Math

Evaluate the integral of sin(x) * cos(x) around the unit circle

To evaluate the integral of $ \sin(x) * \cos(x) $ around the unit circle, we can use the double-angle identity:

$$ \sin(x) \cos(x) = \frac{1}{2} \sin(2x) $$

Now, we need to integrate from $ 0 $ to $ 2\pi $:

$$ \int_{0}^{2\pi} \sin(x) \cos(x) \, dx = \int_{0}^{2\pi} \frac{1}{2} \sin(2x) \, dx $$

Let $ u = 2x $, hence $ du = 2 \, dx $ and $ dx = \frac{1}{2} du $:

$$ \int_{0}^{2\pi} \frac{1}{2} \sin(2x) \, dx = \frac{1}{2} \int_{0}^{4\pi} \sin(u) \frac{1}{2} \, du $$

Combining constants:

$$ \frac{1}{4} \int_{0}^{4\pi} \sin(u) \, du $$

The integral of $ \sin(u) $ over one period is zero, and here we have two periods:

$$ \frac{1}{4} \left(0\right) = 0 $$

The integral evaluates to $ 0 $.

Find the terminal point on the unit circle for an angle of pi/6 radians

To find the terminal point on the unit circle for an angle of $ \frac{\pi}{6} $ radians, we use the unit circle definition:

The coordinates are given by $ ( \cos( \theta ), \sin( \theta ) ) $.

For $ \theta = \frac{\pi}{6} $:

$$ \cos( \frac{\pi}{6} ) = \frac{\sqrt{3}}{2} $$

$$ \sin( \frac{\pi}{6} ) = \frac{1}{2} $$

So, the terminal point is:

$$( \frac{\sqrt{3}}{2}, \frac{1}{2} )$$

Identify the sine and cosine values for the angle π/4 on the unit circle

To find the sine and cosine values for the angle $ \frac{\pi}{4} $ on the unit circle, we use the definitions of sine and cosine:

$$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

$$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

Find the cosine of the angle at 3π/4 radians on the unit circle

The unit circle helps us find the cosine of an angle. For an angle of $ \frac{3π}{4} $ radians:

The reference angle is $ \x0crac{π}{4} $, and in the second quadrant, the cosine is negative.

So, $ \cos(\frac{3π}{4}) = -\cos(\frac{π}{4}) $

We know that $ \cos(\frac{π}{4}) = \frac{\sqrt{2}}{2} $

Therefore, $ \cos(\frac{3π}{4}) = -\frac{\sqrt{2}}{2} $

Prove that the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle for a given angle \theta

To prove that the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle for a given angle \theta , we start with the unit circle definition:

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On the unit circle, the coordinates of a point corresponding to an angle $ \theta $ are $ (\cos(\theta), \sin(\theta)) $.

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Consider the equation $ x^2 + y^2 = 1 $.

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Substitute $ x = \cos(\theta) $ and $ y = \sin(\theta) $:

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$$ \cos^2(\theta) + \sin^2(\theta) = 1 $$

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This identity is known as the Pythagorean identity, and it holds true for all values of $ \theta $. Therefore, the equation $ x^2 + y^2 = 1 $ is satisfied by the coordinates of any point on the unit circle.

Evaluate the cosecant of an angle in the unit circle when its sine is equal to a rational value

To evaluate $ \csc(\theta) $ when $ \sin(\theta) $ is a rational value, let

Find the coordinates of a point on the unit circle given the angle

To find the coordinates of a point on the unit circle given an angle $ \theta $, we use the formulas for sine and cosine:

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$$ x = \cos(\theta) $$

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$$ y = \sin(\theta) $$

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For example, if $ \theta = \frac{\pi}{4} $:

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$$ x = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

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$$ y = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

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So the coordinates are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

Determine the coordinates of $\frac{3\pi}{4}$ on the unit circle

The angle \( \frac{3\pi}{4} \) is in the second quadrant of the unit circle. To find its coordinates, we start by noting that the reference angle for \( \frac{3\pi}{4} \) is \( \frac{\pi}{4} \). The coordinates for \( \frac{\pi}{4} \) are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Since \( \frac{3\pi}{4} \) is in the second quadrant, the x-coordinate will be negative, and the y-coordinate will be positive. Therefore:

$$ \text{Coordinates of } \frac{3\pi}{4} = \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $$

Calculate the length of the arc intercepted by a central angle theta on a unit circle

To calculate the length of the arc intercepted by a central angle $ \theta $ on a unit circle, you can use the formula:

$$ s = r \theta $$

Since the radius $ r $ of the unit circle is 1, the formula simplifies to:

$$ s = \theta $$

Thus, the length of the arc is:

$$ s = \theta $$

Find the angles on the unit circle

Given a point on the unit circle at coordinates (1/2, √3/2), find the corresponding angle in degrees.

The point (1/2, √3/2) corresponds to an angle of 60 degrees.