Problem 43 - Trigonometry

Question

Find all radian solutions using exact values only. $$\sin x+\cos x=0$$

Step-by-Step Solution

Answer

The radian solutions are \( x = \frac{\pi}{4} + \frac{n\pi}{2} \) for any integer \( n \).

Step 1: Move Terms to One Side

Start by rearranging the equation to isolate one of the trigonometric functions. The equation given is \( \sin x + \cos x = 0 \). To isolate \( \cos x \), move \( \sin x \) to the other side, resulting in: \( \cos x = -\sin x \).

Step 2: Use Trigonometric Identity

Recall a useful trigonometric identity: \( \sin x = \cos(\frac{\pi}{2} - x) \). Substitute \( \sin x \) with this identity in the equation from Step 1 to get: \( \cos x = -\cos(\frac{\pi}{2} - x) \).

Step 3: Solve Basic Equation Form

Now, solve the equation \( \cos x = -\cos(\frac{\pi}{2} - x) \). This implies that \( x \) can be expressed using: \[ x = \frac{\pi}{2} - x + n \pi, \] where \( n \) is any integer, because the cosine of an angle equals the negative cosine of the supplementary angles. Simplifying gives \[ 2x = \frac{\pi}{2} + n\pi. \]

Step 4: Solve for x

Divide both sides of the equation \( 2x = \frac{\pi}{2} + n\pi \) by 2 to find \( x \): \[ x = \frac{\pi}{4} + \frac{n\pi}{2}. \] This is a general solution formula for this trigonometric equation.

Step 5: Verify the Solution Range

Since trigonometric functions are periodic, verify the solutions \( x = \frac{\pi}{4} + \frac{n\pi}{2} \) encompass all possible angles with the initial constraint. Adding integer multiples of \( \pi \), the general solution accounts for all possible radian solutions.

Key Concepts

Exact Radian Solutions

When solving trigonometric equations, we often seek exact radian solutions rather than decimal approximations. In the context of the equation \( \sin x + \cos x = 0 \), finding an exact solution involves understanding the periodic nature of sine and cosine functions in the unit circle.
We start by rearranging the given equation: \( \cos x = -\sin x \). This lays the groundwork for applying a strategic approach to find exact values:

• Recognize the potential identities between sine and cosine. They are complements on the unit circle.
• Explore symmetric relationships using \( \cos x = -\cos(\frac{\pi}{2} - x) \), indicating symmetrical angles across quadrants.

By solving \( x = \frac{\pi}{4} + \frac{n\pi}{2} \), where \( n \) is any integer, we account for the infinite and periodic nature of these angles on the circle. Each solution manifests within our expected set of angles, offering an elegant set of exact radian solutions that capture the full essence of the trigonometric function behavior.

Trigonometric Identities

Trigonometric identities are tools that simplify complex equations, particularly useful in converting or simplifying expressions. For our equation, \( \cos x = -\sin x \), invoking the identity \( \sin x = \cos(\frac{\pi}{2} - x) \) is key.
This identity shows how sine can be expressed in terms of cosine, allowing:

• A clear transformation from sine to cosine. This change of perspective can simplify your problem when working in non-standard forms.
• Helps in handling equations where expressions appear both as \( \sin \) and \( \cos \), offering a unified framework to work with.

Importance of these identities:

• They enable switching between perspectives easily, revealing underlying symmetries.
• Enable problem-solving without relying on graphing or numerical methods, highlighting mathematical elegance.

Understanding and applying these identities provides a profound advantage, helping break down equations into solvable forms.

Solution Verification

Verification confirms that solutions indeed satisfy the original equation. For the equation \( \sin x + \cos x = 0 \), we derived a general formula \( x = \frac{\pi}{4} + \frac{n\pi}{2} \). Now, let's ensure it holds.
Verification involves a few steps:

• Substitute \( x = \frac{\pi}{4} + \frac{n\pi}{2} \) back into the original equation.
• Check each possible result by ensuring the equation balances to zero.
• For comprehensive verification, consider a range of values for \( n \) to confirm no possible solutions are missed.

By doing this, we make certain each solution is valid, and the process aids in instilling confidence that every derived angle functions correctly in the original context. This reinforces the understanding that trigonometric solutions often manifest in expected patterns and cycles, completing a robust check for your solutions.