Problem 57 - Precalculus: Functions and Graphs

Question

Exer. \(57-62:\) Use an addition or subtraction formula to find the solutions of the equation that are in the Interval \([0, \pi)\) $$\sin 4 t \cos t=\sin t \cos 4 t$$

Step-by-Step Solution

Answer

The solutions satisfy \(\sin(5t) = \sin(3t)\).

Step 1: Rewriting the Equation

Notice that both sides of the equation involve trigonometric functions: \(\sin 4t \cos t = \sin t \cos 4t\). This is conducive to using trigonometric identities. Start by rewriting the equation using a subtraction formula: \(\sin(A) \cos(B) = \frac{1}{2}[ ext{sin}(A + B) + ext{sin}(A - B)]\). Apply this to both sides.

Step 2: Applying the Addition and Subtraction Formulas

Apply the subtraction formula on each side:\[\frac{1}{2}[\sin(4t+t) + \sin(4t-t)] = \frac{1}{2}[\sin(t+4t) + \sin(t-4t)]\] Simplifying this, we have:\[\frac{1}{2}[\sin(5t) + \sin(3t)] = \frac{1}{2}[\sin(5t) + \sin(3t)]\]. This shows that the equation is true if \(\sin(5t) = \sin(3t)\).

Key Concepts

Addition Formula

The addition formula in trigonometry helps to break down complex expressions into more manageable parts. It states that for any angles \(A\) and \(B\),

• \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
• \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

When solving trigonometric equations like \(\sin 4t \cos t = \sin t \cos 4t\), using these identities allows the equation to be rewritten in terms of simpler expressions.
This forms the basis for finding the solutions within a specified interval. Recognizing the patterns or pairs in the equation lets you decide whether the addition formula is applicable.
For instance, comparing \(\sin 5t\) and \(\sin 3t\), seeing these as results of manipulated trigonometric expressions gives insights into solving the equation.

Subtraction Formula

The subtraction formula in trigonometry is used similarly to the addition formula and helps simplify expressions where angle differences are involved. It states that for any angles \(A\) and \(B\),

• \(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
• \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

In the given exercise, the subtraction formula helps to transform \(\sin 4t \cos t = \sin t \cos 4t\) into an expression like \(\sin(5t) + \sin(3t)\) which simplifies the process of solving for \(t\).
By rewriting each side of the equation using this formula, we can equate or compare simplified terms, which might otherwise seem complex at first glance. This step is crucial in determining equivalences needed to isolate variables and find potential solutions inside the set interval.

Trigonometric Equations

Trigonometric equations are those that involve trigonometric functions like sine and cosine. Solving them requires understanding of identities and formulas that transform the equation into simpler forms.
In practical problems, such as finding solutions to \(\sin 5t = \sin 3t\), the core idea is to look for angles \(t\) where these equations match within a given interval.
This often results in solving basic trigonometric expressions, which can be done using known angle properties or sometimes numerical methods. Trigonometric equations often allow for multiple solutions due to the periodic nature of the functions.
Recognizing cycles and knowing when solutions repeat goes a long way to ensure all solutions in the interval are found without repeating or missing any.

Solution Interval

In trigonometry, a solution interval defines the range within which we need to find the solutions for an equation. The interval \([0, \pi)\) is commonly used.

• \([0, \pi)\) means that we consider all angles \(t\) starting at \(0\) and going up to just but not including \(\pi\).
• This interval captures half a complete cycle of sine or cosine functions, highlighting unique essential solutions other than full repetitions.

For the exercise at hand, once the trigonometric equation is simplified, it becomes necessary to check which solutions for \(t\) exist within \([0, \pi)\).
This may require checking solved angles against the interval limits, discarding those that lie outside. Using tools like unit circles or graphs can assist in visualizing and verifying which solutions are indeed viable within the specified range. Thus, ensuring comprehensive and accurate results.