Problem 67 - Precalculus Mathematics for Calculus

Question

What makes the equation \(\sin (\cos x)=0\) different from all the other equations we've looked at in this section? Find all solutions of this equation.

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{2} + k\pi \), \( k \in \mathbb{Z} \).

Step 1: Understand the Equation Context

The given equation is \( \sin(\cos x) = 0 \). This means we are looking for values of \( x \) for which the cosine of \( x \) results in an angle for which sine is zero.

Step 2: Analyze the Inner Function

Since \( \sin(\theta) = 0 \) for \( \theta = n\pi \), where \( n \) is an integer, we need \( \cos x = n\pi \) for some integer \( n \). However, \( \cos x \) can only take values between -1 and 1.

Step 3: Check Possible Values of \( \cos x \)

The possible values \( n\pi \) that fit in the range \(-1\leq \cos x \leq 1\) are \( \theta = 0 \), since \( \cos x = -1, 0, 1 \) fall within this interval. Thus, we need to solve \( \cos x = 0 \).

Step 4: Solve \( \cos x = 0 \)

The general solution to \( \cos x = 0 \) is \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer, because these are the values where the cosine function is zero.

Step 5: Verify the Understanding of the Problem's Difference

What makes this equation different is its nested nature and that it restricts the values of \( \cos x \) to only specific possibilities (whole multiples of \( \pi \)) due to the sine function requirement.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry, describing the relationship between an angle in a right triangle and the ratio of the opposite side to the hypotenuse. Its general characteristics include:

• Periodic nature with a period of \(2\pi\).
• Range restricted between -1 and 1.
• Zeros at integer multiples of \(\pi\).

These zeros imply that for any angle \(\theta\), if \(\sin(\theta) = 0\), then \(\theta\) must be an integer multiple of \(\pi\). Understanding this property helps solve equations involving sine, like \(\sin(\cos x)=0\). Here, our task is to find values of \(x\) such that \(\cos x\) aligns with these zeros.

Cosine Function

The cosine function is another cornerstone of trigonometry, closely related to the sine function. It represents the ratio of the adjacent side to the hypotenuse in a right triangle. Some key features include:

• Periodicity with a cycle every \(2\pi\).
• Range restricted from -1 to 1.
• Zeros occur at \(x = \frac{\pi}{2} + k\pi\) where \(k\) is an integer.

In the given problem, the cosine function takes on values that are possible inputs for the sine function, leading us to check when \(\cos x = n\pi\). The constraint is that since \(n\pi\) must be within -1 and 1, the viable option is \(n = 0\), making the cosine of \(x\) zero.

General Solution

Finding a general solution in trigonometric equations involves identifying all possible values of the variable that satisfy the equation. For \(\cos x = 0\), the solution set is characterized by:

• Locations where cosine achieves a value of zero.
• These occur at multiples of half revolutions \(\left(\frac{\pi}{2} + k\pi\right)\).

Hence, the general solution is expressed as \(x = \frac{\pi}{2} + k\pi\), encompassing all potential \(x\) values where cosine equals zero and meeting the criteria set by the sine function zeros.

Nested Functions

Nested functions, like \(\sin(\cos x)\), present unique challenges. The function inside (cosine) informs the input to the sine function, necessitating extra steps to pinpoint solutions. The concept involves:

• Evaluating the inner function separately.
• Determining its output before applying the outer function.
• Applying constraints from the outer function to refine solutions of the inner function.

In this case, the nested nature means starting with the output of \(\cos x\), assessing its possible values under sine's constraint (which are \(n\pi\) values), and ensuring compatibility with the limited range of valid cosine values. The nested feature highlights how multi-layered trigonometric relationships can influence problem-solving strategies.