Unit Circle

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) $:

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

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):

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

Find the coordinates of point on the unit circle

Given a point $(x, y)$ on the unit circle, we know that the equation of the circle is:

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

If $ x = \frac{1}{2} $, then we can find $ y $ by solving:

$$ (\frac{1}{2})^2 + y^2 = 1 $$

$$ \frac{1}{4} + y^2 = 1 $$

Solving for $ y $:

$$ y^2 = 1 – \frac{1}{4} $$

$$ y^2 = \frac{3}{4} $$

$$ y = \pm \frac{\sqrt{3}}{2} $$

So the coordinates are:

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