Explore the unit circle and its relationship to angles, radians, trigonometric ratios, and coordinates in the coordinate plane.
Evaluate the integral of cos^3(x)sin(x) with respect to x
To evaluate the integral of $ \cos^3(x)\sin(x) $ with respect to $ x $, we use a substitution method:
Let $ u = \cos(x) $, then $ du = -\sin(x) dx $. Consequently:
$$ \int \cos^3(x)\sin(x) dx = \int u^3 (-du) = -\int u^3 du $$
Now integrate:
$$ -\int u^3 du = -\frac{u^4}{4} + C $$
Substitute back $ \cos(x) $ for $ u $:
$$ -\frac{\cos^4(x)}{4} + C $$
The final answer is:
$$ -\frac{\cos^4(x)}{4} + C $$
Determine the angle in radians of the point on the unit circle in the first quadrant with an x-coordinate of 1/2
To find the angle in radians with an $x$-coordinate of $\frac{1}{2}$ in the first quadrant, we use the unit circle definition of cosine.
For $\cos(\theta) = \frac{1}{2}$, the corresponding angle is:
$$ \theta = \frac{\pi}{3} $$
Determine the cosine of an angle given in radians and convert it to degrees
Given an angle $ \theta = \frac{7\pi}{6} $ radians, we need to determine its cosine and convert the angle to degrees.
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First, convert the angle to degrees:\n
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$$ \theta = \frac{7\pi}{6} \cdot \frac{180^\circ}{\pi} = 210^\circ $$
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The angle $ 210^\circ $ lies in the third quadrant where the cosine is negative.
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Using the unit circle, we know:
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$$ \cos(210^\circ) = \cos(180^\circ + 30^\circ) = -\cos(30^\circ) $$
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Since $ \cos(30^\circ) = \frac{\sqrt{3}}{2} $, we have:
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$$ \cos(210^\circ) = -\frac{\sqrt{3}}{2} $$
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Thus, the cosine of $ \frac{7\pi}{6} $ radians is $ -\frac{\sqrt{3}}{2} $ and the angle in degrees is $ 210^\circ $.
Find the value of sec(θ) for θ in the unit circle
To find the value of $ \sec(\theta) $ for $ \theta $ in the unit circle, we need to recall the definition of secant. The secant function is the reciprocal of the cosine function:
$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$
Given that $ \theta $ is an angle in the unit circle, let’s consider $ \theta = \frac{\pi}{4} $ as an example. For this angle:
$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
Thus,
$$ \sec\left(\frac{\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} $$
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} $:
$$ A = \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8} $$
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