Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arccos 0 $$

Step-by-Step Solution

Answer

\( \theta = 90^\circ \) or \( 270^\circ \) (Degrees); \( \theta = \frac{\pi}{2} \) or \( \frac{3\pi}{2} \) (Radians).

Step 1: Understand the Problem

You need to find the angle \( \theta \) such that the cosine of \( \theta \) is 0. Specifically, you'll find \( \theta \) in both degrees and radians.

Step 2: Use the Cosine Function for Angles

Cosine reaches a value of 0 at certain angles. Recall that \( \cos(90^\circ) = 0 \) and \( \cos(270^\circ) = 0 \). So, the angles in degrees where cosine equals 0 are \( 90^\circ \) and \( 270^\circ \).

Step 3: Convert Degree Values to Radians

To express \( 90^\circ \) and \( 270^\circ \) in radians, use the conversion factor \( \frac{\pi}{180^\circ} \). Therefore, \( 90^\circ = \frac{\pi}{2} \) and \( 270^\circ = \frac{3\pi}{2} \).

Step 4: Determine the Values of \( \theta \)

Combine the results: when \( \cos(\theta) = 0 \), then \( \theta = 90^\circ \) or \( 270^\circ \) in degrees, and \( \theta = \frac{\pi}{2} \) or \( \frac{3\pi}{2} \) in radians.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \( \cos \theta \). It relates an angle in a right triangle to the lengths of the adjacent side and the hypotenuse. When the cosine of an angle is 0, it pinpoints specific angles where the line from the origin behaves in certain characteristic ways in the unit circle representation of angles.

• Cosine reflects the width or horizontal aspect of an angle when situated on the unit circle.
• It indicates the x-coordinate of where the terminating side of an angle intercepts the unit circle.
• Common right angles where cosine hits 0 are \( 90^\circ \) and \( 270^\circ \), directly aligning with the y-axis.

Understanding where \( \cos \theta = 0 \) is crucial as it affects wave patterns, oscillations, and function transformations in various mathematical and engineering contexts. This is why functions like \( \cos(\theta) \) are extensively used in subjects ranging from physics to signal processing.

Radian to Degree Conversion

Converting between radians and degrees is essential in mathematics, as these are the two main units to measure angles. Radians are often used in calculus and when angles are part of circular functions, while degrees are more common in everyday use and elementary geometry.

• A full circle represents \( 360^\circ \) or \( 2\pi \) radians.
• The conversion rule is: 1 radian equals approximately \( 57.2958^\circ \), but we often use the exact conversion ratio of \( \frac{\pi}{180^\circ} \).
• Therefore, to convert angle measurements from degrees to radians, multiply the degree measurement by \( \frac{\pi}{180} \).

Let's take an example: to convert \( 90^\circ \) to radians, use \( 90 \times \frac{\pi}{180} = \frac{\pi}{2} \). Similarly, \( 270^\circ \) converts to \( 270 \times \frac{\pi}{180} = \frac{3\pi}{2} \). This intuitive method ensures angles can easily be switched from one unit of measure to another during calculations.

Angle Measurement

Angle measurement is a vital part of understanding geometry and trigonometry. It involves specifying the size of an angle, allowing us to navigate between the geometrical and analytical worlds efficiently.

• Degrees are more human-interpretive, with \( 360^\circ \) allowing for a clean division of a circle into fractions (quarters, thirds, etc.).
• Radians offer a natural way of grouping angles based on the circle's radius, providing a straightforward link to the arc length.
• Specific angles like \( 90^\circ \) (or \( \frac{\pi}{2} \) radians) and \( 270^\circ \) (or \( \frac{3\pi}{2} \) radians) are significant in mathematics as they describe particular geometric orientations — essentially vertical positions on the unit circle.

When considering trigonometric values, knowing how to measure and convert angles means better insights into how functions like sine and cosine will behave. Practically, this aids in areas from drafting to digital animation, wherever rotational and wave-like properties are necessary.