Problem 100 - Precalculus : Building Concepts and Connections

Question

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{11 \pi}{6}$$

Step-by-Step Solution

Answer

The angle \( s \) for which \( \cos s = \cos t \), where \( t = \frac{11 \pi}{6} \), is \( s = \frac{23 \pi}{6} \).

Step 1: Calculate -t

As the cosine function has the property that cos(θ) = cos(-θ), calculate -t from the given t. Here, \( -t = - \frac{11 \pi}{6} \) .

Step 2: Adjust the solution to fit in the required domain

Since we know that \(2 \pi\) is a full round in radians, we can add it to the negative angle \( -t \) without changing its cosine value. So, we have \( s = -t + 2 \pi \). Substituting the value of \( -t \) we have \( s = -(- \frac{11 \pi}{6}) + 2 \pi = \frac{11 \pi}{6} + 2 \pi \).

Step 3: Simplify the result

Perform addition to simplify the expression and get the final result for \(s\). We have \(s = \frac{11 \pi}{6} + \frac{12 \pi}{6} = \frac{23 \pi}{6}\)

Key Concepts

Understanding the Cosine Function

The cosine function is one of the fundamental trigonometric functions and helps us understand the relationship between angles and lengths in right triangles. It is usually denoted as \( \cos \theta \), where \( \theta \) is the angle. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.

• Mathematically, \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
• This function is periodic, which means its values repeat at regular intervals. For the cosine function, this interval is \( 2\pi \) or 360 degrees.
• The range of the cosine function is from -1 to 1. This means that the highest value it can achieve is 1 and the lowest is -1.

A key property of the cosine function is symmetry. Cosine is an even function, which means \( \cos \theta = \cos(-\theta) \). This property simplifies solving many trigonometric equations, as it helps us find alternative angles that have the same cosine value.

Exploring Angles in Trigonometry

Angles are a fundamental part of trigonometry, helping us measure the steepness or inclination between two lines or planes. They are typically measured in degrees or radians.

• Degrees and radians are two units used to measure angles, though they serve the same purpose.
• One complete circle is 360 degrees, which is equivalent to \( 2\pi \) radians.
• Trigonometric functions like sine and cosine use angles as their input to compute output values.

When dealing with trigonometric functions, it's important to consider angles within their periodic intervals. For cosine, because it repeats every \( 2\pi \) radians, any angle outside of this range can be converted back into it by adding or subtracting \( 2\pi \), helping keep our calculations manageable and intuitive.

The Importance of Radians

Radians are a commonly used unit in trigonometry, particularly because they provide a more natural measure of angles in terms of the radius of a circle.

• In radians, one full circle is \( 2\pi \).
• An angle of \( \pi \) radians represents a half-circle, or 180 degrees.
• Using radians allows for simpler formulas and calculations, particularly in calculus and higher mathematics.

One of the benefits of radians in trigonometry is that many trigonometric identities and equations become more straightforward. For instance, the periodicity of trigonometric functions is expressed in terms of \( 2\pi \), making it easier to understand how functions like cosine behave over repeated cycles.
In our problem, when we find an angle \( s \) such that \( \cos s = \cos t \) within the range \( 0 \leq s < 2\pi \), using radians effectively helps simplify the solution/expression. It ensures the angle calculations adhere to standardized measurement, aligning with the periodic nature of cosine and other trigonometric functions.