Problem 39
Question
\(39-42\) . 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
Verified Answer
\(\theta = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, \pi, \frac{9\pi}{8}, \frac{11\pi}{8}, \frac{13\pi}{8}, \frac{3\pi}{2}, \frac{17\pi}{8} \).
1Step 1: Recognize the Formula
Notice that the expression \( \cos \theta \cos 3 \theta - \sin \theta \sin 3 \theta \) can be rewritten using the cosine addition formula, \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). Here, it is specific to \( \cos(\theta + 3\theta) \), which simplifies to \( \cos(4\theta) \). Thus, the equation becomes \( \cos(4\theta) = 0 \).
2Step 2: Identify Possible Solutions
The equation \( \cos(4\theta) = 0 \) is satisfied when \( 4\theta = \frac{\pi}{2} + n\pi \) where \( n \) is an integer. This comes from the fact that cosine is zero at odd multiples of \( \frac{\pi}{2} \) (i.e., \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)).
3Step 3: Solve for \(\theta\)
Divide through by 4 to solve for \(\theta\):\[\theta = \frac{\pi}{8} + \frac{n\pi}{4} \].
4Step 4: Determine Valid Solutions
Look for values of \(\theta\) in the interval \([0, 2\pi)\). Calculate possible \(\theta\) using positive integers for \(n\):- For \( n = 0 \), \( \theta = \frac{\pi}{8} \).- For \( n = 1 \), \( \theta = \frac{3\pi}{8} \).- For \( n = 2 \), \( \theta = \frac{5\pi}{8} \).- For \( n = 3 \), \( \theta = \frac{7\pi}{8} \).- For \( n = 4 \), \( \theta = \pi \).- For \( n = 5 \), \( \theta = \frac{9\pi}{8} \).- For \( n = 6 \), \( \theta = \frac{11\pi}{8} \).- For \( n = 7 \), \( \theta = \frac{13\pi}{8} \).- For \( n = 8 \), \( \theta = \frac{3\pi}{2} \).- For \( n = 9 \), \( \theta = \frac{17\pi}{8} \).- For \( n = 10 \), \( \theta = \frac{19\pi}{8} \), which exceeds \( 2\pi \), so discard this and higher values.
Key Concepts
Cosine Addition FormulaSolving Trigonometric EquationsTrigonometric Interval Solutions
Cosine Addition Formula
The cosine addition formula is a fundamental tool in trigonometry that simplifies complex expressions by transforming them into simpler terms. It is expressed as:\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]This formula is particularly useful in converting the addition or subtraction of angles into a single trigonometric function. By using this formula, you can easily break down problems involving angles, making calculations more straightforward.
For the expression given in our exercise, \( \cos \theta \cos 3\theta - \sin \theta \sin 3\theta \), the cosine addition formula is applied by recognizing that:- \( A = \theta \)- \( B = 3\theta \)Thus, the expression simplifies to \( \cos(\theta + 3\theta) = \cos(4\theta) \). This transformation allows us to reduce the complexity of solving for \( \theta \) by making the expression more recognizable.
Using the cosine addition formula is not only a skillful way to manage calculations, it also highlights the relationships between different trigonometric functions and their applications in solving problems.
For the expression given in our exercise, \( \cos \theta \cos 3\theta - \sin \theta \sin 3\theta \), the cosine addition formula is applied by recognizing that:- \( A = \theta \)- \( B = 3\theta \)Thus, the expression simplifies to \( \cos(\theta + 3\theta) = \cos(4\theta) \). This transformation allows us to reduce the complexity of solving for \( \theta \) by making the expression more recognizable.
Using the cosine addition formula is not only a skillful way to manage calculations, it also highlights the relationships between different trigonometric functions and their applications in solving problems.
Solving Trigonometric Equations
Solving trigonometric equations involves finding the angle values that satisfy a given trigonometric expression or equation. Once the trigonometric identity is simplified using known formulas, we can proceed to solve the equation.
In the exercise, after applying the cosine addition formula, we have:\[ \cos(4\theta) = 0 \]The equation \( \cos x = 0 \) is satisfied whenever the angle \( x \) is an odd multiple of \( \frac{\pi}{2} \), specifically at:- \( x = \frac{\pi}{2} + n\pi \)
where \( n \) is an integer.This principle allows us to identify potential solutions for \( 4\theta \) as being at these odd multiples of \( \frac{\pi}{2} \). By setting \( 4\theta = \frac{\pi}{2} + n\pi \), we can solve for \( \theta \) by dividing the entire equation by 4.
Hence, we get:\[ \theta = \frac{\pi}{8} + \frac{n\pi}{4} \]Solving trigonometric equations becomes a methodical process by using known trigonometric properties and allows us to track the solutions through simple arithmetic.
In the exercise, after applying the cosine addition formula, we have:\[ \cos(4\theta) = 0 \]The equation \( \cos x = 0 \) is satisfied whenever the angle \( x \) is an odd multiple of \( \frac{\pi}{2} \), specifically at:- \( x = \frac{\pi}{2} + n\pi \)
where \( n \) is an integer.This principle allows us to identify potential solutions for \( 4\theta \) as being at these odd multiples of \( \frac{\pi}{2} \). By setting \( 4\theta = \frac{\pi}{2} + n\pi \), we can solve for \( \theta \) by dividing the entire equation by 4.
Hence, we get:\[ \theta = \frac{\pi}{8} + \frac{n\pi}{4} \]Solving trigonometric equations becomes a methodical process by using known trigonometric properties and allows us to track the solutions through simple arithmetic.
Trigonometric Interval Solutions
When solving trigonometric equations, finding all solutions within a specific interval is crucial to ensuring the completeness of the solution. The interval [0, 2\(\pi\)) represents one full rotation around the unit circle and is a common range for finding solutions.
After solving for \( \theta = \frac{\pi}{8} + \frac{n\pi}{4} \), we focus on determining which values of \( \theta \) fall into this interval. We do this by substituting positive integer values for \( n \).
For example:
After solving for \( \theta = \frac{\pi}{8} + \frac{n\pi}{4} \), we focus on determining which values of \( \theta \) fall into this interval. We do this by substituting positive integer values for \( n \).
For example:
- \( n = 0 \): \( \theta = \frac{\pi}{8} \)
- \( n = 1 \): \( \theta = \frac{3\pi}{8} \)
- \( n = 2 \): \( \theta = \frac{5\pi}{8} \)
- \( n = 3 \): \( \theta = \frac{7\pi}{8} \)
- \( n = 4 \): \( \theta = \pi \)
- \( n = 5 \): \( \theta = \frac{9\pi}{8} \)
- \( n = 6 \): \( \theta = \frac{11\pi}{8} \)
- \( n = 7 \): \( \theta = \frac{13\pi}{8} \)
- \( n = 8 \): \( \theta = \frac{3\pi}{2} \)
- \( n = 9 \): \( \theta = \frac{17\pi}{8} \)
Other exercises in this chapter
Problem 38
\(25-38\) . Find all solutions of the given equation. $$ \csc ^{2} \theta-4=0 $$
View solution Problem 38
Verify the identity. $$ (\sin x+\cos x)^{2}=1+2 \sin x \cos x $$
View solution Problem 39
Prove the identity. $$ \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y $$
View solution Problem 39
\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \csc x=3, \quad 90^{\circ}
View solution