Solving Trigonometric Equations Finding The Smallest Positive Value Of A

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Introduction

In trigonometry, solving equations involving trigonometric functions is a common task. This article focuses on finding the smallest positive value of angle A given the equation $\sin(2A) = \cos(75^{\circ})$. We will explore the trigonometric identities and properties needed to solve this problem and provide a step-by-step solution. This exploration is crucial for students and enthusiasts looking to deepen their understanding of trigonometric functions and their applications. The ability to manipulate trigonometric identities and solve such equations is fundamental in various fields, including physics, engineering, and computer graphics. By understanding the underlying principles and techniques, one can effectively tackle more complex problems involving trigonometric functions.

Understanding Trigonometric Identities

To solve the equation $\sin(2A) = \cos(75^{\circ})$, we need to utilize trigonometric identities that relate sine and cosine functions. A key identity we can use is the cofunction identity, which states that $\sin(\theta) = \cos(90^{\circ} - \theta)$. This identity allows us to express sine in terms of cosine, and vice versa. Applying this identity will help us transform the given equation into a form that is easier to solve. Understanding cofunction identities is essential in simplifying trigonometric expressions and solving equations. These identities are derived from the complementary nature of sine and cosine in a right-angled triangle. The cofunction identities not only aid in solving equations but also in simplifying complex trigonometric expressions, making them a fundamental tool in trigonometry. Moreover, understanding these identities provides a deeper insight into the relationships between different trigonometric functions, enhancing problem-solving skills.

Another important identity to consider is the general solution for trigonometric equations. When we have an equation of the form $\sin(x) = \sin(y)$, the general solution is given by $x = n\pi + (-1)^n y$, where n is an integer. Similarly, if we have $\cos(x) = \cos(y)$, the general solution is $x = 2n\pi \pm y$, where n is an integer. These general solutions help us find all possible values of the angle that satisfy the equation. In the context of our problem, understanding the general solutions will enable us to find all possible values of A and then identify the smallest positive value. The general solutions stem from the periodic nature of trigonometric functions, where the functions repeat their values after certain intervals. These solutions are critical for applications in physics and engineering, where periodic phenomena are modeled using trigonometric functions. Understanding the general solutions ensures that all possible solutions are considered, providing a comprehensive understanding of the problem.

Step-by-Step Solution

Let's solve the equation $\sin(2A) = \cos(75^{\circ})$.

  1. Use the Cofunction Identity: We can rewrite the cosine function in terms of sine using the cofunction identity: $\cos(75^{\circ}) = \sin(90^{\circ} - 75^{\circ}) = \sin(15^{\circ})$. So, our equation becomes $\sin(2A) = \sin(15^{\circ})$.
  2. Apply the General Solution: The general solution for $\sin(x) = \sin(y)$ is given by $x = n\pi + (-1)^n y$, where n is an integer. In our case, $x = 2A$ and $y = 15^{\circ}$. Converting $15^{\circ}$ to radians, we get $15^{\circ} = \frac{15\pi}{180} = \frac{\pi}{12}$. Thus, $2A = n\pi + (-1)^n \frac{\pi}{12}$.
  3. Solve for A: Divide both sides by 2 to isolate A: $A = \frac{n\pi}{2} + (-1)^n \frac{\pi}{24}$.
  4. Find the Smallest Positive Value: We need to find the smallest positive value of A. Let's test different integer values for n:
    • For n = 0: $A = \frac{0\pi}{2} + (-1)^0 \frac{\pi}{24} = \frac{\pi}{24}$ (positive)
    • For n = 1: $A = \frac{1\pi}{2} + (-1)^1 \frac{\pi}{24} = \frac{\pi}{2} - \frac{\pi}{24} = \frac{11\pi}{24}$ (positive)
    • For n = -1: $A = \frac{-1\pi}{2} + (-1)^{-1} \frac{\pi}{24} = -\frac{\pi}{2} - \frac{\pi}{24} = -\frac{13\pi}{24}$ (negative)

The smallest positive value of A occurs when n = 0, which gives us $A = \frac\pi}{24}$. Converting this back to degrees $A = \frac{\pi{24} \cdot \frac{180}{\pi} = \frac{180}{24} = 7.5^{\circ}$.

Alternative Approach

Another approach involves using the identity $\sin(\theta) = \sin(180^{\circ} - \theta)$. Since $\sin(2A) = \sin(15^{\circ})$, we can also write $\sin(2A) = \sin(180^{\circ} - 15^{\circ}) = \sin(165^{\circ})$. Therefore, we have two possibilities:

  1. 2A = 15^{\circ} + 360^{\circ}k$, where *k* is an integer.

  2. 2A = 165^{\circ} + 360^{\circ}k$, where *k* is an integer.

Solving for A in each case:

  1. A=7.5∘+180∘kA = 7.5^{\circ} + 180^{\circ}k

  2. A=82.5∘+180∘kA = 82.5^{\circ} + 180^{\circ}k

For the smallest positive value, let k = 0 in both cases:

  1. A=7.5∘A = 7.5^{\circ}

  2. A=82.5∘A = 82.5^{\circ}

The smallest positive value of A is $7.5^{\circ}$, which matches our previous result. This alternative method reinforces the understanding of the periodic nature of sine function and provides another perspective on solving trigonometric equations.

Conclusion

In conclusion, we found that the smallest positive value of A that satisfies the equation $\sin(2A) = \cos(75^{\circ})$ is $7.5^{\circ}$. We achieved this by using the cofunction identity to relate sine and cosine, applying the general solution for trigonometric equations, and considering different integer values to find the smallest positive solution. Additionally, we explored an alternative approach using the identity $\sin(\theta) = \sin(180^{\circ} - \theta)$, which confirmed our result. This exercise highlights the importance of understanding and applying trigonometric identities and general solutions in solving trigonometric equations. The techniques discussed here are applicable to a wide range of problems in mathematics, physics, and engineering. By mastering these fundamental concepts, students and professionals can confidently tackle more complex challenges involving trigonometric functions.

Key Takeaways

  • Cofunction Identity: $\sin(\theta) = \cos(90^{\circ} - \theta)$ is essential for converting between sine and cosine functions.
  • General Solutions: Understanding the general solutions for trigonometric equations allows for finding all possible solutions.
  • Smallest Positive Value: When solving for angles, it is crucial to consider the smallest positive value, which often corresponds to the simplest solution.
  • Alternative Approaches: Exploring different methods to solve the same problem can reinforce understanding and provide alternative perspectives.

This comprehensive solution not only provides the answer but also elucidates the process, making it a valuable resource for anyone studying trigonometry.

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