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Find the values of sin(π/3), cos(π/3), and tan(π/3)
To find the values of $\sin(\frac{\pi}{3})$, $\cos(\frac{\pi}{3})$, and $\tan(\frac{\pi}{3})$, we use the unit circle.
For $\theta = \frac{\pi}{3}$:
$$ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $$
$$ \cos(\frac{\pi}{3}) = \frac{1}{2} $$
$$ \tan(\frac{\pi}{3}) = \frac{\sin(\frac{\pi}{3})}{\cos(\frac{\pi}{3})} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} $$
Calculate the integral of 1/(1 + x^2) over the unit circle
To calculate the integral of $ \frac{1}{1 + x^2} $ over the unit circle, we first convert to polar coordinates:
$$ x = r \cos(\theta), y = r \sin(\theta) $$
In polar coordinates, the unit circle is defined as:
$$ r = 1 $$
Substituting in the integral:
$$ \int_0^{2\pi} \frac{r}{1 + r^2 \cos^2(\theta)} d\theta $$
Since $r = 1$:
$$ \int_0^{2\pi} \frac{1}{1 + \cos^2(\theta)} d\theta $$
Applying the Weierstrass substitution:
Let $ \tan(\theta/2) = t $, then $ d\theta = \frac{2}{1+t^2} dt $
The integral becomes:
$$ \int_{-\infty}^{\infty} \frac{2}{1 + \cos^2(2 \arctan(t))} \frac{1}{1+t^2} dt $$
After simplification, the integral reduces to:
$$ \pi \int_{-\infty}^{\infty} \frac{2}{2 + t^2} \frac{1}{1+t^2} dt $$
The final answer is:
$$ \pi \ln{2} $$
Determine the cosine and sine values at π/4 on the unit circle
To find the cosine and sine values at $ \frac{\pi}{4} $ on the unit circle:
$$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$ $$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$
The values are:
$$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$ $$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$
Determine the value of cos(θ) and sin(θ) for θ=π/4 on the unit circle
To determine the values of $\cos(\theta)$ and $\sin(\theta)$ for $\theta=\frac{\pi}{4}$ on the unit circle, we use the known coordinates:
At $\theta=\frac{\pi}{4}$, both cosine and sine values are equal to:
$$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$
$$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$
Therefore, the values are:
$$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$
Determine the value of cos(θ) and sin(θ) for θ = π/4
To find the values of $ \cos(\theta) $ and $ \sin(\theta) $ when $ \theta = \frac{\pi}{4} $, we use the unit circle.
On the unit circle, when $ \theta = \frac{\pi}{4} $, both $ \cos(\theta) $ and $ \sin(\theta) $ are equal to:
$$ \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
Find the value of cos(π/3) and sin(π/3)
To find the value of $ \cos(\frac{\pi}{3}) $, we look at the coordinates of the corresponding point on the unit circle.
The coordinate point at $ \frac{\pi}{3} $ is $ (\frac{1}{2}, \frac{\sqrt{3}}{2}) $.
Hence, $ \cos(\frac{\pi}{3}) = \frac{1}{2} $ and $ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $.
Calculate the value of sin(2θ) if cos(θ) = 3/5 and θ is in the first quadrant
To find the value of $ \sin(2\theta) $, we use the double angle identity for sine:
$$ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $$
Given that $ \cos(\theta) = \frac{3}{5} $, we need to find $ \sin(\theta) $. Since $ \theta $ is in the first quadrant, $ \sin(\theta) $ is positive:
$$ \sin(\theta) = \sqrt{1 – \cos^2(\theta)} = \sqrt{1 – \left(\frac{3}{5}\right)^2} = \sqrt{1 – \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} $$
Now we can find $ \sin(2\theta) $:
$$ \sin(2\theta) = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25} $$
Find the angle on the unit circle corresponding to arctan(1/√3)
Given $ \arctan\left(\frac{1}{\sqrt{3}}\right) $, we need to determine the angle $ \theta $ on the unit circle.
We know that $ \arctan(x) $ gives the angle whose tangent is $ x $. Hence:
$$ \tan(\theta) = \frac{1}{\sqrt{3}} $$
We recognize that the angle corresponding to this tangent value is:
$$ \theta = \frac{\pi}{6} $$
Thus, the angle on the unit circle is $ \frac{\pi}{6} $.
Find the cosine of pi/3 using the unit circle
Using the unit circle, the angle $ \frac{\pi}{3} $ corresponds to 60 degrees. On the unit circle, the coordinates of this angle are:
$$ \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) $$
The cosine value is the x-coordinate:
$$ \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} $$
How to memorize the points on the unit circle
To memorize the points on the unit circle, remember that the unit circle has a radius of 1 and is centered at the origin (0,0). The key angles in radians are $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, and so on, up to $2\pi$. Each angle corresponds to coordinates (cosine, sine):
$$\begin{aligned} (0,1) & \quad \text{at} \quad 0 \\ (\frac{1}{2}, \frac{\sqrt{3}}{2}) & \quad \text{at} \quad \frac{\pi}{6} \\ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) & \quad \text{at} \quad \frac{\pi}{4} \\ (\frac{\sqrt{3}}{2}, \frac{1}{2}) & \quad \text{at} \quad \frac{\pi}{3} \\ (1,0) & \quad \text{at} \quad \frac{\pi}{2} \end{aligned}$$
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