Unit Circle

Find the values of sin(θ), cos(θ), and tan(θ) for θ = π/4 using the unit circle

To find the values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ for $\theta = \frac{\pi}{4}$ using the unit circle, we use the following:

On the unit circle, at $\theta = \frac{\pi}{4}$, the coordinates are: $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$.

So,

$$ \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} $$

$$ \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} $$

$$ \tan(\frac{\pi}{4}) = \frac{\sin(\frac{\pi}{4})}{\cos(\frac{\pi}{4})} = 1 $$

Find the exact values of the coordinates of the point where the unit circle intersects the positive x-axis

The unit circle is defined by the equation:

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

The positive x-axis means $ y = 0 $. Substituting $ y = 0 $ into the equation gives:

$$ x^2 + 0^2 = 1 $$

Simplifying, we find:

$$ x^2 = 1 $$

Taking the positive square root (since we are on the positive x-axis), we get:

$$ x = 1 $$

Thus, the coordinates of the point are:

$$ (1, 0) $$

Identify the sine value of an angle corresponding to $3\pi/4$

We start by noting that $ \frac{3\pi}{4} $ is in the second quadrant of the unit circle.

In the second quadrant, the sine value is positive, so we have:

$$ \sin \left( \frac{3\pi}{4} \right) = \sin( \pi – \frac{\pi}{4}) = \sin \left( \frac{\pi}{4} \right) $$

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

$$ \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

Find $ sin(θ) $ and $ cos(θ) $ for θ on the unit circle

To find $ \sin(\theta) $ and $ \cos(\theta) $ when $ \theta $ is on the unit circle:

Recall the unit circle definition: the unit circle is a circle with a radius of 1 centered at the origin. Therefore, if $ (x, y) $ is a point on the unit circle corresponding to the angle $ \theta $, then:

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

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

For example, at $ \theta = \frac{\pi}{4} $, we have:

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

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

Find the values of angles at which sin(θ) = 1/2 on the unit circle

To find the angles $ \theta $ such that $ \sin(\theta) = \frac{1}{2} $, we need to locate where the y-coordinate on the unit circle is $ \frac{1}{2} $.

The angles that satisfy this condition are:

$$ \theta = \frac{\pi}{6} + 2k\pi $$ and $$ \theta = \frac{5\pi}{6} + 2k\pi $$

where $ k $ is any integer.

Determine the pattern of points on the unit circle for $\theta$ within $[0, 2\pi]$

Points on the unit circle are given by the coordinates $(\cos(\theta), \sin(\theta))$, where $\theta$ ranges from $0$ to $2\pi$.

One pattern to observe is that for every angle $\theta$:

$$ \cos(\theta + 2n\pi) = \cos(\theta) $$

$$ \sin(\theta + 2n\pi) = \sin(\theta) $$

where $n$ is an integer. This periodicity shows that the points repeat every $2\pi$.

Calculate the value of tan(θ) at θ = 45° using the unit circle

To calculate $ \tan(\theta) $ at $ \theta = 45° $ using the unit circle, we note that at $ 45° $, the coordinates on the unit circle are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

The formula for $ \tan(\theta) $ is:

$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

Since at $ \theta = 45° $:

$$ \sin(45°) = \frac{\sqrt{2}}{2} $$

$$ \cos(45°) = \frac{\sqrt{2}}{2} $$

The value of $ \tan(45°) $ is:

$$ \tan(45°) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

Determine the value of cos(θ) when sin(θ) = 1/2 in the unit circle

In the unit circle, when $ \sin(\theta) = \frac{1}{2} $, we need to determine $ \cos(\theta) $.

Since $ \sin(\theta) $ relates to the y-coordinate and $ \cos(\theta) $ relates to the x-coordinate in the unit circle, we use the Pythagorean identity:

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

Given $ \sin(\theta) = \frac{1}{2} $, we can substitute:

$$ \left(\frac{1}{2}\right)^2 + \cos^2(\theta) = 1 $$

$$ \frac{1}{4} + \cos^2(\theta) = 1 $$

$$ \cos^2(\theta) = 1 – \frac{1}{4} $$

$$ \cos^2(\theta) = \frac{3}{4} $$

Taking the square root of both sides, we get:

$$ \cos(\theta) = \pm \sqrt{\frac{3}{4}} $$

$$ \cos(\theta) = \pm \frac{\sqrt{3}}{2} $$

Given a point on the unit circle at angle θ, determine the coordinates of the point and the angle θ in radians

The unit circle is defined as the set of points (x,y) such that $x^2 + y^2 = 1$. For a point on the unit circle at an angle $ \theta $, the coordinates of the point are $(\cos(\theta),\sin(\theta))$.

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

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

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

Thus, the coordinates of the point are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. So, the answer is $\theta = \frac{\pi}{4}$ rad.

Find the value of cos(θ) and sin(θ) when θ is an angle on the unit circle

To find the values of $\cos(\theta)$ and $\sin(\theta)$ when $\theta$ is an angle on the unit circle, we use the coordinates of the corresponding point on the unit circle.

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For example, if $\theta = \frac{5\pi}{6}$, then the point on the unit circle is $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.

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Therefore:

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$$\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$$

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$$\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$$