Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-\frac{5 \pi}{3}$$

Step-by-Step Solution

Answer

The reference angle \(\theta'\) for the given angle \(\theta=-\frac{5 \pi}{3}\) is \(\frac{\pi}{3}\).

Step 1: Convert Negative Angle to Positive

The given angle is \(\theta=-\frac{5 \pi}{3}\). Let's first convert this negative angle into a positive angle. To do this, you can add \(2\pi\) (which is equivalent to a full circle) to the given angle until it becomes positive. So the equivalent positive angle is \(\theta_{positive}= -\frac{5 \pi}{3} + 2\pi = \frac{\pi}{3}\).

Step 2: Find the Reference Angle

Now, let's find the reference angle of \(\frac{\pi}{3}\). In the case of an angle measured in radians, the reference angle \(\theta'\) of an angle \(\theta\) which lies in the first quadrant can be found directly as \(\theta'=\theta\). Thus, the reference angle of \(\frac{\pi}{3}\) is \(\theta'=\frac{\pi}{3}\).

Step 3: Sketch the Angles

To help visualize this, it's useful to sketch the angles. On the unit circle, beginning from the positive x-axis, sketch counter-clockwise for the positive angle \(\frac{\pi}{3}\). Mark this angle as \(\theta\). Since \(\theta'\) is also \(\frac{\pi}{3}\), it will coincide with \(\theta\) on the sketch and should also be marked.

Key Concepts

Reference Angle

The reference angle is a key concept in trigonometry. It is the smallest angle that the terminal side of a given angle makes with the x-axis. The reference angle is always positive and is useful in simplifying trigonometric calculations.
When you find a reference angle, you're essentially looking for how much the angle "leans" from a nearest axis line.

• For quadrant I, the reference angle is the angle itself.
• For quadrant II, subtract the angle from \(\pi\).
• In quadrant III, subtract \(\pi\) from the angle.
• For quadrant IV, subtract the angle from \(2\pi\).

By understanding reference angles, you can easily determine trigonometric values by using more familiar angles in the first quadrant.

Standard Position

Angles in trigonometry are often drawn in standard position to provide a consistent way to measure them. An angle is in standard position when its vertex is at the origin, and its initial side lies along the positive x-axis.
The movement from this initial position can be counterclockwise or clockwise:

• Counterclockwise movement results in a positive angle.
• Clockwise movement results in a negative angle.

Our angle, \(-\frac{5\pi}{3}\), was initially negative, so it represents a clockwise movement. Converting it to a positive angle helps in sketching and calculation by following the typical counterclockwise path.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is fundamental in trigonometry because it helps define the sine, cosine, and tangent of an angle. Points on the unit circle correspond to angles, where:

• The x-coordinate is the cosine of the angle.
• The y-coordinate is the sine of the angle.

To visualize angles and their trigonometric values, sketch the angle starting from the positive x-axis. The position on the unit circle will provide the basis for calculating trigonometric functions. It's especially handy for visualizing angles such as \(\frac{\pi}{3}\).

Angle Conversion

Converting angles between different forms can simplify solving trigonometric problems. The most common conversions are between degrees and radians and adjusting negative angles.
To convert a negative angle to a positive one, you can add \(2\pi\) (in radians) or 360 degrees (in degrees) until the angle becomes positive. This process ensures that the angle fits within a standard 0 to \(2\pi\) or 0 to 360-degree range.
In our example, \(-\frac{5\pi}{3} + 2\pi = \frac{\pi}{3}\). This conversion is crucial for identifying the equivalent angle within one rotation of the circle, making the problem easier to solve.