Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=208^{\circ}$$

Step-by-Step Solution

Answer

The reference angle \(\theta^{'}\) for the angle of 208° is 28°. The reference angle is located in the third quadrant and is the acute angle formed by the terminal side of \(\theta\) and the x-axis.

Step 1: Identify the quadrant of the given angle

It is known that angle \(\theta=208^{\circ}\). Since the value of \(\theta\) is between 180° and 270°, the angle lies in the third quadrant.

Step 2: Calculate the reference angle

The reference angle \(\theta^{'}\) for an angle in the third quadrant is found by subtracting the angle from 180°. Hence, \(\theta^{'}= \theta - 180^{\circ} = 208^{\circ} - 180^{\circ} = 28^{\circ}\).

Step 3: Sketch the angle and the reference angle

Draw an angle of 208° in standard position wherein the initial side is along positive x-axis and the terminal side lies in third quadrant. Mark the acute angle of 28° starting from the terminal side toward negative x-axis, and label it as the reference angle \(\theta^{'}\).

Key Concepts

Standard Position

In the world of trigonometry, understanding the concept of an angle in standard position is crucial.
When we talk about an angle being in standard position, we mean that its vertex is located at the origin of a coordinate plane, and its initial side is aligned along the positive x-axis.
This standard placement allows us to consistently define and interpret angles.To build a clearer picture:

• The **initial side** is fixed and starts on the positive x-axis.
• The **terminal side** is where the angle ends after it sweeps from the initial side.
• Angles are measured counterclockwise from the initial side in degrees or radians for positive angles and clockwise for negative angles.

So, when you see \(\theta = 208^{\circ}\), it means that the terminal side has rotated 208 degrees from the initial side, moving counterclockwise. Keeping this consistent approach helps in determining not just positions but also reference angles and quadrants effectively.

Quadrants

The coordinate plane is divided into four regions called quadrants. Each quadrant represents an area where angles can lie, helping us determine the measures and signs of the trigonometric functions for those angles.
When we calculate where an angle ends up after being placed in standard position, identifying the correct quadrant it belongs to is essential.Here’s how quadrants are structured:

• **First Quadrant**: Between 0° and 90°. In this quadrant, all trigonometric functions are positive.
• **Second Quadrant**: Between 90° and 180°. Here, only sine and cosecant are positive.
• **Third Quadrant**: Between 180° and 270°. In this quadrant, tangent and cotangent are positive, which **\(\theta = 208^{\circ}\)** falls into.
• **Fourth Quadrant**: Between 270° and 360°. Here, cosine and secant are positive.

For \(\theta = 208^{\circ}\), since it's more than 180° but less than 270°, it comfortably resides in the third quadrant. Understanding this helps us anticipate how the standard position of the angle interacts with the axis at the specific rotational measure.

Acute Angle

The term "acute angle" refers to an angle that is less than 90°. In trigonometry, when dealing with reference angles, which is effectively what we calculate after identifying the quadrant, the acute nature of these angles shines through.A **reference angle** is essentially the smallest angle between the terminal side of a given angle and the x-axis.
These angles are always positive and because they are "acute," they help simplify calculations in the analysis of trigonometric functions.Here's a closer look:

• For angles in the **First Quadrant**, the reference angle is the same as the standard angle, \(\theta^{\prime} = \theta \).
• For **Second Quadrant**, the reference angle can be calculated by \(\theta^{\prime} = 180^{\circ} - \theta \).
• In the **Third Quadrant** like our example \(\theta = 208^{\circ}\), we find \(\theta^{\prime}\) by \(\theta^{\prime} = \theta - 180^{\circ} = 28^{\circ}\).
• For **Fourth Quadrant**, it's \(\theta^{\prime} = 360^{\circ} - \theta \).

For our specific problem, the reference angle \(\theta^{\prime}\) is 28°, ensuring it remains acute as expected. By using reference angles, we can often simplify complex trigonometric problems into more manageable forms.