ u = x^2 $. Then:
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$$ \frac{du}{dx} = \frac{d}{dx}(x^2) = 2x $$
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Now apply the chain rule:
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$$ \frac{d}{dx} \cos(x^2) = -\sin(x^2) \cdot 2x $$
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The final derivative is:
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$$ \frac{d}{dx} \cos(x^2) = -2x \sin(x^2) $$
Determine the tangent values for the primary angles on the unit circle
To determine the tangent values for the primary angles on the unit circle, we need to evaluate the tangent function at $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ and $2\pi$.
$$ \text{tan}(0) = 0 $$
$$ \text{tan}\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} $$
$$ \text{tan}\left(\frac{\pi}{4}\right) = 1 $$
$$ \text{tan}\left(\frac{\pi}{3}\right) = \sqrt{3} $$
$$ \text{tan}\left(\frac{\pi}{2}\right) = \text{undefined} $$
$$ \text{tan}(\pi) = 0 $$
$$ \text{tan}\left(\frac{3\pi}{2}\right) = \text{undefined} $$
$$ \text{tan}(2\pi) = 0 $$
How to find sine, cosine, and tangent for an angle using the unit circle?
To find the sine, cosine, and tangent of an angle using the unit circle, follow these steps:
1. Locate the angle on the unit circle.
2. Identify the coordinates $(x, y)$ of the point where the terminal side of the angle intersects the unit circle.
3. The coordinates correspond to $\cos(\theta)$ and $\sin(\theta)$ respectively:
$$ \cos(\theta) = x $$
$$ \sin(\theta) = y $$
4. Calculate $\tan(\theta)$ as follows:
$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} $$
Determine the coordinates of the point where the angle θ = π/3 on the unit circle
First, recall that the unit circle has a radius of 1. For the angle $ \theta = \frac{\pi}{3} $, we use the definitions of sine and cosine:
$$ x = \cos(\theta) $$
$$ y = \sin(\theta) $$
When $ \theta = \frac{\pi}{3} $, we have:
$$ x = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$
$$ y = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$
Therefore, the coordinates of the point are:
$$ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $$
Find the values of sin(θ) and cos(θ) where θ is 5π/4 radians on the unit circle
Given $\theta = \frac{5\pi}{4}$, we need to find the values of $\sin(\theta)$ and $\cos(\theta)$ on the unit circle.
The angle $\frac{5\pi}{4}$ is in the third quadrant where both sine and cosine are negative.
In the third quadrant, for an angle of $\frac{5\pi}{4}$,
$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$
$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$
Compute the integral of cos^2(t) on the unit circle
To compute the integral of $\cos^2(t)$ on the unit circle, we can use the double-angle identity for cosine:
$$\cos^2(t) = \frac{1 + \cos(2t)}{2}$$
Now, integrate:
$$\int_0^{2\pi} \cos^2(t) \, dt = \int_0^{2\pi} \frac{1 + \cos(2t)}{2} \, dt$$
Separate the integral:
$$= \frac{1}{2} \int_0^{2\pi} (1 + \cos(2t)) \, dt$$
Split it into two integrals:
$$= \frac{1}{2} \left( \int_0^{2\pi} 1 \, dt + \int_0^{2\pi} \cos(2t) \, dt \right)$$
The first integral is straightforward:
$$= \frac{1}{2} \left( 2\pi + 0 \right)$$
The second integral of cosine over a full period is zero:
$$= \pi$$
Find the coordinates of the point on the unit circle where the angle is 5π/6
To find the coordinates of the point on the unit circle where the angle is $\frac{5\pi}{6}$, we use the unit circle trigonometric identities for sine and cosine.
Since $\frac{5\pi}{6}$ is in the second quadrant:
The x-coordinate is:
$$ x = \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} $$
The y-coordinate is:
$$ y = \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} $$
So, the coordinates are:
$$ \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$
Find the coordinates on the unit circle for an angle of pi/3
To find the coordinates on the unit circle for an angle of $ \frac{\pi}{3} $, we use the cosine and sine functions:
$$ x = \cos(\frac{\pi}{3}) $$
$$ y = \sin(\frac{\pi}{3}) $$
The values are:
$$ \cos(\frac{\pi}{3}) = \frac{1}{2} $$
$$ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $$
Thus, the coordinates are:
$$ (\frac{1}{2}, \frac{\sqrt{3}}{2}) $$
Find the coordinates of the point on the unit circle for an angle of 3π/4 radians
To find the coordinates of the point on the unit circle for an angle of $ \frac{3\pi}{4} $ radians, we need to use the unit circle definition:
For an angle $ \theta $, the coordinates are given by:
$$ (\cos(\theta), \sin(\theta)) $$
Here, $ \theta = \frac{3\pi}{4} $
So,
$$ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} $$
and
$$ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
The coordinates are:
$$ \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $$
Determine the value of cos(π/4) using the unit circle
To determine the value of $\cos(\frac{\pi}{4})$, we use the unit circle. At the angle $\frac{\pi}{4}$, the coordinates on the unit circle are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.
Since the x-coordinate represents the cosine value, we have:
$$ \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $$
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