Solving 9 Sin 2t - 28 Cos 2t = 0 A Step By Step Guide

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Introduction to Trigonometric Equations

In the realm of mathematics, solving trigonometric equations is a fundamental skill, crucial for understanding periodic phenomena and various applications in physics, engineering, and other scientific fields. Trigonometric equations involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. These equations often appear in models describing oscillations, waves, and other cyclical behaviors. This article delves into solving a specific trigonometric equation: $9 ext{sin} 2t - 28 ext{cos} 2t = 0$. This equation requires a systematic approach, leveraging trigonometric identities and algebraic manipulation to isolate the variable t. The process not only demonstrates the solution to this particular equation but also illustrates broader techniques applicable to a wide range of trigonometric problems. Understanding these techniques is essential for anyone seeking to master trigonometry and its applications.

Trigonometric equations, unlike algebraic equations, can have infinitely many solutions due to the periodic nature of trigonometric functions. For example, the sine function repeats its values every $2[Pi]$ radians. This means that if $t$ is a solution to a trigonometric equation, then $t + 2[Pi]k$ is also a solution for any integer $k$. Therefore, when solving trigonometric equations, it is often necessary to find a general solution that encompasses all possible solutions. This involves identifying the principal solutions within a single period and then adding integer multiples of the period to account for all other solutions. The techniques used to solve these equations include algebraic manipulation, trigonometric identities, and inverse trigonometric functions. The choice of technique depends on the specific form of the equation, but the underlying goal is always to isolate the trigonometric function and then the variable.

The importance of trigonometric identities cannot be overstated when solving trigonometric equations. Identities such as the Pythagorean identity ($\text{sin}^2 x + \text{cos}^2 x = 1$), double-angle formulas ($\text{sin} 2x = 2 \text{sin} x \text{cos} x$, $\text{cos} 2x = \text{cos}^2 x - \text{sin}^2 x$), and sum-to-product formulas are invaluable tools. These identities allow us to rewrite equations in more manageable forms, simplify complex expressions, and ultimately isolate the variable. Furthermore, understanding the unit circle and the graphs of trigonometric functions is essential for visualizing solutions and ensuring that all possible solutions are accounted for. This article will demonstrate how to apply these concepts and identities to solve the given equation, providing a clear and step-by-step approach that can be generalized to other trigonometric problems.

Step-by-Step Solution

To effectively solve the trigonometric equation $9 \text{sin} 2t - 28 \text{cos} 2t = 0$, we need to employ a combination of algebraic manipulation and trigonometric understanding. The primary goal is to isolate the trigonometric functions and express the equation in a form that is easier to solve. One common strategy for equations of this type is to manipulate the equation into a form where we have the ratio of sine and cosine, which is the tangent function. By achieving this, we can then use the inverse tangent function to find the solutions for $2t$, and subsequently, for $t$. This approach is particularly useful when dealing with linear combinations of sine and cosine functions, as it simplifies the equation and allows for a straightforward application of inverse trigonometric functions. Throughout the solution process, it is crucial to keep in mind the periodic nature of trigonometric functions and ensure that we find all possible solutions within a given interval or the general solution.

Here's a detailed, step-by-step breakdown of how to solve the given equation:

  1. Isolate the terms: The first step involves rearranging the equation to isolate the sine and cosine terms on opposite sides. This is achieved by adding $28 \text{cos} 2t$ to both sides of the equation. This manipulation sets the stage for creating a ratio of trigonometric functions, which will be pivotal in simplifying the problem. The isolation step is a fundamental algebraic technique applicable to various equation-solving scenarios, and it is particularly effective in this context for leading to the tangent function.

    9sin2t=28cos2t9 \text{sin} 2t = 28 \text{cos} 2t

  2. Divide to form the tangent function: Next, we divide both sides of the equation by $ \text{cos} 2t$ to create the tangent function, since $\frac{\text{sin} 2t}{\text{cos} 2t} = \text{tan} 2t$. It is crucial to note that this step assumes $ \text{cos} 2t \neq 0$. We also divide both sides by 9 to further isolate the tangent function. This step is a key maneuver in solving trigonometric equations involving both sine and cosine, as it transforms the equation into a more manageable form involving a single trigonometric function. This technique is not only efficient but also highlights the interconnectedness of trigonometric functions and their ratios.

    9sin2t9cos2t=28cos2t9cos2t\frac{9 \text{sin} 2t}{9 \text{cos} 2t} = \frac{28 \text{cos} 2t}{9 \text{cos} 2t}

    tan2t=289\text{tan} 2t = \frac{28}{9}

  3. Find the principal solution: Now, we use the inverse tangent function (arctan or tan-1) to find the principal value of $2t$. This gives us the angle whose tangent is $\frac{28}{9}$. The inverse tangent function provides a solution within the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$, which is the principal range for the arctangent function. This step is critical in identifying one of the solutions, but it's important to remember that the tangent function has a period of $[Pi]$, so there are infinitely many solutions.

