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Find the values of sin(θ) at specific angles on the unit circle
To find the values of $ \sin(\theta) $ at specific angles on the unit circle, we can use the known values for common angles:
At $ \theta = 0 $, $$ \sin(0) = 0 $$
At $ \theta = \frac{\pi}{2} $, $$ \sin\left(\frac{\pi}{2}\right) = 1 $$
At $ \theta = \pi $, $$ \sin(\pi) = 0 $$
At $ \theta = \frac{3\pi}{2} $, $$ \sin\left(\frac{3\pi}{2}\right) = -1 $$
At $ \theta = 2\pi $, $$ \sin(2\pi) = 0 $$
Prove that tan(theta) sec(theta) = sin(theta) where theta is an angle in the unit circle
We start with the definitions of the trigonometric functions on the unit circle.
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$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
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$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$
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Multiplying these two expressions, we have:
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$$ \tan(\theta) \sec(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \cdot \frac{1}{\cos(\theta)} $$
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Simplifying, we get:
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$$ \tan(\theta) \sec(\theta) = \frac{\sin(\theta)}{\cos^2(\theta)} \cdot \cos(\theta) $$
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Since $ \cos^2(\theta) \cos(\theta) = \cos(\theta) $, we have:
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$$ \tan(\theta) \sec(\theta) = \sin(\theta) \cdot \frac{1}{\cos^2(\theta)} = \sin(\theta) $$
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Thus, it is proven that:
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$$ \boxed{\tan(\theta) \sec(\theta) = \sin(\theta)} $$
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 the exact values of arcsec(2)
To find the exact value of $ \text{arcsec}(2) $, we need to determine the angle $ \theta $ such that $ \sec(\theta) = 2 $ and $ \theta $ lies within the range of secant
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:
$$ \cos(\theta) = \pm \sqrt{\frac{3}{4}} $$
$$ \cos(\theta) = \pm \frac{\sqrt{3}}{2} $$
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