Problem 62 - 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 \([0, \pi\) ). $$ \tan t-\tan 4 t=1+\tan 4 t \tan t $$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{\pi}{20}, \frac{\pi}{4}, \frac{9\pi}{20}, \frac{13\pi}{20}, \frac{17\pi}{20} \).

Step 1: Recognize the Trigonometric Identity

We need to recognize that the equation \( \tan a - \tan b = 1 + \tan a \tan b \) is equivalent to the tangent addition identity for \( \tan(a + b) \), which states: \[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}.\] Rearranging and substituting, we see that the expression fits the form of \( \tan(a+b) = \tan \frac{\pi}{4} \) or \( \tan 4t - \tan t = \tan \left( \frac{\pi}{4} \right) \). Hence, we realize that the equation simplifies to \( t + 4t = \frac{\pi}{4} + k\pi \) where \( k \) is an integer.

Step 2: Simplify and Solve the Equation

Using the identity from Step 1, simplify the equation to find \( 5t = \frac{\pi}{4} + k\pi \). This implies:\[5t = \frac{\pi}{4} + k\pi \]Solve for \( t \) as follows:\[t = \frac{\pi}{20} + \frac{k\pi}{5}\]

Step 3: Determine Solutions within the Interval

We need to find the integer values for \( k \) such that \( t \) is within the interval \( [0, \pi) \). Hence, calculate:\[0 \leq \frac{\pi}{20} + \frac{k\pi}{5} < \pi\]For \( k = 0 \): \[ t = \frac{\pi}{20} \approx 0.157 \] (valid)For \( k = 1 \):\[ t = \frac{5\pi}{20} = \frac{\pi}{4} \approx 0.785 \] (valid)For \( k = 2 \):\[ t = \frac{9\pi}{20} \approx 1.413 \] (valid)For \( k = 3 \):\[ t = \frac{13\pi}{20} \approx 2.042 \] (valid)For \( k = 4 \):\[ t = \frac{17\pi}{20} \approx 2.670 \] (valid)For \( k = 5 \): \[ t = \frac{21\pi}{20} \approx 3.298 \] (not valid since it's over \( \pi \))Thus, \( k = 0, 1, 2, 3, 4 \) are valid solutions.

Key Concepts

Tangent Addition Formula

The tangent addition formula is a trigonometric identity that helps simplify expressions involving the sum of two tangent values. It states:\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\] This formula can be incredibly helpful when solving equations like the one in this exercise, where terms of tangents of angles are involved.
To use it effectively, recognize when the structure of your problem matches the form of the tangent addition or subtraction identity. In the given exercise, this is exactly what happens when you equate both sides of the main equation, leading you to set up an angle equation, resulting in \( t + 4t = \frac{\pi}{4} + k\pi \).
Using addition formulas simplifies solving trigonometric equations by converting them into algebraically manageable forms, and allows solutions over predetermined intervals. This facilitates checking and evaluating potential solutions efficiently.

Angle Sum Identities

Angle sum identities, like the tangent addition formula used here, provide equations for calculating the trigonometric functions of sums or differences of angles. These identities are crucial as they:

• Break complex expressions into simpler forms.
• Allow you to use known values like \(\frac{\pi}{4}\) to find other angles.

The angle sum identity for tangent helped us rearrange and identify the structure needed to solve the equation by establishing that:\[\tan(5t) = \tan\left(\frac{\pi}{4} + k\pi \right)\]
This transformation simplifies the original equation into a simpler one where you can solve for \(t\) more directly, using common trigonometric values.
Angle sum identities are not limited to tangent functions; similar identities exist for sine and cosine as well, allowing broad and versatile application in different trigonometric problems.

Solution of Trigonometric Equations

Solving trigonometric equations involves finding angles that satisfy given trigonometric conditions over a specific interval. Here, you're tasked with finding solutions in the interval \([0, \pi)\).
The key steps involve:

• Using identities to simplify the equations (e.g., using the tangent addition formula).
• Rearranging the equation to a standard form (like \(5t = \frac{\pi}{4} + k\pi\)).
• Solving algebraically to find possible angle solutions.
• Verifying which solutions fit the required interval.

In this problem, after simplification, the solution involves calculating \(t\) values that satisfy \(0 \leq \frac{\pi}{20} + \frac{k\pi}{5} < \pi\). By iterating different integer values of \(k\), you're able to determine multiple acceptable solutions within the desired range.
Trigonometric equations require attention to detail and a solid understanding of trigonometric identities; they can often involve multiple solutions due to the periodic nature of trigonometric functions.