    2t=arctan2892t = \text{arctan} \frac{28}{9}

    Using a calculator, we find:

    2t≈1.2626 radians2t \approx 1.2626 \text{ radians}

  4. General solution: Since the tangent function has a period of $[Pi]$, we add integer multiples of $[Pi]$ to the principal solution to find all possible solutions for $2t$. This is because the tangent function repeats its values every $[Pi]$ radians. The general solution captures all angles that have the same tangent value, accounting for the periodic nature of the function. This step is essential for ensuring that we have found all possible solutions to the trigonometric equation.

    2t=1.2626+k\[Pi],k∈Z2t = 1.2626 + k\[Pi], \quad k \in \mathbb{Z}

  5. Solve for t: Finally, we divide both sides by 2 to solve for $t$. This gives us the general solution for $t$, which includes all possible values of $t$ that satisfy the original equation. The division by 2 is the last step in isolating the variable $t$, providing the complete solution set. This final step completes the process of solving the trigonometric equation, yielding a comprehensive solution that accounts for the periodic behavior of the tangent function.

    t=1.2626+k\[Pi]2,k∈Zt = \frac{1.2626 + k\[Pi]}{2}, \quad k \in \mathbb{Z}

    t≈0.6313+k\[Pi]2,k∈Zt \approx 0.6313 + \frac{k\[Pi]}{2}, \quad k \in \mathbb{Z}

Therefore, the general solution for the equation $9 \text{sin} 2t - 28 \text{cos} 2t = 0$ is $t \approx 0.6313 + \frac{k[Pi]}{2}$, where $k$ is any integer. This solution represents an infinite set of values for $t$ that satisfy the original equation, reflecting the periodic nature of trigonometric functions.

Alternative Methods

While the previous method provides a straightforward approach, there are alternative ways to tackle the trigonometric equation $9 \text{sin} 2t - 28 \text{cos} 2t = 0$. Exploring these different methods not only enhances our problem-solving skills but also deepens our understanding of trigonometric functions and their properties. One such alternative involves transforming the equation into a form that uses the auxiliary angle method. This method is particularly useful for equations of the form $a \text{sin} x + b \text{cos} x = 0$, where $a$ and $b$ are constants. Another approach involves using trigonometric identities to rewrite the equation in a different form, which might lead to a simpler solution process. Each method offers a unique perspective and set of techniques, making it valuable to explore them to broaden our mathematical toolkit.

Auxiliary Angle Method

This method involves expressing the given equation in the form $R \text{sin}(2t - \alpha) = 0$, where $R$ is the amplitude and $\alpha$ is the phase angle. The process begins by rewriting the equation as:

9sin2t−28cos2t=Rsin(2t−α)9 \text{sin} 2t - 28 \text{cos} 2t = R \text{sin}(2t - \alpha)

Expanding the right side using the sine subtraction formula, we get:

Rsin(2t−α)=R(sin2tcosα−cos2tsinα)R \text{sin}(2t - \alpha) = R(\text{sin} 2t \text{cos} \alpha - \text{cos} 2t \text{sin} \alpha)

Comparing coefficients, we have:

Rcosα=9R \text{cos} \alpha = 9

Rsinα=28R \text{sin} \alpha = 28

Squaring and adding these equations gives us:

R2(cos2α+sin2α)=92+282R^2(\text{cos}^2 \alpha + \text{sin}^2 \alpha) = 9^2 + 28^2

R2=81+784=865R^2 = 81 + 784 = 865

R=865R = \sqrt{865}

Now, we can find $\alpha$ by dividing the second equation by the first:

RsinαRcosα=289\frac{R \text{sin} \alpha}{R \text{cos} \alpha} = \frac{28}{9}

tanα=289\text{tan} \alpha = \frac{28}{9}

α=arctan289≈1.2626 radians\alpha = \text{arctan} \frac{28}{9} \approx 1.2626 \text{ radians}

Thus, the equation becomes:

865sin(2t−1.2626)=0\sqrt{865} \text{sin}(2t - 1.2626) = 0

sin(2t−1.2626)=0\text{sin}(2t - 1.2626) = 0

The solutions for $2t - 1.2626$ are integer multiples of $[Pi]$:

2t−1.2626=k\[Pi],k∈Z2t - 1.2626 = k\[Pi], \quad k \in \mathbb{Z}

2t=1.2626+k\[Pi],k∈Z2t = 1.2626 + k\[Pi], \quad k \in \mathbb{Z}

t=1.2626+k\[Pi]2,k∈Zt = \frac{1.2626 + k\[Pi]}{2}, \quad k \in \mathbb{Z}

This method yields the same general solution as before.

