Problem 76 - Precalculus

Question

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

Step-by-Step Solution

Answer

\( \theta = 180^{\circ} \)

Step 1: Understand the Cosine Function

The cosine function, for an angle \( \theta \), gives us the horizontal coordinate of the point on a unit circle at that angle. The range of values for cosine is from -1 to 1.

Step 2: Identify Where Cosine Equals -1

Cosine equals -1 at the angle where the horizontal coordinate on the unit circle is farthest to the left. This occurs at the angle \( \theta = 180^{\circ} \), which is directly opposite the positive x-axis.

Step 3: Confirm Angle in Given Range

The problem asks for \( \theta \) within the range \( 0^{\circ} \leq \theta \leq 360^{\circ} \). The angle \( 180^{\circ} \) fits within this specified range.

Key Concepts

Unit Circle

When exploring trigonometry, the unit circle is an essential tool. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Understanding the unit circle is crucial because it allows us to visualize the angles and their trigonometric values.
A concept to grasp is how angles are measured around the circle, starting from the positive x-axis and moving counterclockwise. This path represents the increasing angle \( \theta \)\ that we are familiar with in trigonometry.

• The unit circle shows the relationship between angles and the coordinates of points.
• For any angle \( \theta \), the point on the unit circle can be represented as \((\cos \theta, \sin \theta)\).
• This visualization helps in understanding where the cosine and sine functions derive their values.

Overall, the unit circle is an invaluable aid in grasping how trigonometric functions such as cosine work.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and relates the angle \( \theta \) to the horizontal coordinate of the point on the unit circle. In simpler terms, it tells us how far left or right a point is when an angle is drawn from the center of the circle.
The cosine of an angle can range from -1 to 1:

• \(\cos \theta = 1 \) when \( \theta = 0^{\circ} \) or \(360^{\circ}\), meaning the point is farthest to the right on the circle.
• \(\cos \theta = -1 \) when \( \theta = 180^{\circ} \), signifying the point is farthest to the left.
• This maximum left position occurs because the full circle spans 360 degrees, creating a symmetry around the origin.

For example, the exercise shows us that the angle \( \theta = 180^{\circ} \) gives us \( \cos \theta = -1\), and this happens because the point on the circle at this angle is at its leftmost position.
Remember, understanding the cosine function's behavior on the unit circle helps you solve trigonometric problems more effectively.

Angle Measurement

To master trigonometry, understanding how angles are measured is key. Angle measurement is typically done in degrees or radians, with each system having its uses. However, degrees are more intuitive and frequently used in introductory trigonometry.
Angles start at the positive x-axis:

• As you move counterclockwise, the angle \(\theta\) increases from \(0^{\circ}\) to \(360^{\circ}\).
• This range represents a full circle, allowing us to express every possible position on the circle.
• Specific angles, like \(90^{\circ}, 180^{\circ}, 270^{\circ}, \) and \(360^{\circ} \), are pivotal points around the circle, often referenced in trigonometric identities.

For our exercise, understanding that \( \theta = 180^{\circ} \) is a critical angle as it points directly left from the origin helps us know why \( \cos \theta = -1 \) at this position.
By grasping these fundamental angle measurements, it becomes easier to comprehend how other trigonometric concepts interact and solve related problems.