Problem 54 - Precalculus with Calculus Previews

Question

Find all angles \(\theta\), where \(0^{\circ} \leq \theta<\) \(360^{\circ}\), that satisfy the given condition. $$ \cos \theta=1 $$

Step-by-Step Solution

Answer

The angle is \( 0^{\circ} \).

Step 1: Understanding the Problem

We are asked to find angles \( \theta \) in the interval \( [0^{\circ}, 360^{\circ}) \) for which \( \cos \theta = 1 \). The cosine function is the x-coordinate on the unit circle, and we need to find where it equals 1.

Step 2: Identifying Key Angles

On the unit circle, \( \cos \theta = 1 \) occurs when the angle \( \theta \) points to the rightmost point, which is equivalent to a full rotation of \( 0^{\circ} \) or \( 360^{\circ} \). However, since \( \theta \) must be less than \( 360^{\circ} \), we focus on \( 0^{\circ} \).

Step 3: Eliminating Overlapping Angles

Although \( 360^{\circ} \) returns the cosine value back to the starting position (\(1\)), it is not included in the solution set because \( \theta < 360^{\circ} \). Therefore, \( 0^{\circ} \) is the only angle within the defined range.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. Its simplicity makes it an excellent tool for visualizing angles and trigonometric functions.

The unit circle allows us to see the relationship between an angle and its trigonometric values like sine and cosine.

• The circle itself is defined by the equation: \(x^2 + y^2 = 1\).
• The cosine of an angle is related to the x-coordinate of the corresponding point on the unit circle.
• For any angle \(\theta\), it traverses the circle in a counterclockwise direction starting from the positive x-axis.

Understanding the unit circle helps in identifying the key points where trigonometric functions attain values like \(-1, 0,\) and \(1\), which are crucial for solving trig equations involving functions such as cosine.

Angle Measurement

Angle measurement is a way to quantify the rotation or orientation from one point or line to another. It is usually measured in degrees or radians, where one full circle is either \(360^{\circ}\) or \(2\pi\) radians.
When working with angles in trigonometry, it's common to refer to standard positions:

• The initial side is the starting ray along the positive x-axis.
• The terminal side is the ray where the angle measurement ends after rotation.

In the context of the unit circle, angles are often measured in degrees between \(0^{\circ}\) and \(360^{\circ}\) to correspond with the full circular rotation.

When dealing with exercises that restrict \(\theta\) within specific boundaries, like \(0^{\circ} \leq \theta < 360^{\circ}\), it highlights one complete rotation without repetition, ensuring each angle is uniquely defined.

Trigonometric Identity

Trigonometric identities are equations that hold true for all angles, connecting various trigonometric functions. Such identities help simplify complex trigonometric equations and find specific angle measures.
The cosine function has specific values it typically returns at key points on the unit circle. Important identities and properties related to cosine include:

• Cosine has a maximum value of 1 and a minimum of -1.
• For \(\cos \theta = 1\), this occurs when \(\theta = 0^{\circ}, 360^{\circ}, 720^{\circ}\), etc., which align with full circle rotations.

In exercises dealing with specific conditions like \(\cos \theta = 1\), knowing these identities aids in rapidly pinpointing potential solutions, allowing for an efficient problem-solving process by identifying critical points like \(0^{\circ}\) in the unit circle.