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Find the value of sec(π/3) on the unit circle
To find the value of $\sec(\frac{\pi}{3})$, we need to first determine the cosine of $\frac{\pi}{3}$.
On the unit circle, $\cos(\frac{\pi}{3}) = \frac{1}{2}$.
The secant function is the reciprocal of the cosine function, so
$$\sec(\frac{\pi}{3}) = \frac{1}{\cos(\frac{\pi}{3})} = \frac{1}{\frac{1}{2}} = 2$$
Find the equation of a circle with center at (h, k) and radius 1
The general equation of a circle with center $(h, k)$ and radius $r$ is given by:
$$ (x – h)^2 + (y – k)^2 = r^2 $$
For a unit circle, the radius $r = 1$. Therefore, the equation becomes:
$$ (x – h)^2 + (y – k)^2 = 1 $$
Calculating the Total Cost of a Stay at a Hotel in San Diego
Let’s assume you are planning to stay at a hotel in Hotel Circle, San Diego, CA, United States. The nightly rate for the hotel is $150, and you plan to stay for 5 nights. Additionally, there is a city tax of 10% and a service charge of $20 per night. Calculate the total cost of your stay.
First, calculate the total cost before tax and service charge:
$$150 \text{ (nightly rate)} \times 5 \text{ (nights)} = 750 \text{ dollars}$$
Next, calculate the city tax:
$$750 \text{ dollars} \times 0.10 = 75 \text{ dollars}$$
Then, calculate the service charge:
$$20 \text{ dollars/night} \times 5 \text{ nights} = 100 \text{ dollars}$$
Finally, add all these amounts together to find the total cost:
$$750 \text{ dollars} + 75 \text{ dollars} + 100 \text{ dollars} = 925 \text{ dollars}$$
Therefore, the total cost of your stay is $925.
Find the value of sin(π/3) using the unit circle
$$\text{The angle } \frac{\pi}{3} \text{ is equivalent to } 60^{\circ}.$$
$$\text{On the unit circle, the coordinates for } 60^{\circ} \text{ are } \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right).$$
$$\sin(\frac{\pi}{3}) \text{ is the y-coordinate, which is } \frac{\sqrt{3}}{2}.$$
Find the sine and cosine of 𝜋/6 radians on the unit circle
To find the sine and cosine of $\frac{\pi}{6}$ radians on the unit circle, we need to recall the standard angle values:
At $\frac{\pi}{6}$ radians:
$$cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$$
$$sin(\frac{\pi}{6}) = \frac{1}{2}$$
Thus, the cosine of $\frac{\pi}{6}$ radians is $\frac{\sqrt{3}}{2}$, and the sine of $\frac{\pi}{6}$ radians is $\frac{1}{2}$.
Calculate the circumference of a unit circle
The unit circle has a radius of 1 unit. The formula for the circumference of a circle is given by
$$C = 2 \pi r$$
Substitute \( r = 1 \):
$$C = 2 \pi \times 1 = 2 \pi$$
The circumference of the unit circle is \(2\pi\).
Find the coordinates and trigonometric values for an angle on the unit circle
Consider an angle $ \theta = \frac{7\pi}{6} $ on the unit circle. We need to find the coordinates of the point on the unit circle corresponding to this angle, as well as the sine and cosine values.
First, identify the reference angle: $$ \theta_{ref} = \pi – \frac{7\pi}{6} = \frac{\pi}{6} $$
Next, find the coordinates for the reference angle $ \frac{\pi}{6} $:
$$ \left( \cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right) \right) = \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$
Since $ \theta = \frac{7\pi}{6} $ is in the third quadrant, both sine and cosine are negative:
$$ \left( \cos\left(\frac{7\pi}{6}\right), \sin\left(\frac{7\pi}{6}\right) \right) = \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $$
How to Remember the Unit Circle Fast
$$\text{To remember the unit circle, focus on key angles and their coordinates. Start with } 0^\circ, 30^\circ, 45^\circ, 60^\circ, \text{ and } 90^\circ.$$
$$\text{For example, at } 0^\circ, \text{ the coordinates are } (1, 0).$$
$$\text{At } 30^\circ, \text{ the coordinates are } \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right).$$
$$\text{At } 45^\circ, \text{ the coordinates are } \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right).$$
$$\text{At } 60^\circ, \text{ the coordinates are } \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right).$$
$$\text{At } 90^\circ, \text{ the coordinates are } (0, 1).$$
$$\text{Memorize these points, and use symmetry to fill in the rest of the circle.}$$
Find the value of tan(θ) on the unit circle for θ = 5π/4
To find the value of $ \tan(\theta) $ on the unit circle for $ \theta = \frac{5\pi}{4} $, we first determine the coordinates of the point on the unit circle that corresponds to this angle.
The angle $ \frac{5\pi}{4} $ is in the third quadrant, where both sine and cosine values are negative. The reference angle is $ \pi/4 $.
The coordinates for $ \theta = \frac{5\pi}{4} $ are $ (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $.
Since $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $, we have:
$$ \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $$
What does sin represent on the unit circle?
On the unit circle, the sine of an angle represents the y-coordinate of the point on the unit circle that corresponds to that angle.
$$\text{Given an angle } \theta, \text{the coordinates of the point P on the unit circle are } (\cos(\theta), \sin(\theta)).$$
This means:
$$\sin(\theta) = y.$$
For example, if \(\theta = 30^\circ\):
$$\sin(30^\circ) = \frac{1}{2}.$$
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