Question

a) Explain, with reference to the unit circle, what the interval \(-2 \pi \leq \theta<4 \pi\) represents. b) Use your explanation to determine all values for \(\theta\) in the interval \(-2 \pi \leq \theta<4 \pi\) such that \(\mathrm{P}(\theta)=\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right).\) c) How do your answers relate to the word "coterminal"?

Short answer

\theta = \frac{2 \pi}{3}, \frac{4 \pi}{3}, -\frac{4 \pi}{3}, -\frac{10 \pi}{3}, \frac{8 \pi}{3}

Step-by-step solution

Understand the Unit Circle and Interval \(-2 \pi \leq \theta < 4 \pi\))

The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Angles in the unit circle are measured in radians, starting from the positive x-axis. The interval \(-2 \pi \leq \theta < 4 \pi\)) includes all angles from two full rotations counter-clockwise (\(\-2 \pi\) radians) to four full rotations counter-clockwise (\(4 \pi\) radians). This interval covers a complete range of angles, including multiple full rotations.

Identify all Equivalent Angles in the Given Interval

To find all angles \(\theta \) within \(-2 \pi \leq \theta < 4 \pi\)) that correspond to a specific point on the unit circle, recognize that angles differing by \(2 \pi\) are coterminal. Coterminal angles share the same terminal side on the unit circle.

Find Angles Corresponding to the Point \(\left(\-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)

The point \(\left(\-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) corresponds to \(\theta = \frac{2 \pi}{3}\) and \(\theta = \frac{8 \pi}{3}\) in the first full rotation. To find all such \(\theta\) within the interval \(\left(\left(2k+1\) \frac{2\pi}{3} | k \in \mathbb{Z}, \-2 \pi \leq \theta < 4 \pi\right)\right), calculate additional angles by adding or subtracting \(2\pi\) from these primary angles.

Calculate All Relevant Angles

Using \(\theta = \frac{2 \pi}{3}\) and \(\theta = \frac{4 \pi}{3}\left(\frac{8 \pi}{3} = 2 \pi + \frac{2 \pi}{3}\right)\), include all coterminal angles: \left(\text{\=repeat all angles respected to interval length sets additional calculations if needed including middle of interval \region matched counts}\right).

Define Coterminal Angles and Relate to Found Solutions

Coterminal angles are those that differ by an integer multiple of \(2 \pi\). Solutions within the interval \-2 \pi \leq \theta < 4 \pi with ends respected relate as cos and sin repetition results.

Key concepts

radian measure

The concept of radian measure is integral to understanding angles on the unit circle. A radian is a way to measure angles based on the radius of the circle. Specifically, one radian is the angle created when the arc length is equal to the radius of the circle.
Most students are more familiar with degrees, but radians offer a more natural measure for many mathematical purposes. Here, we focus on the interval \(-2 \pi \leq \theta < 4 \pi\), which means we're considering all angles from \(-2 \pi\) (two complete rotations in the clockwise direction) to \(4 \pi\) (four complete rotations in the counter-clockwise direction).
In the context of the unit circle, this range includes multiple full circles. Understanding how angles operate in this interval is crucial for working with periodic functions such as sine and cosine, which repeat every \(2\pi\).

coterminal angles

Coterminal angles are angles that share the same terminal side on the unit circle. They are essentially the same angle but represented differently by adding or subtracting full circle rotations (multiples of \(2\pi\)).
If you have an angle \(\theta\), any angle of the form \(\theta + 2k\pi\), where \(k\) is an integer, is coterminal with \(\theta\). This happens because each time you add \(2\pi\) (a full circle), you end up at the same position on the unit circle.
In the given problem, they ask for angles within \(-2 \pi \leq \theta < 4 \pi\) that correspond to the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). To solve this, find a primary angle like \(\frac{2\pi}{3}\) that hits the point, then add and subtract multiples of \(2\pi\) to get all coterminal angles within the given range. This gives an array of angles such as \(\frac{2\pi}{3}, \frac{8\pi}{3}, -\frac{4\pi}{3}\).

trigonometric functions

Trigonometric functions like sine, cosine, and tangent describe relationships between angles and sides in right triangles, but they also extend to any angles through the unit circle.
The unit circle helps to visualize these functions. Every point \( (x, y) \) on the unit circle corresponds to an angle \(\theta\), with \(x\) being the cosine and \(y\) being the sine of the angle.
For the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), its coordinates tell us that \(\cos(\theta) = -\frac{1}{2}\) and \(\sin(\theta) = \frac{\sqrt{3}}{2}\). These values are associated with the angle \(\theta = \frac{2\pi}{3}\) and its coterminal angles. Understanding these functions and how they relate to the unit circle is key to solving many problems in trigonometry.
Remember, trigonometric functions repeat periodically, which aligns with our work on coterminal angles. This means knowing one angle and its reference on the unit circle lets us know all other angles sharing the same trigonometric values.