Unit Circle

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.

Find the angle θ in radians for a point on the unit circle that satisfies given conditions

Given a point $ P $ on the unit circle, where the coordinates of $ P $ are $ ( \cos(\theta), \sin(\theta) ) $.

If the coordinates of $ P $ are given as $ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $, we need to determine the angle $ \theta $.

On the unit circle, these coordinates correspond to:

$$ \cos(\theta) = \frac{1}{2} \quad \text{and} \quad \sin(\theta) = \frac{\sqrt{3}}{2} $$

From the unit circle, we know that:

$$ \theta = \frac{\pi}{3} $$

Since the angle $ \theta $ can also be in the second quadrant, we have:

$$ \theta = \frac{5\pi}{3} $$

Find the value of sin(θ), cos(θ), and tan(θ) for θ = π/3 on the unit circle

When $θ = \fracπ3$, we can find the values of $\sin(θ)$, $\cos(θ)$, and $\tan(θ)$ from the unit circle:

$$\sin(\fracπ3) = \frac{\sqrt3}2$$

$$\cos(\fracπ3) = \frac12$$

$$\tan(\fracπ3) = \frac{\sin(\fracπ3)}{\cos(\fracπ3)} = \sqrt3$$

Evaluate the integral of cos(2x) from 0 to pi/2

To evaluate the integral of $ \cos(2x) $ from $ 0 $ to $ \frac{\pi}{2} $:

$$ \int_0^{\frac{\pi}{2}} \cos(2x) \, dx $$

Use the substitution $ u = 2x $, then $ du = 2dx $ or $ dx = \frac{1}{2} du $:

$$ \int_0^{\frac{\pi}{2}} \cos(2x) \, dx = \frac{1}{2} \int_0^{\pi} \cos(u) \, du $$

The integral of $ \cos(u) $ is $ \sin(u) $:

$$ \frac{1}{2} \left[ \sin(u) \right]_0^{\pi} $$

Evaluate the definite integral:

$$ \frac{1}{2} \left( \sin(\pi) – \sin(0) \right) = \frac{1}{2} (0 – 0) = 0 $$

Therefore, the final answer is:

$$ 0 $$

Determine the coordinates of the point on the unit circle corresponding to a given angle

To determine the coordinates of the point on the unit circle corresponding to the angle $\theta$, we use the following formulas for the unit circle:

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

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

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

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

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

So, the coordinates are:

$$ \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$