Unit Circle

Calculate the exact value of sin(7π/6) using the unit circle

To determine the exact value of $\sin\left(\frac{7\pi}{6}\right)$ using the unit circle, first note that $\frac{7\pi}{6}$ is in the third quadrant.

In the third quadrant, the sine function is negative.

Now, find the reference angle for $\frac{7\pi}{6}$:

$$ 7\pi / 6 – \pi = \pi / 6 $$

The reference angle is $\pi / 6$, whose sine value is $\frac{1}{2}$.

Since sine is negative in the third quadrant:

$$ \sin \left( \frac{7\pi}{6} \right) = -\frac{1}{2} $$

Determine the cotangent of an angle on the unit circle

The cotangent of an angle $ \theta $ on the unit circle is given by:

$$ \cot( \theta ) = \frac{1}{\tan( \theta )} = \frac{\cos( \theta )}{\sin( \theta )} $$

Let

Determine the equation of a unit circle and explain the geometric significance

The equation of a unit circle centered at the origin is given by:

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

This equation signifies that any point $ (x, y) $ on the unit circle is at a distance of 1 unit from the origin. The radius of the circle is always 1.

Find the value of tan(θ) using the unit circle when θ = 3π/4

We need to find the value of $ \tan(\theta) $ where $ \theta = \frac{3\pi}{4} $ using the unit circle. The coordinates of the point on the unit circle corresponding to $ \theta = \frac{3\pi}{4} $ are:

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

Recall that $ \tan(\theta) = \frac{y}{x} $. Therefore:

$$ \tan \left( \frac{3\pi}{4} \right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 $$

Determine the trigonometric identity of sin(θ) using the unit circle

To determine the trigonometric identity of $ \sin(\theta) $ using the unit circle, we start by understanding the unit circle definition:

The unit circle is a circle with a radius of $1$ centered at the origin $(0, 0)$.

For any angle $\theta$ measured from the positive x-axis, the coordinates of the point where the terminal side of $\theta$ intersects the unit circle are given by $(\cos(\theta), \sin(\theta))$.

Therefore, the identity for $\sin(\theta)$ is the y-coordinate of this intersection point:

$$ \sin(\theta) = y $$

Where $y$ is the y-coordinate of the intersection point.

To provide a concrete example, if $\theta = \frac{\pi}{4}$, the coordinates of the intersection point are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so:

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

Find the values of sin(θ), cos(θ), and tan(θ) for θ = 7π/6 using the unit circle

To find the values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ for $ \theta = \frac{7\pi}{6} $ using the unit circle, we start by locating the angle on the unit circle:

$ \theta = \frac{7\pi}{6} $ corresponds to an angle in the third quadrant, where both sine and cosine values are negative.

In the unit circle, for $ \theta = \frac{7\pi}{6} $:

$$ \sin\left( \frac{7\pi}{6} \right) = -\frac{1}{2} $$

$$ \cos\left( \frac{7\pi}{6} \right) = -\frac{\sqrt{3}}{2} $$

To find the tangent, use: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

$$ \tan\left( \frac{7\pi}{6} \right) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $$

Find the value of cos(θ) when θ is on the Unit Circle at specific points

To find the value of $ \cos(\theta) $ on the Unit Circle at specific points, consider the following:

• When $ \theta = 0 $:

$$ \cos(0) = 1 $$

• When $ \theta = \frac{\pi}{2} $:

$$ \cos\left(\frac{\pi}{2}\right) = 0 $$

• When $ \theta = \pi $:

$$ \cos(\pi) = -1 $$

• When $ \theta = \frac{3\pi}{2} $:

$$ \cos\left(\frac{3\pi}{2}\right) = 0 $$

• When $ \theta = 2\pi $:

$$ \cos(2\pi) = 1 $$

Find the value of sin(θ) and cos(θ) at different points on the unit circle

To find the value of $ \sin(\theta) $ and $ \cos(\theta) $ at different points on the unit circle, consider the following angles:

1. $\theta = \frac{\pi}{6}$:

$$ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2}, \quad \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} $$

2. $\theta = \frac{\pi}{4}$:

$$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}, \quad \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

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

$$ \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}, \quad \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} $$

Fill in the blank: The value of sin(____) at the unit circle when the angle is pi/2 is 1

Fill in the blank: The value of $\sin(____)$ at the unit circle when the angle is $\frac{\pi}{2}$ is 1.

Find the values of sine, cosine, and tangent of an angle theta when the angle is 225 degrees

To find the values of sine, cosine, and tangent of an angle $\theta$ when the angle is $225^\circ$, we use the unit circle:

The angle $225^\circ$ lies in the third quadrant, where sine and cosine are both negative:

$$\sin(225^\circ) = \sin(180^\circ + 45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$$

$$\cos(225^\circ) = \cos(180^\circ + 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}$$

$$\tan(225^\circ) = \frac{\sin(225^\circ)}{\cos(225^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$$