Problem 49 - Precalculus: Functions and Graphs

Question

Solve the equation for \(x\) in terms of \(y\) if \(x\) is restricted to the given interval. $$y=15-2 \cos x ; \quad[0, \pi]$$

Step-by-Step Solution

Answer

\(x = \cos^{-1} \left( \frac{15 - y}{2} \right)\) for \(13 \leq y \leq 17\).

Step 1: Rearrange the Equation

Start by isolating the cosine term in the given equation. The equation is: \[ y = 15 - 2 \cos x \]Rearrange to find:\[ 2 \cos x = 15 - y \]

Step 2: Solve for Cosine

Divide both sides by 2 to solve for \( \cos x \):\[ \cos x = \frac{15 - y}{2} \]

Step 3: Use Inverse Cosine Function

To find \( x \), use the inverse cosine function. Thus, we have:\[ x = \cos^{-1} \left( \frac{15 - y}{2} \right) \]Ensure \(x\) is in the interval \([0, \pi]\).

Step 4: Check Restrictions and Solution Validity

The cosine value \( \cos x \) must be within the domain of [-1, 1] for the inverse cosine function. Therefore, check:\[-1 \leq \frac{15 - y}{2} \leq 1 \]Solve these inequalities to determine the valid range for \( y \).

Key Concepts

Cosine Function

The cosine function is a fundamental element of trigonometry, describing the relationship between the sides of a right triangle relative to an angle. In a unit circle, it refers to the horizontal coordinate of a point. The function \(\cos x\) outputs values ranging from -1 to 1. It's periodic with a repeating cycle every \(2\pi\) radians. This periodic nature is crucial for operations such as finding angle solutions across various intervals.

In the context of solving equations involving cosine, it is essential to remember:

• Cosine values represent the horizontal distance from the origin to a point on the circle.
• The cosine function decreases from 1 to -1 as the angle \(x\) moves from 0 to \(\pi\), then increases as it completes its cycle.
• You can use the inverse cosine, represented as \(\cos^{-1}\), to retrieve an angle if the cosine value is known.

Understanding how cosine behaves allows you to manipulate it to solve equations, such as rearranging an equation to isolate the cosine term, as done in the solution provided.

Inequalities

Inequalities are expressions that establish a relationship of non-equality between two values or expressions. In trigonometric contexts, they often limit the possible values function outputs can take, ensuring valid and meaningful solutions.

Here's why understanding inequalities is key when working with functions like the cosine function:

• They help to ascertain the valid range of a function's output. For example, since cosine values only exist between -1 and 1, inequalities guide these limits.
• We use them to check whether computed values fit within these bounds, validating results.
• In the context of this problem, inequalities work to verify if the rearranged equation's solutions were plausible for all given conditions.

Consider the equation from the solution, \( -1 \leq \frac{15 - y}{2} \leq 1 \), an inequality that reflects the permissible range for output values when using the \( \cos^{-1}\) function, guaranteeing solutions are possible within the specified interval.

Restricting Domain

Restricting the domain of a function is necessary to create accurate and expected outputs for various trigonometric equations.

Trigonometric functions naturally have broad domains, resulting in multiple possible solutions. Restricting the domain narrows this field, allowing more precise problem-solving.

• In trigonometry, it helps to set an interval, such as \( [0, \pi] \), to limit potential solutions.
• The domain restriction ensures we find a single valid output of the inverse trig function, minimizing ambiguity.
• Such constraints also align calculations with real-world limitations or geometric interpretations, ensuring realistic solution outcomes.

In our example equation, restricting \( x \)to \[0, \pi \] makes it possible to define one unambiguous solution for the inverse cosine function, \( x = \cos^{-1}\left(\frac{15 - y}{2}\right) \), reinforcing the logic that any result stays within a real and interpretable path.