Homework

Find the coordinates of a point on the unit circle at an angle of π/6

To find the coordinates of a point on the unit circle at an angle of $ \frac{\pi}{6} $, we use the fact that the coordinates are given by $ ( \cos(\theta), \sin(\theta)) $ where $ \theta $ is the angle:

$$ \theta = \frac{\pi}{6} $$

Therefore:

$$ \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} $$

$$ \sin(\frac{\pi}{6}) = \frac{1}{2} $$

The coordinates are:

$$ \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$

Calculate the area of the shaded region in a unit circle with central angles

Let’s calculate the area of the shaded region in a unit circle with central angles $ \theta $ and $ \alpha $.

The area of a sector of a circle is given by:

$$ A = \frac{1}{2} r^2 \theta $$

For a unit circle, $ r = 1 $, so the above formula simplifies to:

$$ A = \frac{1}{2} \theta $$

The area of the shaded region is then the difference between two sector areas:

$$ A_{shaded} = \frac{1}{2} (\theta – \alpha) $$

Find the coordinates on the unit circle for the angles

Find the coordinates on the unit circle for the angle $ \theta = \frac{\pi}{4} $:

The coordinates are given by $ (\cos(\theta), \sin(\theta)) $.

For $ \theta = \frac{\pi}{4} $:

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

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

The coordinates are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

Find the tangent values at 0, π/4, and π/3 on the unit circle

To find the tangent values at points $0$, $\frac{\pi}{4}$, and $\frac{\pi}{3}$ on the unit circle, we use the tangent function $tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$:

1. At $\theta = 0$:

$$ \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 $$

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

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

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

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

Find the coordinates where $ \cos(\theta) = \sin(\theta) $ on the unit circle

To find the coordinates where $ \cos(\theta) = \sin(\theta) $ on the unit circle, we start from the equation:

$$ \cos(\theta) = \sin(\theta) $$

Since both cosine and sine are equal, we can express this as:

$$ \cos(\theta) = \sin(\theta) $$

Divide both sides by $ \cos(\theta) $:

$$1 = \tan(\theta) $$

This implies

$$ \theta = \frac{\pi}{4} + n\pi $$

for integer values of n. The corresponding coordinates on the unit circle are:

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

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:

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