Math

Find the sine and cosine of the angle where the terminal side intersects the unit circle at the point (-1/2, sqrt(3)/2)

To find the sine and cosine of the angle whose terminal side intersects the unit circle at the point $ \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) $, we start by identifying the coordinates of the point on the unit circle.

The x-coordinate, $ x = -\frac{1}{2} $, represents the cosine of the angle.

The y-coordinate, $ y = \frac{\sqrt{3}}{2} $, represents the sine of the angle.

Therefore, the cosine of the angle is:

$$ \cos(\theta) = -\frac{1}{2} $$

And the sine of the angle is:

$$ \sin(\theta) = \frac{\sqrt{3}}{2} $$

Find the angle in the unit circle

Given a point on the unit circle, find the angle such that $\sin(\theta) = \frac{\sqrt{3}}{2}$ and $\cos(\theta) = \frac{1}{2}$.

First, recognize that the coordinates given correspond to the point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle.

This point is in the first quadrant, where all trigonometric functions are positive.

The angle $\theta$ which satisfies this condition is $\theta = \frac{\pi}{3}$.

Therefore, the angle is $$\theta = \frac{\pi}{3}$$.

What is the tangent of 45 degrees on the unit circle?

The tangent of an angle in the unit circle is given by $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $.

For $ \theta = 45^{\circ} $ or $ \theta = \frac{\pi}{4} $ rad:

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

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

Therefore,

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

Find the values of sin, cos, and tan for 45 degrees

To find the values of $\sin$, $\cos$, and $\tan$ for $45^\circ$, we use the unit circle.

For $45^\circ$:

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

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

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

Thus,

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

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

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

Find the value of sec(π/3) on the unit circle

To find $\sec(\frac{\pi}{3})$, we first need to find $\cos(\frac{\pi}{3})$ since $\sec(\theta) = \frac{1}{\cos(\theta)}$.

On the unit circle, for $\theta = \frac{\pi}{3}$, we have $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

Therefore, $\sec(\frac{\pi}{3}) = \frac{1}{\cos(\frac{\pi}{3})} = \frac{1}{\frac{1}{2}} = 2$.

So, $\sec(\frac{\pi}{3}) = 2$.

Find the sine, cosine, and tangent values of the angle π/4 on the unit circle

To find the sine, cosine, and tangent values of the angle \( \frac{\pi}{4} \) on the unit circle, we need to recall the coordinates of the corresponding point on the unit circle. The coordinates of the point corresponding to the angle \( \frac{\pi}{4} \) are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

The sine value is the y-coordinate: $$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

The cosine value is the x-coordinate: $$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

The tangent value is the ratio of the sine and cosine: $$ \tan \left( \frac{\pi}{4} \right) = \frac{ \sin \left( \frac{\pi}{4} \right) }{ \cos \left( \frac{\pi}{4} \right) } = \frac{ \frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } = 1 $$

Find the sine, cosine, and tangent values for the angle $\frac{\pi}{4}$ on the unit circle

To find the sine, cosine, and tangent values for the angle $\frac{\pi}{4}$ on the unit circle, we first recognize that $\frac{\pi}{4}$ is a standard angle.

The coordinates for this angle on the unit circle are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore, the sine value is the y-coordinate:

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

The cosine value is the x-coordinate:

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

The tangent value is the ratio of the sine to the cosine:

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

Find the angle where the tangent is equal to 1/√3 on the unit circle

To find the angle where the tangent is equal to \( \frac{1}{\sqrt{3}} \) on the unit circle, we need to find the angles θ that satisfy this condition.

From trigonometric identities, we know that:

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

Given:

$$\tan(\theta) = \frac{1}{\sqrt{3}}$$

We recognize that:

$$\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}$$

Since the tangent function has a period of \( \pi \), the general solution for θ is:

$$\theta = \frac{\pi}{6} + k\pi\ (k \in \mathbb{Z})$$

Find all possible equations for circles on the unit circle

The equation of a unit circle is:

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

Any circle equation that lies on the unit circle must satisfy this equation. Therefore, an example of such an equation is:

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

which indicates the circle with radius 1 centered at the origin.

Find the sine and cosine of 45 degrees using the unit circle

To find the sine and cosine of $45^{\circ}$ using the unit circle, we need to locate $45^{\circ}$ on the unit circle chart.

The coordinates of the point where the $45^{\circ}$ angle intersects the unit circle are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Thus, the sine of $45^{\circ}$ is the y-coordinate:

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

And the cosine of $45^{\circ}$ is the x-coordinate:

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