Question

Algebraically determine the number of solutions for the equation \(\cos 2 x \cos x-\sin 2 x \sin x=0\) over the domain \(-360^{\circ}

Short answer

There are 8 solutions within the given domain.

Step-by-step solution

- Recognize Trigonometric Identity

Identify that the given equation \(\cos 2x \cos x - \sin 2x \sin x = 0\) can be simplified using the product-to-sum identities. Specifically, note that \(\cos a \cos b - \sin a \sin b = \cos(a + b)\).

- Apply the Identity

Rewrite the equation using the trigonometric identity: \(\cos 2x \cos x - \sin 2x \sin x = \cos(2x + x) \). This simplifies to \(\cos 3x = 0\).

- Solve for x

Solve the equation \(\cos 3x = 0\). The cosine function equals zero at odd multiples of \(\frac{\pi}{2}\). Therefore, \(3x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer.

- Isolate x

Isolate \(x\) by dividing by 3: \(x = \frac{\pi}{6} + \frac{k\pi}{3}\).

- Determine Solutions in Given Domain

Identify the values of \(k\) that fit within the domain \(-360^{\circ} < x \leq 360^{\circ}\). Convert the domain to radians: \(-2\pi < x \leq 2\pi\). Calculate the corresponding values: \(x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}, \-\frac{5\pi}{6}, \-\frac{\pi}{6}, \cdots\).

- Count the Solutions

Count all unique solutions that fall within the domain \(-2\pi < x \leq 2\pi\). These are \(\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}, \-\frac{5\pi}{6}, \-\frac{\pi}{6}, \-\frac{7\pi}{6}, \-\frac{11\pi}{6}}\). Thus, eight solutions exist.

Key concepts

cosine function

The cosine function is one of the fundamental trigonometric functions, commonly abbreviated as \(\text{cos}\). It allows you to relate the angle of a right triangle to the lengths of its adjacent side and hypotenuse. Cosine is crucial in solving various trigonometric equations and is periodic, meaning it repeats its values in regular intervals. In the cosine function, \(\text{cos}(x)\), \(\text{x}\) represents the angle, and the output is the ratio of the length of the adjacent side to the hypotenuse.
The cosine function has several important properties:

• Periodicity: \(\text{cos}(x) = \text{cos}(x + 2\pi)\)
• Range: The output values of cosine lie between -1 and 1
• Even Function: \(\text{cos}(-x) = \text{cos}(x)\)
• Zeros: \(\text{cos}(x) = 0\) at \(\frac{\pi}{2} + k\pi\), where \(\text{k}\) is any integer

Understanding these properties allows you to manipulate and solve trigonometric equations involving the cosine function efficiently.

product-to-sum identities

Product-to-sum identities are powerful tools in trigonometry used to transform products of trigonometric functions into a sum or difference of trigonometric functions. These identities can often simplify complex expressions and make equations easier to solve.
The product-to-sum identities are derived from the angle addition formulas and are as follows:

• \(\text{cos}(A) \text{cos}(B) = \frac{\text{cos}(A + B) + \text{cos}(A - B)}{2}\)
• \(\text{sin}(A) \text{sin}(B) = \frac{\text{cos}(A - B) - \text{cos}(A + B)}{2}\)
• \(\text{sin}(A) \text{cos}(B) = \frac{\text{sin} (A + B) + \text{sin} (A - B)}{2}\)
• \(\text{cos}(A) \text{sin}(B) = \frac{\text{sin} (A + B) - \text{sin} (A - B)}{2}\)

For instance, in this specific exercise, the identity used is \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). By recognizing the form of the given equation and applying this identity, we significantly simplify the equation, making it easier to solve. Apply the product-to-sum formulas whenever you encounter a trigonometric expression in a product form, transforming it into an addition form that is often more manageable.

solving trigonometric equations

Solving trigonometric equations involves finding the values of the variable (usually an angle) that satisfy the given equation. Here's a step-by-step approach to solving trigonometric equations using the example from the exercise:
1. **Identify the Equation Form**: Recognize patterns or identities that might simplify the equation. In this example, we used \(\cos(2x) \cos(x) - \sin(2x) \sin(x) = 0\), which simplifies using a product-to-sum formula to \( \cos(3x) = 0 \).
2. **Transform and Simplify**: Apply known identities to simplify the equation. This might involve expressing everything in terms of a single trigonometric function.
3. **Solve for the Angle**: Once simplified, solve for the angle. For \( \cos(3x) = 0 \), the cosine function is zero at odd multiples of \( \frac{\pi}{2}\), giving us \(3x = \frac{\pi}{2} + k\pi\).
4. **Isolate the Variable**: Divide or manipulate the equation to isolate the variable \(x\), resulting in \(x = \frac{\pi}{6} + \frac{k\pi}{3}\).
5. **Find Solutions in the Given Domain**: Determine the values of the variable that fit within any specified domain. Convert the domain to appropriate units if necessary, and solve for values of \(\text k\) that fit within the range. For instance, in the domain \(-2\pi < x \leq 2\pi\), we identify the values and count the number of allowed solutions.
Practicing these steps with various trigonometric equations helps in mastering the techniques and recognizing patterns quickly.