Math

Find the coordinates of the point on the unit circle for an angle of 3π/4 radians

To find the coordinates of the point on the unit circle for an angle of $ \frac{3\pi}{4} $ radians, we need to use the unit circle definition:

For an angle $ \theta $, the coordinates are given by:

$$ (\cos(\theta), \sin(\theta)) $$

Here, $ \theta = \frac{3\pi}{4} $

So,

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

and

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

The coordinates are:

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

Determine the value of cos(π/4) using the unit circle

To determine the value of $\cos(\frac{\pi}{4})$, we use the unit circle. At the angle $\frac{\pi}{4}$, the coordinates on the unit circle are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Since the x-coordinate represents the cosine value, we have:

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

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

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

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

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

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) $:

$$ \sin(2\theta) = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25} $$

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):

$$\begin{aligned} (0,1) & \quad \text{at} \quad 0 \\ (\frac{1}{2}, \frac{\sqrt{3}}{2}) & \quad \text{at} \quad \frac{\pi}{6} \\ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) & \quad \text{at} \quad \frac{\pi}{4} \\ (\frac{\sqrt{3}}{2}, \frac{1}{2}) & \quad \text{at} \quad \frac{\pi}{3} \\ (1,0) & \quad \text{at} \quad \frac{\pi}{2} \end{aligned}$$