Math

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 values of x that satisfy sin(x) = 05 on the unit circle

To find the values of $ \sin(x) = 0.5 $ on the unit circle, we need to determine the angles where the sine function equals 0.5. From the unit circle, we know that:

$$ \sin(\frac{\pi}{6}) = 0.5 $$

$$ \sin(\frac{5\pi}{6}) = 0.5 $$

So the values of $x$ are:

$$ x = \frac{\pi}{6} + 2k\pi \text{ or } x = \frac{5\pi}{6} + 2k\pi $$

where $k$ is any integer.

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:

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

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

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

Find the exact values of sin(3pi/4) and cos(3pi/4) using the unit circle

To find the exact values of $ \sin\left(\frac{3\pi}{4}\right) $ and $ \cos\left(\frac{3\pi}{4}\right) $, we use the unit circle:

$$ \sin\left(\frac{3\pi}{4}\right) $$ is located in the second quadrant, where the sine value is positive and the corresponding reference angle is $ \frac{\pi}{4} $. Therefore,

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

Similarly, $$ \cos\left(\frac{3\pi}{4}\right) $$ is also in the second quadrant, where the cosine value is negative:

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

Calculate the sine and cosine values at 45 degrees using the unit circle

To calculate the sine and cosine values at $45^\circ$ using the unit circle, we recognize that a $45^\circ$ angle forms an isosceles right triangle in the unit circle.

The coordinates of the point where the angle intersects the unit circle are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Therefore, the values are:

$$\sin(45^\circ) = \frac{\sqrt{2}}{2}$$

$$\cos(45^\circ) = \frac{\sqrt{2}}{2}$$

Find the angle corresponding to the point (1/2, -√3/2) on the unit circle

To find the angle that corresponds to the point $ \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) $ on the unit circle, we look at the coordinates.

The x-coordinate is $ \frac{1}{2} $ and the y-coordinate is $ -\frac{\sqrt{3}}{2} $. These values correspond to an angle in the fourth quadrant.

The reference angle with these coordinates is $ \frac{\pi}{3} $ because:

$$ \cos \theta = \frac{1}{2} \text{ and } \sin \theta = -\frac{\sqrt{3}}{2} $$

Since the angle is in the fourth quadrant, the actual angle is:

$$ \theta = 2\pi – \frac{\pi}{3} = \frac{5\pi}{3} $$