Problem 47 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{11 \pi}{3} $$

Step-by-Step Solution

Answer

Reference angle: \(\frac{\pi}{3}\); Terminal point: \((\frac{1}{2}, \frac{\sqrt{3}}{2})\).

Step 1: Normalize the Angle

Since the angle is given as \(-\frac{11\pi}{3}\), first convert this angle to an equivalent angle between \(0\) and \(2\pi\). To do this, keep adding \(2\pi\) (equivalent to \(\frac{6\pi}{3}\)) until the angle is in the desired range. \[-\frac{11\pi}{3} + 2\pi = -\frac{11\pi}{3} + \frac{6\pi}{3} = -\frac{5\pi}{3}\]\[-\frac{5\pi}{3} + 2\pi = -\frac{5\pi}{3} + \frac{6\pi}{3} = \frac{\pi}{3}\]So an equivalent positive angle is \(\frac{\pi}{3}\).

Step 2: Determine the Reference Angle

The reference angle is the acute angle the terminal side makes with the x-axis. Since \( \frac{\pi}{3}\) is already less than \(\pi\), it is in the first quadrant. Thus, the reference angle is simply itself:Reference Angle = \( \frac{\pi}{3} \).

Step 3: Find the Terminal Point

The terminal point for an angle \( t \) on the unit circle can be found using the coordinates \( (\cos(t), \sin(t)) \). For \( t = \frac{\pi}{3} \), calculate:\( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)\( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)Thus, the terminal point determined by \( t = -\frac{11\pi}{3} \) is \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \).

Key Concepts

Reference Angle

The reference angle is a crucial concept when studying trigonometry. It represents the smallest angle formed between the terminal side of the original angle and the x-axis. Understanding this helps us work more easily with trigonometric functions, regardless of where the angle lies on the coordinate plane.
To find a reference angle, especially in radians, first ensure that your angle is within the interval of 0 to \(2\pi\). If the angle is negative or exceeds \(2\pi\), add or subtract \(2\pi\) as necessary until it's within this range. This conversion is always effective as it shifts the angle into a more manageable context. Here's how this happens with \(-\frac{11\pi}{3}\): converting it into the equivalent positive angle \(\frac{\pi}{3}\).

• An angle in the first quadrant (0 to \(\pi/2\)) has its reference angle equal to itself.
• In the second quadrant, the reference angle is \(\pi - \theta\).
• For the third quadrant, use \(\theta - \pi\).
• In the fourth quadrant, calculate \(2\pi - \theta\).

This concept simplifies complex trigonometric problems by providing an acute angle that is easily managed and understood.

Unit Circle

The Unit Circle is a foundational tool in trigonometry. With a radius of 1, it allows students to directly find the trigonometric values of any angle based on its position on the circle. It’s a staple in understanding how sine, cosine, and tangent functions work together.
Each angle on the unit circle corresponds to a specific point \((x, y)\) on the circumference, where:

• \(x = \cos(\theta)\)
• \(y = \sin(\theta)\)

The simplicity of the Unit Circle stems from its consistent radius, which makes calculating these trigonometric functions straightforward.
For example, for the angle \(\frac{\pi}{3}\), its position on the Unit Circle gives:

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

This means the specific terminal point determined by \(\frac{\pi}{3}\) on the Unit Circle is \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), offering a robust tool for solving various trigonometric problems.

Terminal Point

The terminal point of an angle is where its terminal side intersects the Unit Circle. Identifying this point is crucial for understanding the behavior and outcome of trigonometric functions associated with an angle.
To determine the terminal point for any angle, use its cosine and sine values as coordinates. So, for an angle \(t\) on the Unit Circle, the terminal point is at \((\cos(t), \sin(t))\).
In our case, when considering the angle \(-\frac{11\pi}{3}\), first convert it into a positive angle equivalent to \(\frac{\pi}{3}\). Using the Unit Circle, calculate:

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

Thus, the terminal point associated with the angle \(-\frac{11\pi}{3}\) is \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). This provides crucial insight into the angle's properties and its position on the Unit Circle.