Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-\frac{11 \pi}{5}$$

Step-by-Step Solution

Answer

The reference angle \(\theta'\) for given \(\theta=-\frac{11 \pi}{5}\) is \(\frac{4 \pi}{5}\) or in degrees it's 144°. It lies in the third quadrant.

Step 1: Convert to equivalent positive angle

Since \(-\frac{11 \pi}{5}\) is negative, add full circle rotations (2\(\pi\)) until you have a positive angle. Adding \(2\pi\) to \(-\frac{11 \pi}{5}\) gives \(-\frac{\pi}{5}\). You'll need to add \(2\pi\) one more time to get a positive result, which is \(\frac{9 \pi}{5}\).

Step 2: Identify the quadrant

The angle \(\frac{9 \pi}{5}\) lies in the third quadrant (between \(\pi\) and \(\frac{3 \pi}{2}\)) since \(\frac{9 \pi}{5}\) is greater than \(\pi\) and less than \(\frac{3 \pi}{2}\).

Step 3: Calculate reference angle

To find the reference angle \(\theta'\), subtract \(\pi\) from the angle when it's in the third quadrant. So \(theta' = \frac{9 \pi}{5} - \pi = \frac{4 \pi}{5}\).

Step 4: Graph it!

Next, sketch the angle \(\theta=-\frac{11 \pi}{5}\) in standard position, a simple line rotating about the origin beginning from the positive X-axis. Since we know it lies in the third quadrant, the rotation will be of 180° + whatever the reference angle is. In this case, it's 180° + reference angle 144° ( \(\frac{4 \pi}{5} \times \frac{180°}{\pi} = 144° \) ). Label the reference angle \(\theta'\).

Key Concepts

Reference Angle

A reference angle is a useful tool in trigonometry as it helps us find equivalent angles that are easier to work with. The reference angle for any given angle is the smallest angle to the X-axis, which is always positive and between 0 and 90 degrees, or 0 and \( \frac{\pi}{2} \) radians. By finding the reference angle, we can solve many trigonometric problems more conveniently.

• To find the reference angle, first determine which quadrant the terminal side of the given angle falls in. Angles in the third quadrant, for example, require subtracting \( \pi \) radians from the given angle.
• In our problem, the angle \( \frac{9\pi}{5} \) lies in the third quadrant. Thus, the reference angle \( \theta' \) is given by \( \theta' = \frac{9\pi}{5} - \pi = \frac{4\pi}{5} \).

This calculation simplifies the problem, allowing us to convert this angle into degrees if needed or directly use it in trigonometric functions.

Standard Position

When discussing angles in trigonometry, understanding 'standard position' is vital. An angle is said to be in standard position when its vertex is located at the origin of a coordinate plane, with its initial side aligned along the positive X-axis. The terminal side will then determine the angle's measure by rotating counterclockwise or clockwise.

• If the angle is positive, it rotates counterclockwise from the X-axis. With a negative angle, the rotation is clockwise.
• For the angle \( \theta = -\frac{11\pi}{5} \), observe that it starts on the positive X-axis and rotates clockwise, continuing into the third quadrant after converting it to a positive angle \( \frac{9\pi}{5} \).

Sketching the standard position offers a visual insight into the angle, making it easier to grasp its size and direction.

Radians to Degrees

Converting between radians and degrees is essential for comparing angles, especially when visualizing or performing specific calculations. Radians and degrees are two units of measuring angles, with \( 360^{\circ} = 2\pi \) radians.

• To convert from radians to degrees, use the formula: \( \text{Degrees} = \text{Radians} \times \frac{180^{\circ}}{\pi} \).
• In our example, to convert the reference angle \( \frac{4\pi}{5} \) radians to degrees, multiply: \( \frac{4\pi}{5} \times \frac{180^{\circ}}{\pi} = 144^{\circ} \).

Understanding this conversion helps align angle measures with everyday contexts and can prove beneficial in fields like engineering and physics, where degrees are more commonly used.