Problem 41 - Precalculus

Question

Evaluate each expression, if possible. $$\cos (3 \pi)-\sec (-3 \pi)$$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Understand the Cosine Function

The cosine function, \(\cos(\theta)\), gives the x-coordinate of the point on the unit circle subtended by angle \(\theta\) from the positive x-axis. \(\cos(3\pi)\) is the cosine of angle \(3\pi\) radians.

Step 2: Calculate \(\cos(3\pi)\)

Since \(3\pi\) radians is equivalent to \(\pi + 2\pi\), it places us at the same point as \(\pi\) on the unit circle, which is (-1, 0). Therefore, \(\cos(3\pi) = -1\).

Step 3: Understand the Secant Function

The secant function, \(\sec(\theta)\), is the reciprocal of the cosine function: \(\sec(\theta) = \frac{1}{\cos(\theta)}\). \(\sec(-3\pi)\) requires evaluating the cosine at \(-3\pi\) radians.

Step 4: Calculate \(\sec(-3\pi)\)

\(-3\pi\) radians places us at the same point as \(\pi\) on the unit circle. Therefore, \(\cos(-3\pi) = \cos(\pi) = -1\). Thus, \(\sec(-3\pi) = \frac{1}{\cos(-3\pi)} = -1\).

Step 5: Subtract the Values

Now that we know \(\cos(3\pi) = -1\) and \(\sec(-3\pi) = -1\), we can compute the expression: \[-1 - (-1) = -1 + 1 = 0.\]

Key Concepts

Understanding the Cosine Function

The cosine function is a crucial component of trigonometry, offering a way to describe the position of a point on the unit circle. In trigonometry, the cosine of an angle, denoted as \( \cos(\theta) \), gives the x-coordinate of a point that correlates to this angle's position on the circle.
Angles on the unit circle are measured in radians. A radian is the standard unit of angular measure, where \( 2\pi \) radians circle the entire unit circle, equating to 360 degrees. For example, \( \pi \) radians correspond to 180 degrees.

• \( \cos(0) = 1 \), facing the positive x-axis.
• \( \cos(\pi) = -1 \), directly on the negative x-axis.
• \( \cos(2\pi)= 1 \), completing the circle back to the positive x-axis.

In the exercise, evaluating \( \cos(3\pi) \) involves understanding that \( 3\pi \) is equivalent to completing one and a half circles around. So, it yields the same value as \( \cos(\pi) \), which is \( -1 \).
The periodic nature of the cosine function means it repeats its pattern every \( 2\pi \) radians. This understanding allows convenient calculations even for larger angles.

Exploring the Secant Function

The secant function, represented as \( \sec(\theta) \), serves as one of the reciprocal trigonometric functions. As the reciprocal of the cosine function, its definition is \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This relationship implies several essential characteristics.

• If \( \cos(\theta) = 0 \), \( \sec(\theta) \) is undefined, since division by zero is not possible.
• \( \sec(\theta) \) carries the sign of \( \cos(\theta) \).
• For \( \theta = 0 \), since \( \cos(0) = 1 \), \( \sec(0) = 1 \). Similarly, \( \sec(\pi) = -1 \).

In the problem at hand, calculating \( \sec(-3\pi) \) means finding the reciprocal of \( \cos(-3\pi) \). Since \( \cos(-3\pi) = \cos(\pi) = -1 \), the secant function here equals \( \sec(-3\pi) = -1 \).
Understanding secant's behavior is vital for grasping its broader applications in solving complex trigonometric equations and identities.

The Role of the Unit Circle

The unit circle is an indispensable tool in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane, typically used to define the trigonometric functions. It helps visualize and comprehend how angles translate to coordinates, crucial for understanding functions like cosine and secant.

• Every point on the unit circle can be described as \((\cos(\theta), \sin(\theta)) \).
• The x-coordinate \( \cos(\theta) \) represents the cosine of the angle, and the y-coordinate \( \sin(\theta) \) the sine.
• The circle smoothens seeing the cycle of trigonometric functions as angles progress through \( 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \).

The unit circle aids in immediately seeing that the cosine function at multiples of \( \pi \) places points on the x-axis, such as \((-1, 0)\) at \( \pi \) and \( \cos(\pi)=-1 \).
It embraces the symmetry of trigonometric functions, giving clarity on how angles beyond \( 2\pi \) circle back into earlier positions. Recognizing this can significantly simplify solving equations like \( \cos(3\pi) \) and \( \sec(-3\pi) \).
Overall, the unit circle is the cornerstone of tying together angle measurements and their trigonometric values.