Using Trigonometric Identities

Another approach is to manipulate the equation using trigonometric identities to simplify it. However, in this specific case, directly applying identities doesn't lead to a more straightforward solution than the initial method. The key to solving this equation efficiently lies in recognizing the relationship between sine and cosine and transforming the equation into a tangent function, as demonstrated in the primary solution. While this method doesn't provide a distinct alternative solution process for this particular equation, it reinforces the importance of understanding trigonometric relationships and their applications in problem-solving.

Importance of Understanding Trigonometric Solutions

The importance of understanding trigonometric solutions extends far beyond the realm of academic mathematics. Trigonometric functions and their solutions are fundamental in modeling periodic phenomena, which are ubiquitous in the natural and engineered world. From the oscillations of a pendulum to the propagation of electromagnetic waves, trigonometric functions provide a mathematical framework for describing and analyzing these cyclical behaviors. Mastering the techniques for solving trigonometric equations is therefore essential for anyone working in fields such as physics, engineering, computer science, and even economics, where periodic trends and cycles are frequently encountered. A deep understanding of these concepts enables professionals to build accurate models, make predictions, and design systems that interact effectively with the world around them.

In physics, trigonometric functions are used to describe simple harmonic motion, the behavior of waves (such as sound and light), and the motion of projectiles. Understanding the solutions to trigonometric equations allows physicists to calculate the amplitudes, frequencies, and phases of these phenomena, which are crucial for predicting their behavior. For example, in electrical engineering, alternating current (AC) circuits are modeled using trigonometric functions, and the solutions to trigonometric equations are used to determine the voltage and current at any given time. This knowledge is vital for designing efficient and reliable electrical systems. Similarly, in acoustics, understanding the behavior of sound waves, which are also modeled using trigonometric functions, is essential for designing concert halls, musical instruments, and noise-canceling technologies.

In engineering, trigonometric solutions play a critical role in structural analysis, signal processing, and control systems. Civil engineers use trigonometric functions to analyze the forces acting on bridges and buildings, ensuring their stability and safety. Mechanical engineers use these functions to design mechanisms and machines, such as engines and turbines, that involve rotating or oscillating parts. In signal processing, trigonometric functions are used to analyze and manipulate signals, such as audio and video, which is essential for telecommunications and multimedia applications. Control systems, which are used to regulate the behavior of dynamic systems, often rely on trigonometric functions to model and control the system's response to various inputs. The ability to solve trigonometric equations is therefore a cornerstone of engineering practice, enabling the design and implementation of complex and efficient systems.

Furthermore, in computer science, trigonometric functions are used in graphics, simulations, and data analysis. Computer graphics relies heavily on trigonometric functions to perform rotations, translations, and scaling of objects in 2D and 3D space. Simulations, such as those used in video games and scientific research, often involve modeling physical phenomena that are described by trigonometric functions. In data analysis, trigonometric functions can be used to identify and analyze periodic trends in data sets, which is valuable in fields such as finance, meteorology, and epidemiology. For example, the stock market exhibits cyclical patterns that can be analyzed using trigonometric techniques, and weather patterns can be predicted using models that incorporate trigonometric functions to represent seasonal variations. In all these applications, a strong understanding of trigonometric solutions is essential for developing accurate and effective computational tools.

Conclusion

In conclusion, solving the trigonometric equation $9 \text{sin} 2t - 28 \text{cos} 2t = 0$ demonstrates the power and versatility of trigonometric functions and techniques. The step-by-step solution, involving algebraic manipulation and the use of the inverse tangent function, highlights a general approach applicable to a wide range of trigonometric problems. Furthermore, exploring alternative methods, such as the auxiliary angle method, broadens our understanding and provides additional tools for tackling complex equations. The general solution $t \approx 0.6313 + \frac{k[Pi]}{2}$, where $k$ is an integer, underscores the periodic nature of trigonometric functions and the importance of capturing all possible solutions.

Moreover, the broader discussion of the importance of understanding trigonometric solutions emphasizes their critical role in various scientific and engineering disciplines. From modeling physical phenomena in physics to designing complex systems in engineering and developing computational tools in computer science, trigonometric functions are indispensable. A solid grasp of these concepts not only enables us to solve specific equations but also equips us with the ability to analyze and model periodic behaviors in the world around us. Therefore, mastering trigonometric solutions is a valuable investment for anyone pursuing a career in STEM fields or seeking a deeper understanding of the mathematical foundations of our world. This article has aimed to provide a comprehensive guide to solving the given equation and to underscore the significance of trigonometric functions in a broader context.