Math

Find the coordinates of a point on the unit circle at angle pi/3

The unit circle has a radius of 1. The coordinates of a point on the circle at angle $ \frac{\pi}{3} $ are given by:

$$ (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) $$

Therefore,

$$ \cos(\frac{\pi}{3}) = \frac{1}{2} $$

and

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

Thus, the coordinates are:

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

Determine the values of sin(θ), cos(θ), and tan(θ) for θ in the second quadrant of the unit circle

In the second quadrant, the angle $ \theta $ ranges from $ \frac{\pi}{2} $ to $ \pi $. Here, $ \sin(\theta) $ is positive, $ \cos(\theta) $ is negative, and $ \tan(\theta) $ is negative.

Using the unit circle, for $ \theta = \frac{2\pi}{3} $:

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

$$ \cos(\frac{2\pi}{3}) = -\frac{1}{2} $$

$$ \tan(\frac{2\pi}{3}) = -\sqrt{3} $$

Find the value of sin(π/3) on the unit circle

To find the value of $ \sin\left(\frac{\pi}{3}\right) $ on the unit circle, we look at the reference angle for $ \frac{\pi}{3} $.

The reference angle is $ 60^{\circ} $, and the sine value for $ 60^{\circ} $ is:

$$ \sin\left(60^{\circ}\right) = \frac{\sqrt{3}}{2} $$

Therefore:

$$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$

Find the area of a sector of a unit circle with a central angle of θ radians

To find the area of a sector of a unit circle with a central angle of $ \theta $, we use the formula for the area of a sector:

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$$ A = \frac{1}{2} r^2 \theta $$

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

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$$ A = \frac{1}{2} \cdot 1^2 \cdot \theta = \frac{1}{2} \theta $$

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The area of the sector is:

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$$ A = \frac{\theta}{2} $$

Find the value of sin(π/6), cos(π/3), and tan(π/4) using the unit circle chart

Using the unit circle chart, we find:

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

$$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$

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

Find the value of cos(π/4) and sin(π/4) using the unit circle

To find the values of $ \cos(\frac{\pi}{4}) $ and $ \sin(\frac{\pi}{4}) $ using the unit circle, we need to identify the coordinate point on the unit circle that corresponds to the angle $ \frac{\pi}{4} $.

The angle $ \frac{\pi}{4} $ is located in the first quadrant where both sine and cosine values are positive. This angle corresponds to the point $ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $ on the unit circle.

Therefore:

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

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

Find the equation of the tangent line to the unit circle at (1, 0)

The unit circle is given by the equation:

$$ x^2 + y^2 = 1 $$

To find the equation of the tangent line at $(1, 0)$, we first find the slope of the tangent. Differentiate the equation implicitly with respect to $x$:

$$ 2x + 2y \x0crac{dy}{dx} = 0 $$

At the point $(1, 0)$, substitute $x = 1$ and $y = 0$:

$$ 2(1) + 2(0) \x0crac{dy}{dx} = 0 $$

So, the slope $\x0crac{dy}{dx}$ at $(1, 0)$ is $0$. The equation of the tangent line using point-slope form is:

$$ y – 0 = 0(x – 1) $$

Therefore, the equation is:

$$ y = 0 $$

What does the sine function represent on the unit circle?

On the unit circle, the sine function $ \sin(\theta) $ represents the y-coordinate of a point on the circle corresponding to the angle $ \theta $ measured from the positive x-axis.

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Mathematically, if a point on the unit circle is given by $ (x, y) $, then for any angle $ \theta $, the coordinates can be expressed as:

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

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

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This implies that the sine of an angle is the vertical distance from the x-axis to the point on the circle.

Determine the values of trigonometric functions for specific angles on the unit circle

Given the angle $ \theta = \frac{5\pi}{4} $ radians, find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $.

The coordinates of $ \frac{5\pi}{4} $ on the unit circle are $ \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) $.

So, $ \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} $, $ \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} $, and

$$ \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = 1. $$

Find the value of arcsin(1/2) using the unit circle

To find the value of $ \arcsin(\frac{1}{2}) $ using the unit circle, we need to determine the angle whose sine is $ \frac{1}{2} $.

On the unit circle, the sine of an angle is the y-coordinate of the corresponding point.

The angle that has a sine of $ \frac{1}{2} $ in the range $ -\frac{\pi}{2} $ to $ \frac{\pi}{2} $ is $ \frac{\pi}{6} $.

Therefore, the value of $ \arcsin(\frac{1}{2}) $ is:

$$ \frac{\pi}{6} $$