Solving Trigonometric Equations Sin Β = Cos(β + 2θ) And Sin 2θ = Cos 3θ

$$$$

In the realm of mathematics, trigonometric equations hold a significant place, often posing intriguing challenges that demand a blend of conceptual understanding and analytical prowess. This article delves into the intricate process of solving two specific trigonometric equations: (i) ${\sin \beta = \cos(\beta + 2\theta)}$ and (iv) ${\sin 2\theta = \cos 3\theta}$. Our exploration will not only provide step-by-step solutions but also illuminate the underlying principles and techniques that govern the world of trigonometric equations. Whether you are a student grappling with trigonometric concepts or a seasoned mathematician seeking a refresher, this guide aims to offer clarity and insights into the art of solving these equations.

(i) Solving ${\sin \beta = \cos(\beta + 2\theta)}$

Transforming the Equation

Our journey begins with the equation ${\sin \beta = \cos(\beta + 2\theta)}$. The key to unraveling this equation lies in recognizing the fundamental trigonometric identity that connects sine and cosine: ${\cos(x) = \sin(\frac{\pi}{2} - x)}$. By applying this identity, we can express the cosine term in terms of sine, thereby paving the way for simplification. The equation then transforms into:

${\sin \beta = \sin(\frac{\pi}{2} - (\beta + 2\theta))}$

General Solutions for Sine Equations

Now that we have expressed both sides of the equation in terms of sine, we can leverage the general solution for sine equations. If ${\sin x = \sin y}$, then the general solutions are given by:

1. ${x = n\pi + y}$, where n is an integer.
2. ${x = n\pi + (\pi - y)}$, where n is an integer.

Applying this principle to our transformed equation, we obtain two sets of solutions:

Case 1: ${\beta = n\pi + (\frac{\pi}{2} - (\beta + 2\theta))}$

In this case, we have:

${\beta = n\pi + \frac{\pi}{2} - \beta - 2\theta}$

Rearranging the terms, we get:

${2\beta + 2\theta = n\pi + \frac{\pi}{2}}$

Dividing both sides by 2, we arrive at:

${\beta + \theta = \frac{n\pi}{2} + \frac{\pi}{4}}$

Thus, one set of solutions is given by:

${\theta = \frac{n\pi}{2} + \frac{\pi}{4} - \beta}$, where n is an integer.

Case 2: ${\beta = n\pi + (\pi - (\frac{\pi}{2} - (\beta + 2\theta)))}$

In the second case, we have:

${\beta = n\pi + \pi - \frac{\pi}{2} + \beta + 2\theta}$

Simplifying the equation, we get:

${0 = n\pi + \frac{\pi}{2} + 2\theta}$

Solving for ${\theta}$, we obtain:

${\theta = -\frac{n\pi}{2} - \frac{\pi}{4}}$, where n is an integer.

Combining the Solutions

Therefore, the general solutions for the equation ${\sin \beta = \cos(\beta + 2\theta)}$ are:

1. ${\theta = \frac{n\pi}{2} + \frac{\pi}{4} - \beta}$
2. ${\theta = -\frac{n\pi}{2} - \frac{\pi}{4}}$

where n is an integer. These solutions encapsulate all possible values of ${\theta}$ that satisfy the given equation for any value of ${\beta}$.

(iv) Solving ${\sin 2\theta = \cos 3\theta}$

Transforming the Equation using Trigonometric Identities

Next, we turn our attention to the equation ${\sin 2\theta = \cos 3\theta}$. Similar to the previous case, we can utilize the identity ${\cos(x) = \sin(\frac{\pi}{2} - x)}$ to transform the cosine term into sine:

${\sin 2\theta = \sin(\frac{\pi}{2} - 3\theta)}$

Applying General Solutions for Sine Equations

Again, we invoke the general solution for sine equations. If ${\sin x = \sin y}$, then the general solutions are given by:

1. ${x = n\pi + y}$, where n is an integer.
2. ${x = n\pi + (\pi - y)}$, where n is an integer.

Applying this to our transformed equation, we derive two cases:

Case 1: ${2\theta = n\pi + (\frac{\pi}{2} - 3\theta)}$

In this scenario, we have:

${2\theta = n\pi + \frac{\pi}{2} - 3\theta}$

Combining the ${\theta}$ terms, we get:

${5\theta = n\pi + \frac{\pi}{2}}$

Dividing both sides by 5, we obtain one set of solutions:

${\theta = \frac{n\pi}{5} + \frac{\pi}{10}}$, where n is an integer.

Case 2: ${2\theta = n\pi + (\pi - (\frac{\pi}{2} - 3\theta))}$

In the second case, we have:

${2\theta = n\pi + \pi - \frac{\pi}{2} + 3\theta}$

Simplifying the equation, we get:

${2\theta = n\pi + \frac{\pi}{2} + 3\theta}$

Rearranging the terms, we have:

${-\theta = n\pi + \frac{\pi}{2}}$

Multiplying both sides by -1, we obtain another set of solutions:

${\theta = -n\pi - \frac{\pi}{2}}$, where n is an integer.

This can also be written as:

${\theta = m\pi - \frac{\pi}{2}}$, where m is an integer (by substituting m = -n).

Combining the Solutions for ${\sin 2\theta = \cos 3\theta}$

Thus, the general solutions for the equation ${\sin 2\theta = \cos 3\theta}$ are:

1. ${\theta = \frac{n\pi}{5} + \frac{\pi}{10}}$
2. ${\theta = m\pi - \frac{\pi}{2}}$

where n and m are integers. These solutions represent the complete set of values for ${\theta}$ that satisfy the equation.

Conclusion

In summary, we have meticulously solved two trigonometric equations, ${\sin \beta = \cos(\beta + 2\theta)}$ and ${\sin 2\theta = \cos 3\theta}$, by employing fundamental trigonometric identities and the general solutions for sine equations. The solutions obtained provide a comprehensive understanding of the values that satisfy these equations. Mastering these techniques is crucial for tackling a wide range of trigonometric problems and solidifying one's grasp of mathematical concepts. The ability to transform equations, apply general solutions, and manipulate trigonometric identities is a testament to mathematical proficiency. As you continue your journey in mathematics, remember that practice and a deep understanding of the underlying principles are the keys to success. Whether you are solving equations for academic purposes or exploring the beauty of mathematical relationships, the world of trigonometry offers endless opportunities for intellectual growth and discovery.