Solving Cos(2x - 10°) ⋅ Sec(x + 30°) = 1 A Trigonometric Equation

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Introduction to Trigonometric Equations

Trigonometric equations are fundamental in mathematics, particularly in fields like physics, engineering, and computer graphics. These equations involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. Solving these equations often requires a deep understanding of trigonometric identities, algebraic manipulation, and the periodic nature of trigonometric functions. This article delves into the solution of a specific trigonometric equation: cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 }. By exploring this equation, we will cover essential techniques for solving trigonometric problems and gain valuable insights into the behavior of trigonometric functions.

In the realm of mathematical problem-solving, trigonometric equations hold a significant position due to their wide-ranging applications. Trigonometric functions describe the relationships between angles and sides of triangles, which are crucial in various scientific and engineering disciplines. For instance, in physics, these functions are used to model oscillations, waves, and periodic phenomena. In engineering, they are essential for analyzing alternating currents, mechanical vibrations, and structural stability. Moreover, trigonometric equations are integral to computer graphics for tasks such as rotations, scaling, and projections. The ability to solve these equations accurately is, therefore, a critical skill for anyone working in these fields.

The challenge in solving trigonometric equations lies in the periodic nature of trigonometric functions and the multiple solutions that can arise within a given interval. Unlike algebraic equations, which typically have a finite set of solutions, trigonometric equations often have an infinite number of solutions due to the cyclical behavior of trigonometric functions. This characteristic necessitates a methodical approach that combines algebraic techniques with a thorough understanding of trigonometric identities and the unit circle. In this article, we will demonstrate such an approach by dissecting the equation cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 } step by step.

Understanding the intricacies of solving trigonometric equations not only enhances one's mathematical proficiency but also fosters a deeper appreciation for the interconnectedness of mathematical concepts. The process of solving these equations involves drawing upon a variety of mathematical tools and techniques, including algebraic manipulation, trigonometric identities, and graphical analysis. By mastering these tools, one can approach a wide range of mathematical problems with confidence and creativity. In the following sections, we will embark on a detailed exploration of the given equation, uncovering the underlying principles and strategies that lead to its solution. This journey will not only provide a solution to the specific equation at hand but also equip readers with a valuable skill set for tackling other trigonometric challenges.

Problem Statement and Initial Transformation

To begin, let's restate the trigonometric equation we aim to solve: cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 }. The presence of both cosine and secant functions suggests a strategic approach involving trigonometric identities. Recognizing that secant is the reciprocal of cosine, we can rewrite the equation to consolidate the trigonometric functions, making it easier to manipulate and solve. This initial transformation is a crucial step in simplifying the equation and revealing its underlying structure.

The key to simplifying this equation lies in the fundamental trigonometric identity that relates secant and cosine: sec⁡(θ)=1cos⁡(θ){ \sec(\theta) = \frac{1}{\cos(\theta)} }. Applying this identity to our equation, we can replace sec⁡(x+30∘){ \sec(x + 30^\circ) } with 1cos⁡(x+30∘){ \frac{1}{\cos(x + 30^\circ)} }. This substitution transforms the original equation into a form that involves only cosine functions, which simplifies the subsequent algebraic manipulations. The transformed equation becomes:

cos⁡(2x−10∘)⋅1cos⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \frac{1}{\cos(x + 30^\circ)} = 1 }

Now, we have an equation that is more amenable to algebraic manipulation. The next step involves multiplying both sides of the equation by cos⁡(x+30∘){ \cos(x + 30^\circ) } to eliminate the fraction. This operation leads to a new equation that relates the cosine functions directly, setting the stage for further simplification and the application of trigonometric identities. The process of transforming the equation in this way is a common technique in solving trigonometric problems, as it allows us to work with simpler expressions and uncover relationships between different trigonometric functions.

The multiplication yields:

cos⁡(2x−10∘)=cos⁡(x+30∘){ \cos(2x - 10^\circ) = \cos(x + 30^\circ) }

This form of the equation is significantly more manageable than the original. It presents a direct relationship between the cosine of two different angles, which opens the door to the use of trigonometric identities to find the values of x{ x } that satisfy the equation. In the following sections, we will explore the trigonometric identities that are applicable to this situation and how they can be used to solve for x{ x }. This step-by-step approach allows us to break down a complex problem into smaller, more manageable parts, making the solution process more transparent and accessible.

Applying Trigonometric Identities

Now that we have the equation cos⁡(2x−10∘)=cos⁡(x+30∘){ \cos(2x - 10^\circ) = \cos(x + 30^\circ) }, we can leverage the properties of the cosine function to find the solutions for x{ x }. The cosine function has the characteristic that cos⁡(A)=cos⁡(B){ \cos(A) = \cos(B) } if and only if A=B+360∘k{ A = B + 360^\circ k } or A=−B+360∘k{ A = -B + 360^\circ k }, where k{ k } is an integer. This property stems from the periodic and even nature of the cosine function, which repeats its values every 360∘{ 360^\circ } and is symmetric about the y-axis. Applying this property is a crucial step in solving the equation, as it allows us to express the relationship between the angles in a way that can be solved algebraically.

