Problem 47 - Precalculus: A Concise Course

Question

Find the reference angle \(\boldsymbol{\theta}^{\prime}\), and sketch \(\boldsymbol{\theta}\) and \(\boldsymbol{\theta}^{\prime}\) in standard position. $$ \theta=-125^{\circ} $$

Step-by-Step Solution

Answer

The reference angle \(\theta^{\prime}\) for \(\theta = -125^\circ\) is \(55^\circ\). They have been sketched on the standard position with the given procedure.

Step 1: Understanding the Angle Orientation

In standard position, positive angles are measured counterclockwise from the positive X-axis whilst negative angles are measured clockwise from the positive X-axis. Therefore, the given angle \(\theta = -125^\circ\) was measured clockwise from the positive X-axis.

Step 2: Calculate for Reference Angle

A reference angle \(\theta^{\prime}\) of a given angle \(\theta\) is the smallest angle to the X-axis. It is always positive and less than or equal to \(90^\circ\). To find \(\theta^{\prime}\), add the absolute of \(\theta\) with \(180^\circ\) if \(\theta\) is less than \(-90^\circ\), subtract \(180^\circ\) from \(\theta\) if \(\theta\) is greater than \(90^\circ\) and finally, take the absolute of \(\theta\) if \(-90^\circ< \theta < 90^\circ\). Since \(\theta = -125^\circ\) is less than \(-90^\circ\), \(\theta^{\prime} = 180^\circ - |-125^\circ| = 55^\circ\).

Step 3: Sketching the Angle and Reference Angle

Using a compass, draw a circle to represent the path of the angle. To sketch \(\theta\), start from the positive X-axis (indicated as \(0^\circ\)) then move clockwise \(-125^\circ\) reaching into the second quadrant. The reference angle \(\theta^{\prime} = 55^\circ\) is the acute angle formed between the terminal side of \(\theta = -125^\circ\) and the x-axis in the second quadrant.

Key Concepts

Standard Position Angles

Angles in trigonometry are often described in what's called standard position. This means that an angle's initial side lies along the positive x-axis and its vertex is located at the origin of a Cartesian coordinate system. Visualizing angles in standard position is crucial for understanding their measurements in trigonometry.

For an angle to be in standard position, it does not matter whether it opens counterclockwise or clockwise; what is essential to note is that counterclockwise opening angles are considered positive, while clockwise opening ones are negative. This distinction is significant when it comes to calculating reference angles and analyzing functions such as sine, cosine, and tangent.

For example, the angle in the exercise, \( \theta = -125^\text{o} \) is measured by moving clockwise (negative direction) from the positive x-axis, starting at the 0-degree mark and swinging to the -125-degree location. Recognizing this aspect of standard position angles is vital for understanding the rest of the steps that involve sketching and finding reference angles.

Measuring Angles in Trigonometry

The measurement of angles in trigonometry follows certain conventions. One of these is the size of the reference angle, always an acute angle between 0 and 90 degrees, regardless of the original angle's size or quadrant. The reference angle helps us relate different angles to the basic right-angled triangle properties.

Reference angles can be found by considering their absolute distance to the nearest x-axis, making sure it's always going to be a positive value. If the original angle is negative and greater than -180 degrees, we find the reference angle by subtracting the angle from 180 degrees. Conversely, for angles above 90 degrees, we find the reference angle by subtracting the angle from 180 degrees as well.

In our exercise, the angle \( \theta = -125^\text{o} \) dictates that you take the absolute value and subtract it from 180 degrees, resulting in the reference angle \( \theta' = 55^\text{o} \) which lies within the required range of a reference angle.

Sketching Angles

The skill of sketching angles helps provide a visual representation and understanding of how angles function within a coordinate system. First, we draw a circle to represent the path that the angle follows from the start point. To sketch an angle in standard position, one must begin at the positive x-axis and move in the proper direction according to the angle's positivity or negativity.

To clearly sketch the angle \( \theta = -125^\text{o} \) the starting point is along the positive x-axis, then we move 125 degrees clockwise to represent the negative angle. For the reference angle \( \theta' = 55^\text{o} \), it is the smallest angle formed between the terminal side of the sketched angle \( \theta \) and the x-axis. It is important to notice that the reference angle is always positive and is drawn from the terminal side of \( \theta \) to the x-axis, whether moving in the positive or negative direction.

This visual context is crucial for grasping the relationship between the reference angle and the standard position angle, playing a fundamental role in the comprehension of many trigonometric concepts.