Unit Circle

Find the values of tan(θ) at various angles and verify using the unit circle

To find the values of $ \tan(\theta) $ at various angles and verify using the unit circle, we consider the following angles: $ \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $

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

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

2. For $ \theta = \frac{3\pi}{4} $:

$$ \tan(\frac{3\pi}{4}) = -1 $$

3. For $ \theta = \frac{5\pi}{4} $:

$$ \tan(\frac{5\pi}{4}) = 1 $$

4. For $ \theta = \frac{7\pi}{4} $:

$$ \tan(\frac{7\pi}{4}) = -1 $$

Verification: Using the unit circle, we observe that at these angles, the tangent value is consistent with the coordinates (x, y) where $ \tan(\theta) = \frac{y}{x} $.

Determine the coordinates of a point on the unit circle where the tangent line has a slope of 3/4

To determine the coordinates of a point on the unit circle where the tangent line has a slope of $\frac{3}{4}$, we start with the equation of the unit circle:

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

The slope of the tangent line at a point $(x, y)$ on the circle can be found by differentiating implicitly:

$$2x + 2y\frac{dy}{dx} = 0$$

Solving for $\frac{dy}{dx}$, we get:

$$\frac{dy}{dx} = -\frac{x}{y}$$

We need the slope to equal $\frac{3}{4}$:

$$-\frac{x}{y} = \frac{3}{4}$$

This implies:

$$y = -\frac{4x}{3}$$

Substitute $y = -\frac{4x}{3}$ back into the unit circle equation:

$$x^2 + \left(-\frac{4x}{3}\right)^2 = 1$$

$$x^2 + \frac{16x^2}{9} = 1$$

$$\frac{25x^2}{9} = 1$$

$$x^2 = \frac{9}{25}$$

$$x = \pm \frac{3}{5}$$

Substitute $x$ back into $y = -\frac{4x}{3}$:

$$y = \mp \frac{4 \cdot \frac{3}{5}}{3} = \mp \frac{4}{5}$$

The coordinates are:

$$\left(\frac{3}{5}, -\frac{4}{5}\right)$$ and $$\left(-\frac{3}{5}, \frac{4}{5}\right)$$

Determine the angle in radians of the point on the unit circle in the first quadrant with an x-coordinate of 1/2

To find the angle in radians with an $x$-coordinate of $\frac{1}{2}$ in the first quadrant, we use the unit circle definition of cosine.

For $\cos(\theta) = \frac{1}{2}$, the corresponding angle is:

$$ \theta = \frac{\pi}{3} $$

Find the value of sec(θ) for θ in the unit circle

To find the value of $ \sec(\theta) $ for $ \theta $ in the unit circle, we need to recall the definition of secant. The secant function is the reciprocal of the cosine function:

$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$

Given that $ \theta $ is an angle in the unit circle, let’s consider $ \theta = \frac{\pi}{4} $ as an example. For this angle:

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

Thus,

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

Find the secant line to the unit circle that is equidistant from the x-axis

To find the secant line to the unit circle that is equidistant from the $x$-axis, we use the equation of the unit circle

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

and the general equation of a line

$$ y = mx + b $$

Since the secant line is equidistant from the $x$-axis, the $y$-intercept $b$ must satisfy the condition that the distances from $b$ to the points of intersection with the circle are equal. So, we solve:

Substitute $y = mx + b$ into the circle

Determine the exact values of sine and cosine for the angle π/4 using the unit circle

To find the exact values of sine and cosine for the angle $ \frac{\pi}{4} $, we use the unit circle.

For $ \theta = \frac{\pi}{4} $, the coordinates on the unit circle are:

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

Since $ \frac{\pi}{4} $ is an angle in the first quadrant where sine and cosine values are positive, we use the 45-degree reference angle values. We have:

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

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

Thus, the exact values are:

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

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

Identify the coordinates of specific angles on the unit circle

To find the coordinates of specific angles on the unit circle, remember that the unit circle has a radius of 1.

For the angle $\theta = \frac{\pi}{4}$, the coordinates are:

$$(\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$$

Find the area of a sector in a unit circle with a central angle of θ radians

The formula to find the area of a sector in a unit circle is:

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

where $ \theta $ is the central angle in radians.

For example, if $ \theta = \frac{\pi}{4} $:

$$ A = \frac{1}{2} \times \frac{\pi}{4} = \frac{\pi}{8} $$

Identify the quadrant in which the angle lies

To identify the quadrant in which the angle $ \theta $ lies, follow these steps:

1. If $ 0 \leq \theta < \frac{\pi}{2} $, then the angle is in the first quadrant.

2. If $ \frac{\pi}{2} \leq \theta < \pi $, then the angle is in the second quadrant.

3. If $ \pi \leq \theta < \frac{3\pi}{2} $, then the angle is in the third quadrant.

4. If $ \frac{3\pi}{2} \leq \theta < 2\pi $, then the angle is in the fourth quadrant.

Find the value of cos(π/3)

The value of $ \cos\left( \frac{\pi}{3} \right) $ can be found using the unit circle. The angle $ \frac{\pi}{3} $ corresponds to 60 degrees. On the unit circle, the coordinates for the angle 60 degrees are:

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

Therefore, the value of $ \cos\left( \frac{\pi}{3} \right) $ is:

$$ \frac{1}{2} $$