This leads to two distinct cases that we must consider. Each case represents a set of possible solutions for x{ x }, and it is essential to analyze both cases to find all solutions within a given interval or to express the general solution. The first case arises from the direct equality of the angles, while the second case accounts for the symmetry of the cosine function. By considering both cases, we ensure that we capture all possible solutions that satisfy the original equation. This meticulous approach is a hallmark of rigorous mathematical problem-solving.

Case 1: 2x−10∘=x+30∘+360∘k{ 2x - 10^\circ = x + 30^\circ + 360^\circ k }

This case represents the scenario where the angles are equal, plus integer multiples of 360∘{ 360^\circ }, accounting for the periodicity of the cosine function. Simplifying this equation involves basic algebraic manipulation to isolate x{ x }. Subtracting x{ x } from both sides and adding 10∘{ 10^\circ } yields:

x=40∘+360∘k{ x = 40^\circ + 360^\circ k }

Case 2: 2x−10∘=−(x+30∘)+360∘k{ 2x - 10^\circ = -(x + 30^\circ) + 360^\circ k }

This case accounts for the symmetry of the cosine function, where the cosine of an angle is equal to the cosine of its negative. Expanding and simplifying this equation requires a bit more algebraic manipulation. First, distribute the negative sign:

2x−10∘=−x−30∘+360∘k{ 2x - 10^\circ = -x - 30^\circ + 360^\circ k }

Next, add x{ x } and 10∘{ 10^\circ } to both sides:

3x=−20∘+360∘k{ 3x = -20^\circ + 360^\circ k }

Finally, divide by 3 to solve for x{ x }:

x=−203∘+120∘k{ x = -\frac{20}{3}^\circ + 120^\circ k }

General Solutions and Specific Examples

From the previous section, we have derived two sets of solutions for the equation cos⁡(2x−10∘)=cos⁡(x+30∘){ \cos(2x - 10^\circ) = \cos(x + 30^\circ) }. These solutions are expressed in terms of an integer k{ k }, representing the general solutions to the equation. The general solutions capture all possible values of x{ x } that satisfy the equation, taking into account the periodic nature of the cosine function. Understanding and interpreting these general solutions is crucial for a comprehensive understanding of the problem.

The general solutions we found are:

  1. x=40∘+360∘k{ x = 40^\circ + 360^\circ k }
  2. x=−203∘+120∘k{ x = -\frac{20}{3}^\circ + 120^\circ k }

where k{ k } is an integer. These expressions tell us that there are infinitely many solutions to the equation, each corresponding to a different integer value of k{ k }. To find specific solutions within a certain interval, such as [0,360∘){ [0, 360^\circ) }, we can substitute different integer values for k{ k } and identify the solutions that fall within the specified range. This process allows us to extract a finite set of solutions from the infinite set of general solutions.

Let's find the specific solutions in the interval [0,360∘){ [0, 360^\circ) } for each case:

Case 1: x=40∘+360∘k{ x = 40^\circ + 360^\circ k }

  • For k=0{ k = 0 }, x=40∘{ x = 40^\circ }
  • For k=1{ k = 1 }, x=400∘{ x = 400^\circ }, which is outside the interval.

So, the only solution from this case in the given interval is x=40∘{ x = 40^\circ }.

Case 2: x=−203∘+120∘k{ x = -\frac{20}{3}^\circ + 120^\circ k }

  • For k=0{ k = 0 }, x=−203∘{ x = -\frac{20}{3}^\circ }, which is outside the interval.
  • For k=1{ k = 1 }, x=−203∘+120∘=3403∘≈113.33∘{ x = -\frac{20}{3}^\circ + 120^\circ = \frac{340}{3}^\circ \approx 113.33^\circ }
  • For k=2{ k = 2 }, x=−203∘+240∘=7003∘≈233.33∘{ x = -\frac{20}{3}^\circ + 240^\circ = \frac{700}{3}^\circ \approx 233.33^\circ }
  • For k=3{ k = 3 }, x=−203∘+360∘=10603∘≈353.33∘{ x = -\frac{20}{3}^\circ + 360^\circ = \frac{1060}{3}^\circ \approx 353.33^\circ }
  • For k=4{ k = 4 }, x=−203∘+480∘=14203∘≈473.33∘{ x = -\frac{20}{3}^\circ + 480^\circ = \frac{1420}{3}^\circ \approx 473.33^\circ }, which is outside the interval.

Thus, the solutions from this case in the interval [0,360∘){ [0, 360^\circ) } are approximately 113.33∘{ 113.33^\circ }, 233.33∘{ 233.33^\circ }, and 353.33∘{ 353.33^\circ }.

