Problem 59 - 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 \([0, \pi)\) $$\cos 5 t \cos 2 t=-\sin 5 t \sin 2 t$$

Step-by-Step Solution

Answer

The solutions in the interval \([0, \pi)\) are \( t = \frac{\pi}{14}, \frac{3\pi}{14}, \frac{5\pi}{14}, \text{and} \frac{\pi}{2} \).

Step 1: Recognize the Trigonometric Identity

The given equation is \( \cos 5t \cos 2t = -\sin 5t \sin 2t \). Recognize that this can be expressed using the cosine sum-to-product identity. The identity is \( \cos A \cos B - \sin A \sin B = \cos(A + B) \). Hence, the given equation is equivalent to \( \cos(5t + 2t) = \cos(7t) \).

Step 2: Simplify the Equation to a Basic Trig Equation

From Step 1, the equation \( \cos(7t) = 0 \) is formulated since the right-hand side must equal zero to match the trigonometric identity. This leads to a solution where the angle inside the cosine function equals odd multiples of \( \frac{\pi}{2} \).

Step 3: Solve for t

Since \( \cos(7t) = 0 \), it follows that \( 7t = \frac{(2n+1)\pi}{2} \) where \( n \) is an integer because \( \cos \theta = 0 \) when \( \theta = (2n+1)\frac{\pi}{2} \).

Step 4: Find Possible Values of t within the Interval

To find valid \( t \) in the interval \( [0, \pi) \), solve for \( t \) by plugging \( n \):- \( t = \frac{\pi}{14} \) when \( n=0 \)- \( t = \frac{3\pi}{14} \) when \( n=1 \)- \( t = \frac{5\pi}{14} \) when \( n=2 \)- \( t = \frac{\pi}{2} \) when \( n=3 \),This process continues until \( 7t \) exceeds \( \pi \). Only the values resulting in \( t < \pi \) are valid solutions.

Key Concepts

Cosine Sum-to-Product Identity

The cosine sum-to-product identity is a useful tool in trigonometry for simplifying expressions. It combines multiple trigonometric terms into a single cosine function. This identity is particularly handy when dealing with the products of cosines and sines. Using this identity, you can express \( \cos A \cos B - \sin A \sin B \) as \( \cos(A + B) \).
In this exercise, the original problem \( \cos 5t \cos 2t = -\sin 5t \sin 2t \) aligns perfectly with this identity. By identifying the expression as fitting the form \( \cos A \cos B - \sin A \sin B \), it simplifies to \( \cos(7t) \).
This simplification is a crucial step for solving the equation, as it reduces the problem from a complex combination to a simple trigonometric equation involving just one angle.

Solving Trigonometric Equations

Once a trigonometric equation is simplified, as we've done using the cosine sum-to-product identity, the next step is solving for the variable. In this case, the equation is \( \cos(7t) = 0 \).
The general solution for when \( \cos \theta = 0 \) is \( \theta = (2n+1)\frac{\pi}{2} \), where \( n \) is any integer. This formula gives us many potential solution angles that satisfy the cosine being zero. For this problem, substituting \( \theta = 7t \) allows us to solve for specific values of \( t \) by rearranging to find \( t = \frac{(2n+1)\pi}{14} \).
By working through different integer values of \( n \), we can list possible solutions which we will further refine based on the given interval.

Interval Notation

Finding solutions to an equation often involves checking which solutions fit within a specific range or interval. Interval notation is a mathematical language used to describe this set of numbers.
In this problem, we are interested in the solutions within the interval \([0, \pi)\). This notation signifies that we want the values that \( t \) can take, starting from 0 up to but not including \( \pi \).
Using \( t = \frac{(2n+1)\pi}{14} \) as derived from the trigonometric identity, we substitute consecutive values of \( n \) until \( t < \pi \) is no longer true. The successful values, like \( t = \frac{\pi}{14} \), \( t = \frac{3\pi}{14} \), \( t = \frac{5\pi}{14} \), and \( t = \frac{\pi}{2} \), meet the criteria and fit within the given interval. This process effectively narrows the infinite set of potential solutions to a specific set that is valid for the problem constraints.