Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2 \tan t \csc t+2 \csc t+\tan t+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( t = 0 \) and \( t = \pi \).

Step 1: Substitute Identities

The given equation is \( 2 \tan t \csc t + 2 \csc t + \tan t + 1 = 0 \). Recall the trigonometric identities, \( \tan t = \frac{\sin t}{\cos t} \) and \( \csc t = \frac{1}{\sin t} \). Substitute these into the equation.

Step 2: Simplify the Equation

After substitution, the equation becomes \[ 2 \left( \frac{\sin t}{\cos t} \right) \left( \frac{1}{\sin t} \right) + 2 \left( \frac{1}{\sin t} \right) + \frac{\sin t}{\cos t} + 1 = 0. \] Simplifying this yields \[ \frac{2}{\cos t} + \frac{2}{\sin t} + \frac{\sin t}{\cos t} + 1 = 0. \].

Step 3: Multiply Through by \(\sin t \cos t\)

To eliminate the denominators, multiply each term by \( \sin t \cos t \). This transforms the equation into \[ 2 \sin t + 2 \cos t + \sin^2 t \cos t + \sin t \cos t = 0. \]

Step 4: Organize and Factor the Equation

Reorganize the terms and factor out common factors: \[ (2 \sin t + \sin^2 t \cos t) + (2 \cos t + \sin t \cos t) = 0. \] Factor \( \sin t \) and \( \cos t \) where possible.

Step 5: Solve for Potential Solutions

Solve each part of the factored equation separately. For the sin term solutions, when \( \sin t = 0 \), \( t = 0, \pi \). For terms with \( \cos t \), solve for which leads acceptable results in the interval \([0, 2\pi)\).

Step 6: Verify and Compile Solutions

Double-check each solution \( t = 0 \) and \( t = \pi \) with the original equation to confirm they satisfy the equation. Combinations ensuring the equation balances confirm in the valid interval.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools for simplifying and solving trigonometric equations. They express one trigonometric function in terms of others, often making complex expressions more manageable.
In the given problem, we utilize two main identities:

• Tangent: \( \tan t = \frac{\sin t}{\cos t} \)
• Cosecant: \( \csc t = \frac{1}{\sin t} \)

Substituting these identities into the equation allows us to work with basic trigonometric ratios like sine and cosine. This transforms the original equation into a form that is more convenient for simplification and further manipulation. Understanding these identities is crucial as they pave the way towards solving the equation effectively. Once these trigonometric functions are replaced, the equation becomes simpler, thereby highlighting the power of identities in trigonometry.

Solving Trigonometric Equations

The process of solving trigonometric equations often involves substitution, simplification, and factoring. In our example, after substituting the identities, we simplify the equation by multiplying through by common denominators to eliminate fractions.
This results in an equation with sine and cosine terms:

• The equation is simplified to \( 2 \sin t + 2 \cos t + \sin^2 t \cos t + \sin t \cos t = 0 \).
• Next, rearrange and group similar terms for factoring.
• Factor by grouping if possible, identifying common factors like \( \sin t \) and \( \cos t \).

Through this, each group of terms can lead to simpler equations. Solving for the values of \( t \) usually involves finding angles that satisfy these trigonometric equations within the given interval, which is \([0, 2\pi)\). Successfully solving trigonometric equations requires practice and familiarity with factoring techniques and identities.

Verifying Solutions

After arriving at potential solutions for the trigonometric equation, it is important to verify each solution to ensure they satisfy the original equation. Verification ensures the results are correct and applicable within the specified interval.
Here's how you can verify:

• Substitute each potential solution back into the original equation.
• Check if the equation balances, meaning the left-hand side equals the right-hand side (which is zero in this case).
• For example, if \( t = 0 \) and \( t = \pi \) are solutions, substitute them back to check.
• If substituting these values does not satisfy the equation, reevaluate them.

Through verification, you ensure that all calculated solutions are not only valid mathematically but are also appropriate within the interval \([0, 2\pi)\). This final step is crucial in confirming the accuracy of the trigonometric solutions obtained.