Math

Find the sine, cosine, and tangent values at 45 degrees using the unit circle

To find the sine, cosine, and tangent values at $45^{\circ}$ (or $\frac{\pi}{4}$ radians) using the unit circle, we look at the coordinates of the corresponding point on the circle.

On the unit circle, at $45^{\circ}$, the coordinates are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. The x-coordinate represents the cosine value, and the y-coordinate represents the sine value.

Thus,

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

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

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

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

So, the values are:

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

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

$$ \tan(45^{\circ}) = 1 $$

Given a point on the unit circle with coordinates \((x, y)\), if the point corresponds to an angle \(\theta\) in standard position, find the angle \(\theta\) if \(x = -\frac{1}{2}\) State your answer in radians

Given the point on the unit circle with coordinates $(x, y)$, we need to find $\theta$ if $x = -\frac{1}{2}$.

Since $x = -\frac{1}{2}$ on the unit circle, we can use the cosine function to find the angle. So, $\cos(\theta) = -\frac{1}{2}$.

The angles that satisfy this equation are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$ in the interval $[0, 2\pi)$.

Hence, the angles $\theta$ corresponding to $x = -\frac{1}{2}$ are:

$$ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} $$

What is the cosine and sine value of π/3 on the flipped unit circle?

To find the cosine and sine values of $$\frac{\pi}{3}$$ on the flipped unit circle, we start by recalling the standard unit circle values.

On the standard unit circle,

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

and

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

When flipping the unit circle over the x-axis, the sine value changes its sign:

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

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

Find the values of cos(θ) on the unit circle

Consider the unit circle where the radius is 1. Identify the angles $\theta$ where $\cos(\theta) = \frac{1}{2}$.

Step 1: Recall the unit circle and the corresponding cosine values for common angles.

Step 2: Evaluate the cosine values: $\cos(60^\circ) = \frac{1}{2}$ and $\cos(300^\circ) = \frac{1}{2}$.

Step 3: Convert these angles to radians: $60^\circ = \frac{\pi}{3}$ and $300^\circ = \frac{5\pi}{3}$.

Therefore, the values of $\theta$ where $\cos(\theta) = \frac{1}{2}$ are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.

Find the cosecant of an angle at 30 degrees on the unit circle

The unit circle value for sine at 30 degrees is $\frac{1}{2}$. The cosecant is the reciprocal of sine.

$$ \csc(30^{\circ}) = \frac{1}{\sin(30^{\circ})} = \frac{1}{\frac{1}{2}} = 2 $$

So, the cosecant of 30 degrees is 2.

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

To find the sine, cosine, and tangent of a $45^\circ$ angle, we start by remembering that on the unit circle:

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

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

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

Therefore, the sine, cosine, and tangent of $45^\circ$ are $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$, and $1$ respectively.

Find the Angle and Length of the Arc in Different Quadrants of the Unit Circle

Given a unit circle centered at the origin, consider a point that makes an angle of \( \theta = \frac{7\pi}{6} \) radians with the positive x-axis. Find the quadrant in which this point lies and the length of the arc from the point where \( \theta = 0 \) to this point.

Solution:

1. Determine the quadrant:

The angle \( \theta = \frac{7\pi}{6} \) can be converted to degrees:

$$ \theta = \frac{7\pi}{6} \times \frac{180}{\pi} = 210^{\circ} $$

Since 210 degrees lies between 180 and 270 degrees, the point is in the third quadrant.

2. Calculate the length of the arc:

The length of an arc (s) in a unit circle is given by:

$$ s = r \cdot \theta $$

Because it is a unit circle (\( r = 1 \)):

$$ s = 1 \cdot \frac{7\pi}{6} = \frac{7\pi}{6} $$

Thus, the length of the arc is \( \frac{7\pi}{6} \) units.

Find the coordinates of the point at 45 degrees on the unit circle

The unit circle has a radius of 1. At an angle of $45^\circ$, the coordinates of the point can be found using the cosine and sine functions:

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

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

Therefore, the coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Find all circle equations that lie on the unit circle

The unit circle is defined by the equation:

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

To find all circles that lie on the unit circle, we consider the general equation of a circle:

$$ (x – a)^2 + (y – b)^2 = r^2 $$

For the circle to lie on the unit circle, the radius of this circle must be zero, as any larger radius would extend beyond the unit circle. Therefore:

$$ r = 0 $$

Thus, the equation simplifies to a point:

$$ (x – a)^2 + (y – b)^2 = 0 $$

Expanding this gives:

$$ x = a, y = b $$

But since it must lie on the unit circle:

$$ a^2 + b^2 = 1 $$

So, all such points (a, b) lie on the unit circle.

Therefore, the equations of all circles on the unit circle are:

$$ (x – a)^2 + (y – b)^2 = 0 $$ where $$ a^2 + b^2 = 1 $$

Find the coordinates of the point on the unit circle at an angle of 45 degrees

To find the coordinates of the point on the unit circle at an angle of $45^\circ$, we use the fact that the unit circle has a radius of 1.

The coordinates for an angle $\theta$ in radians can be given by $(\cos \theta, \sin \theta)$.

Converting $45^\circ$ to radians:

$$\theta = 45^\circ = \frac{45 \pi}{180} = \frac{\pi}{4}$$

Therefore, the coordinates are:

$$ (\cos \frac{\pi}{4}, \sin \frac{\pi}{4}) $$

Since $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, the coordinates are:

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