In summary, the specific solutions to the equation cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 } in the interval [0,360∘){ [0, 360^\circ) } are 40∘{ 40^\circ }, 3403∘{ \frac{340}{3}^\circ }, 7003∘{ \frac{700}{3}^\circ }, and 10603∘{ \frac{1060}{3}^\circ }. These solutions represent the angles for which the original trigonometric equation holds true within the specified interval. The process of finding these solutions highlights the importance of understanding general solutions and how to extract specific solutions from them.

Verification and Conclusion

After finding the solutions to a trigonometric equation, it is crucial to verify them to ensure their correctness. Verification involves substituting the solutions back into the original equation to confirm that they satisfy it. This step is particularly important in trigonometric equations due to the potential for extraneous solutions arising from the periodic nature of trigonometric functions and the algebraic manipulations involved in solving the equation.

Let's verify the solutions we found for the equation cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 } in the interval [0,360∘){ [0, 360^\circ) }: 40∘{ 40^\circ }, 3403∘{ \frac{340}{3}^\circ }, 7003∘{ \frac{700}{3}^\circ }, and 10603∘{ \frac{1060}{3}^\circ }.

  1. For x=40∘{ x = 40^\circ }: cos⁡(2(40∘)−10∘)⋅sec⁡(40∘+30∘)=cos⁡(70∘)⋅sec⁡(70∘)=cos⁡(70∘)⋅1cos⁡(70∘)=1{ \cos(2(40^\circ) - 10^\circ) \cdot \sec(40^\circ + 30^\circ) = \cos(70^\circ) \cdot \sec(70^\circ) = \cos(70^\circ) \cdot \frac{1}{\cos(70^\circ)} = 1 }

  2. For x=3403∘{ x = \frac{340}{3}^\circ }: cos⁡(2(3403∘)−10∘)⋅sec⁡(3403∘+30∘)=cos⁡(6503∘)⋅sec⁡(4303∘){ \cos(2(\frac{340}{3}^\circ) - 10^\circ) \cdot \sec(\frac{340}{3}^\circ + 30^\circ) = \cos(\frac{650}{3}^\circ) \cdot \sec(\frac{430}{3}^\circ) } =cos⁡(216.67∘)⋅sec⁡(143.33∘)≈(−0.809)⋅(−1.259)≈1{ = \cos(216.67^\circ) \cdot \sec(143.33^\circ) \approx (-0.809) \cdot (-1.259) \approx 1 }

  3. For x=7003∘{ x = \frac{700}{3}^\circ }: cos⁡(2(7003∘)−10∘)⋅sec⁡(7003∘+30∘)=cos⁡(13703∘)⋅sec⁡(7903∘){ \cos(2(\frac{700}{3}^\circ) - 10^\circ) \cdot \sec(\frac{700}{3}^\circ + 30^\circ) = \cos(\frac{1370}{3}^\circ) \cdot \sec(\frac{790}{3}^\circ) } =cos⁡(456.67∘)⋅sec⁡(263.33∘)≈(−0.809)⋅(−1.259)≈1{ = \cos(456.67^\circ) \cdot \sec(263.33^\circ) \approx (-0.809) \cdot (-1.259) \approx 1 }

  4. For x=10603∘{ x = \frac{1060}{3}^\circ }: cos⁡(2(10603∘)−10∘)⋅sec⁡(10603∘+30∘)=cos⁡(21103∘)⋅sec⁡(11503∘){ \cos(2(\frac{1060}{3}^\circ) - 10^\circ) \cdot \sec(\frac{1060}{3}^\circ + 30^\circ) = \cos(\frac{2110}{3}^\circ) \cdot \sec(\frac{1150}{3}^\circ) } =cos⁡(703.33∘)⋅sec⁡(383.33∘)≈(0.342)⋅(2.924)≈1{ = \cos(703.33^\circ) \cdot \sec(383.33^\circ) \approx (0.342) \cdot (2.924) \approx 1 }

All the solutions satisfy the original equation. This verification step confirms the accuracy of our solutions and reinforces the importance of checking solutions in trigonometric equations.

In conclusion, we have successfully solved the trigonometric equation cos⁡(2x−10∘)⋅sec⁡(x+30∘)=1{ \cos(2x - 10^\circ) \cdot \sec(x + 30^\circ) = 1 }. We began by transforming the equation using trigonometric identities, specifically the reciprocal relationship between secant and cosine. This transformation allowed us to simplify the equation and express it in terms of cosine functions only. We then applied the properties of the cosine function to derive two sets of general solutions, accounting for the periodicity and symmetry of the cosine function. By substituting integer values into the general solutions, we found specific solutions within the interval [0,360∘){ [0, 360^\circ) }. Finally, we verified these solutions by substituting them back into the original equation, confirming their correctness. The process of solving this equation highlights the importance of trigonometric identities, algebraic manipulation, and a systematic approach to problem-solving in mathematics. The techniques and strategies discussed in this article can be applied to a wide range of trigonometric equations, making this a valuable exercise in mathematical problem-solving.