Problem 59 - Algebra and Trigonometry with Analytic Geometry

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 are \( t = \frac{\pi}{7}, \frac{3\pi}{7}, \frac{5\pi}{7} \).

Step 1: Recognize the Trigonometric Identity

The given equation is \( \cos 5t \cos 2t = -\sin 5t \sin 2t \). We recognize this as a form suitable for the cosine of a sum identity. Specifically, the identity is \( \cos A \cos B - \sin A \sin B = \cos(A+B) \). However, due to the minus sign on the right, this corresponds to \( \cos(A+B+\pi) \), leading to the identity \( \cos(A-B) \).

Step 2: Apply the Identity

Using the identity from Step 1, we rewrite the equation as \( \cos(5t + 2t) = \cos(\pi + 0) = \cos \pi \). Therefore, the equation simplifies to \( \cos 7t = -1 \).

Step 3: Solve the Simplified Equation

The equation \( \cos 7t = -1 \) implies that the angle is \( \pi \) within any integer multiples of \( 2\pi \). Therefore, solve for \( t \) using \( 7t = \pi + 2k\pi \), where \( k \) is an integer, to get \( t = \frac{\pi(1 + 2k)}{7} \).

Step 4: Find Solutions within the Interval

To find solutions within \([0, \pi)\), calculate \( t \) for different integer values of \( k \). Start with \( k = 0 \), yielding \( t = \frac{\pi}{7} \). Then, check for \( k = 1 \) which gives \( t = \frac{3\pi}{7} \). Finally, for \( k = 2 \), we have \( t = \frac{5\pi}{7} \). When \( k = 3 \), \( t = \frac{7\pi}{7} = \pi \), which is not included in the interval \([0, \pi)\).

Step 5: List the Valid Solutions

The solutions of the initial equation \( \cos 5t \cos 2t = -\sin 5t \sin 2t \) in the interval \([0, \pi)\) are \( t = \frac{\pi}{7} \), \( \frac{3\pi}{7} \), and \( \frac{5\pi}{7} \).

Key Concepts

Addition and Subtraction Formulas

In trigonometry, addition and subtraction formulas are critical tools to handle complex expressions. Such formulas can simplify expressions and equations, making them more manageable to solve. The general forms of these formulas are:

• Addition Formula: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• Subtraction Formula: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These formulas help in rewriting the expressions involving sums and differences of angles. For example, the original equation \( \cos 5t \cos 2t = -\sin 5t \sin 2t \) is handled by recognizing it as a variant of the cosine subtraction formula. Such simplification is vital for trigonometric identities and solving equations efficiently.

Cosine of Sum Identity

The cosine of sum identity is a fundamental trigonometric concept used to express the cosine of a sum of two angles. It is given by:

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

In the context of our exercise, this identity is used in the slightly adjusted form to handle the additional minus sign present on the right side of the equation. This alteration led to recognizing that \( \cos 5t \cos 2t = -\sin 5t \sin 2t \) is similar to \( \cos(5t + 2t - \pi) \), or \( \cos(7t - \pi) \). Identifying such patterns allows for transforming complex equations into simpler ones that are easier to solve. When an equation involves sums of angles or specific trigonometric values, using these identities simplifies finding solutions, making them more approachable particularly in integer multiples of \( \pi \).

Interval Solutions

Determining interval solutions in trigonometric equations involves finding values within a specified range. For the equation \( \cos 7t = -1 \), solving in the interval \([0, \pi)\) means we need values of \( t \) within this range.To find the solutions:

• Start from the solved equation \( 7t = \pi + 2k\pi \), where \( k \) is any integer.
• We divide by 7 to isolate \( t \), giving \( t = \frac{\pi(1 + 2k)}{7} \).
• Test integer values to ensure they fall within the desired interval: Starting from \( k = 0 \), giving \( t = \frac{\pi}{7} \), to \( k = 2 \), giving \( t = \frac{5\pi}{7} \).
• Values like \( t = \pi \), while mathematically valid, are not included if they fall outside \([0, \pi)\).

Using the interval properly ensures solutions adhere strictly to the given conditions. Practicing these steps can be helpful to solve these equations systematically.