Unit Circle

Find the values of sine and cosine for an angle of 45 degrees in the unit circle

First, recall that in the unit circle, an angle of 45 degrees corresponds to $\frac{\pi}{4}$ radians.

From trigonometric identities:

$$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$

$$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$

Therefore, the values are:

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

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

Find the values of sin, cos, and tan at 45 degrees on the unit circle

To find the values of $\sin$, $\cos$, and $\tan$ at $45^\circ$ on the unit circle, we start by noting that $45^\circ$ is the same as $\frac{\pi}{4}$ radians.

The coordinates of the point on the unit circle at $\frac{\pi}{4}$ radians are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. Thus,

$$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$

$$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$

$$\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

Therefore, the values are:

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

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

$$\tan 45^\circ = 1$$

Find the cosine of a point on the unit circle in the complex plane

Given a point on the unit circle in the complex plane, represented by the complex number $z = e^{i\theta}$, determine the value of $\cos(\theta)$.

Since $z = e^{i\theta}$, we know that:

$$z = \cos(\theta) + i\sin(\theta)$$

Thus, the real part of $z$ is $\cos(\theta)$. Therefore, the value of $\cos(\theta)$ is simply the real part of $z$.

Hence, if $z = e^{i\theta} = \cos(\theta) + i\sin(\theta)$, then $\cos(\theta) = \text{Re}(z)$.

Find the exact values of sine, cosine, and tangent for the angle that corresponds to the point where the terminal side of angle θ intersects the unit circle at (cosθ, sinθ) Given that θ is in the fourth quadrant and the point on the unit circle is (1/2,

Given that $\theta$ is in the fourth quadrant and the point on the unit circle is $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$, we can find the exact values of $\sin\theta$, $\cos\theta$, and $\tan\theta$.

First, we recognize that $(\cos\theta, \sin\theta)$ directly gives us the cosine and sine values:

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

$$ \sin\theta = -\frac{\sqrt{3}}{2} $$

To find $\tan\theta$, we use the identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$:

$$ \tan\theta = \frac{ -\frac{\sqrt{3}}{2} }{ \frac{1}{2} } $$

$$ \tan\theta = -\sqrt{3} $$

Therefore, the values are:

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

$$ \sin\theta = -\frac{\sqrt{3}}{2} $$

$$ \tan\theta = -\sqrt{3} $$

Find the sine, cosine, and tangent of 45 degrees on the unit circle

We know that at $45^\circ$, the coordinates on the unit circle are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Therefore,

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

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

To find $\tan 45^\circ$, we use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$:

$$\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

Find the sine and cosine of the angle θ when it equals π/4 on the unit circle

To find the sine and cosine of the angle $\theta = \frac{\pi}{4}$ on the unit circle, we use the coordinates of the point where the terminal side of the angle intersects the unit circle.

The unit circle has a radius of 1, and for $\theta = \frac{\pi}{4}$, the coordinates are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore,

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

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

Find the Cartesian coordinates of a point on the unit circle when given an angle and a trigonometric function value

Given the angle $\theta = \frac{7\pi}{6}$ on the unit circle, find the Cartesian coordinates $ (x, y) $ for the corresponding point.

Since the unit circle has a radius of 1, we use the trigonometric identities for sine and cosine:

$$ x = \cos(\theta) $$

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

For $\theta = \frac{7\pi}{6}$:

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

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

Therefore, the Cartesian coordinates are:

$$ (x, y) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) $$

Find the sine, cosine, and tangent of a point on the unit circle

For the point on the unit circle corresponding to the angle $\theta = \frac{\pi}{4}$, find the sine, cosine, and tangent.

Step 1: Recognize that on the unit circle, the radius is 1.

Step 2: Use the angle $\theta = \frac{\pi}{4}$.

Step 3: Find sine and cosine for $\frac{\pi}{4}$. Since $\frac{\pi}{4} = 45^\circ$, $\sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$.

Step 4: Calculate tangent using $\tan \theta = \frac{\sin \theta}{\cos \theta} = 1$.

Answers:

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

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

$$\tan \left( \frac{\pi}{4} \right) = 1$$

Find the value of sec(θ) when θ = π/4 on the unit circle

To find the value of $ \sec(\theta) $ when $ \theta = \frac{\pi}{4} $ on the unit circle, we first recall that $ \sec(\theta) = \frac{1}{\cos(\theta)} $.

At $ \theta = \frac{\pi}{4} $, the cosine of $ \theta $ is $ \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $.

Therefore,

$$ \sec(\frac{\pi}{4}) = \frac{1}{\cos(\frac{\pi}{4})} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} $$

Given a point on the unit circle at $\theta = \frac{5\pi}{6}$, find the coordinates of this point and determine the angle in degrees Additionally, use the graphing calculator TI-Nspire to visualize the unit circle and verify the coordinates

To solve the problem, follow these steps:

1. Identify the coordinates of the point on the unit circle at $\theta = \frac{5\pi}{6}$.

The coordinates can be determined using the unit circle definitions: $$\left(\cos \theta, \sin \theta \right)$$.

2. Calculate the coordinates:

$$\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$$

$$\sin \frac{5\pi}{6} = \frac{1}{2}$$

So, the coordinates are $$\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$$.

3. Convert the angle to degrees:

$$\theta = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^{\circ}$$

4. Verify using TI-Nspire:

– Open the graphing calculator TI-Nspire.

– Plot the unit circle.

– Add a point at the angle $\theta = \frac{5\pi}{6}$ and verify the coordinates.

Final Answer: The coordinates are $$\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$$ and the angle is $150^{\circ}$.