Problem 39 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi)\) $$\cos \theta \cos 3 \theta-\sin \theta \sin 3 \theta=0$$

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{8}, \frac{5\pi}{8}, \frac{9\pi}{8}, \frac{13\pi}{8}\)

Step 1: Identify the Formula to Use

The given expression is \( \cos \theta \cos 3\theta - \sin \theta \sin 3\theta \). This fits the pattern of the cosine addition formula \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). Here, it suggests using the formula for \( \cos(\theta + 3\theta) = \cos 4\theta \).

Step 2: Apply the Cosine Addition Formula

Apply the formula to rewrite the expression. The given expression \( \cos \theta \cos 3\theta - \sin \theta \sin 3\theta \) becomes \( \cos 4\theta = 0 \).

Step 3: Solve the Equation \( \cos 4\theta = 0 \)

The cosine of an angle is zero at \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \). Thus, we have \( 4\theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Step 4: Solve for \( \theta \)

Divide both sides of \( 4\theta = \frac{\pi}{2} + k\pi \) by 4 to solve for \( \theta \). This gives \( \theta = \frac{\pi}{8} + \frac{k\pi}{4} \).

Step 5: Find Solutions in the Interval \([0, 2\pi)\)

Start with the smallest integer value of \( k \) that results in \( \theta \) within \([0, 2\pi)\). Continuing with integer values of \( k \), the allowed solutions for \( \theta \) are \( \frac{\pi}{8}, \frac{5\pi}{8}, \frac{9\pi}{8}, \frac{13\pi}{8} \).

Key Concepts

Cosine Addition Formula

When dealing with trigonometric functions, especially in complex equations, using identities can simplify expressions significantly. One such identity is the **Cosine Addition Formula**. This formula states:\[\cos(A + B) = \cos A \cos B - \sin A \sin B\]This identity allows us to express the cosine of a sum as the difference between the products of cosines and sines of individual angles. In the given exercise, the expression \( \cos \theta \cos 3\theta - \sin \theta \sin 3\theta \) fits directly into this formula. By recognizing this, the expression can be rewritten as \( \cos(\theta + 3\theta) = \cos(4\theta) \). Using the Cosine Addition Formula not only simplifies the equation but also aids in recognizing familiar forms, allowing easier manipulation in solving problems. It eliminates multiple components, simplifying the process down to a single trigonometric expression.

Solving Trigonometric Equations

Once the equation is simplified using the Cosine Addition Formula, the next step is **Solving Trigonometric Equations**. In our case, the simplified expression is \( \cos(4\theta) = 0 \). The cosine of an angle equals zero at specific points:

• \( \frac{\pi}{2} \)
• \( \frac{3\pi}{2} \)
• \( \frac{5\pi}{2} \)
• and so forth.

Therefore, to solve \( \cos 4\theta = 0 \), we equate:\[4\theta = \frac{\pi}{2} + k\pi\]Here, \( k \) represents any integer, and it accounts for the periodic nature of trigonometric functions, allowing for multiple solutions within a cycle. Solving for \( \theta \) involves isolating \( \theta \) by dividing both sides by 4, resulting in:\[\theta = \frac{\pi}{8} + \frac{k\pi}{4}\]This solution provides a general form that can be used to find specific interval solutions as required.

Interval Solutions

Finally, to find all solutions within a specific range, we move to **Interval Solutions**. Our interval for this exercise is \([0, 2\pi)\). To gather the complete set of solutions, we substitute integer values for \( k \) into:\[\theta = \frac{\pi}{8} + \frac{k\pi}{4}\]Starting with the smallest possible values of \( k \):

• For \( k = 0 \), \( \theta = \frac{\pi}{8} \)
• For \( k = 1 \), \( \theta = \frac{5\pi}{8} \)
• For \( k = 2 \), \( \theta = \frac{9\pi}{8} \)
• For \( k = 3 \), \( \theta = \frac{13\pi}{8} \)

By continuing this sequence and ensuring that \( \theta \) remains below \( 2\pi \), we obtain all the valid solutions within the interval. Each solution represents an angle where the original equation \( \cos(4\theta) = 0 \) holds true.