Problem 72 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\cos \theta=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The possible values of \(\theta\) are \(120^{\circ}\) and \(240^{\circ}\).

Step 1: Understand the Range and Equation

We need to find all possible angles \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) for which \(\cos \theta = -\frac{1}{2}\).

Step 2: Determine General Solutions for Cosine

The cosine function is negative in the second and third quadrants. Cosine of an angle is \(-\frac{1}{2}\) at \(120^{\circ}\) and \(240^{\circ}\) (since these angles correspond to \(60^{\circ}\) in the unit circle, where cosine is \(\frac{1}{2}\)).

Step 3: Identify Specific Angles Within 0° to 360°

Since we're looking for angles between \(0^{\circ}\) and \(360^{\circ}\), we directly have the solutions: \(\theta = 120^{\circ}\) and \(\theta = 240^{\circ}\).

Step 4: Verification

Verify that \(\cos 120^{\circ} = -\frac{1}{2}\) and \(\cos 240^{\circ} = -\frac{1}{2}\). The calculations confirm that these are correct, as the coordinates in the unit circle for these angles reflect the cosine values.

Key Concepts

Understanding the Cosine Function

The cosine function, often denoted as \(\cos\theta\), is a fundamental concept in trigonometry that measures the horizontal distance from a point on the unit circle to the origin. It works with angles and helps us understand the relationship between the angle's measurement and its corresponding position on a circle. The cosine of an angle can fall between -1 and 1 because it represents the x-coordinate of a point on the unit circle.

• At \(0^\circ\), cosine is \(1\) because the point is at the far right of the circle.
• At \(90^\circ\), cosine is \(0\) as the point moves to the top of the circle.
• At \(180^\circ\), cosine is \(-1\) when the point is far left.
• At \(270^\circ\), it is back at \(0\).

This periodic nature means the function repeats every \(360^\circ\). When dealing with problems like determining \(\cos\theta = -\frac{1}{2}\), the solution involves understanding in which quadrants this occurs.

Exploring the Unit Circle

A unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is crucial for understanding trigonometric functions like cosine, as angles are measured from the positive x-axis.The unit circle helps visualize how the cosine function behaves. The point where an angle intercepts the circle determines the cosine value:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

For example, given that \(\cos\theta = -\frac{1}{2}\), lookup on the unit circle confirms this occurs in the second and third quadrants (e.g., \(120^\circ\) and \(240^\circ\)). These quadrant rules and positions provide a visual aid in solving trigonometric equations.

Decoding Angle Measurement

Angle measurement in trigonometry is typically expressed in degrees or radians. Degrees are more common in elementary education, where a full circle is \(360^\circ\). For the unit circle, angles are measured counterclockwise starting from the positive x-axis.Understanding angle measurement is essential:

• Angles between \(0^\circ\) and \(90^\circ\) fall in the first quadrant.
• \(90^\circ\) to \(180^\circ\) corresponds to the second quadrant.
• \(180^\circ\) to \(270^\circ\) is the third quadrant.
• The fourth quadrant spans \(270^\circ\) to \(360^\circ\).

In the context of the problem \(\cos\theta = -\frac{1}{2}\), we specifically look at \(120^\circ\) in the second quadrant and \(240^\circ\) in the third quadrant, as these correspond to the angle measurement validating that cosine is negative in these regions. This understanding plays a critical role in correctly identifying where a specific cosine value can occur within a given range.