Solving Trigonometric Equations: Finding Angles A And B
Introduction
In this article, we will dive into solving trigonometric equations, specifically focusing on problems that involve the difference of squares of cosine and sine functions. These types of problems often appear in mathematics, physics, and engineering contexts. Understanding how to manipulate trigonometric identities is crucial for simplifying and solving these equations. We will tackle two specific examples:
(a) If cos²(24°) - sin²(24°) = cos(A°), find A. (b) If cos²(6x) - sin²(6x) = cos(B), find B.
By walking through the solutions step-by-step, we will illustrate the key trigonometric identities and techniques needed to solve such problems. This will not only help in finding the specific values of A and B but also in building a strong foundation for tackling more complex trigonometric equations in the future.
(a) If cos²(24°) - sin²(24°) = cos(A°), find A
Understanding the Problem
The first part of our problem involves finding the value of A in the equation cos²(24°) - sin²(24°) = cos(A°). To solve this, we need to recognize a fundamental trigonometric identity. The expression cos²(θ) - sin²(θ) is a well-known identity that simplifies to cos(2θ). Recognizing this identity is the first key step in simplifying the equation and isolating the variable we want to solve for.
Applying the Trigonometric Identity
Let's start by stating the relevant trigonometric identity:
cos(2θ) = cos²(θ) - sin²(θ)
This identity is a direct result of the cosine double-angle formula. By applying this identity, we can rewrite the given equation in a much simpler form. In our case, θ = 24°, so we can substitute this value into the identity:
cos(2 * 24°) = cos²(24°) - sin²(24°)
This simplifies to:
cos(48°) = cos²(24°) - sin²(24°)
Now, we can substitute this back into the original equation:
cos(48°) = cos(A°)
Solving for A
Now that we have cos(48°) = cos(A°), we can easily solve for A. Since the cosine function is equal for the angles 48° and A°, the most straightforward solution is that A is equal to 48 degrees. However, we should also consider the properties of the cosine function to see if there are other possible solutions.
The cosine function has a period of 360°, meaning that cos(x) = cos(x + 360n) for any integer n. Additionally, cosine is an even function, meaning that cos(x) = cos(-x). Therefore, we need to consider these properties when finding all possible solutions for A.
The general solutions for the equation cos(A°) = cos(48°) can be expressed as:
A° = 48° + 360n
A° = -48° + 360n
where n is an integer. For practical purposes, especially in contexts where we are looking for angles within a single rotation (0° to 360°), we can consider n = 0. This gives us two primary solutions:
A = 48° (when n = 0 in the first equation)
A = -48° + 360° = 312° (when n = 0 in the second equation)
Thus, the two solutions for A within the range of 0° to 360° are 48° and 312°. However, if the context of the problem implies that we are looking for the smallest positive solution, then A = 48° is the most appropriate answer.
Conclusion for Part (a)
In conclusion, by recognizing and applying the cosine double-angle identity, we were able to simplify the given equation and find the value of A. The primary solution for A is 48°, but we also identified another solution, 312°, by considering the periodic and even nature of the cosine function. This illustrates the importance of understanding trigonometric identities and the properties of trigonometric functions when solving trigonometric equations.
(b) If cos²(6x) - sin²(6x) = cos(B), find B
Understanding the Problem
In the second part of our problem, we are given the equation cos²(6x) - sin²(6x) = cos(B) and we need to find the expression for B. Similar to the first part, this problem involves the difference of squares of cosine and sine functions. The key to solving this lies in recognizing and applying the same trigonometric identity we used before: the cosine double-angle identity.
Applying the Trigonometric Identity
The trigonometric identity we will use is:
cos(2θ) = cos²(θ) - sin²(θ)
In this case, our θ is 6x. Substituting 6x for θ in the identity, we get:
cos(2 * 6x) = cos²(6x) - sin²(6x)
This simplifies to:
cos(12x) = cos²(6x) - sin²(6x)
Now, we can substitute this back into the original equation:
cos(12x) = cos(B)
Solving for B
Now that we have cos(12x) = cos(B), we can determine the expression for B. Similar to the previous problem, we need to consider the general solutions for B, taking into account the periodic and even properties of the cosine function.
From the equation cos(12x) = cos(B), we can infer that B must be an angle that has the same cosine value as 12x. The general solutions for this equation can be expressed as:
B = 12x + 360n
B = -12x + 360n
where n is an integer. These solutions account for the periodic nature of the cosine function, where angles that are multiples of 360° apart have the same cosine value, and the even nature of the cosine function, where cos(x) = cos(-x).
If we are looking for the simplest expression for B, we can consider the case when n = 0. This gives us two primary solutions:
B = 12x (when n = 0 in the first equation)
B = -12x (when n = 0 in the second equation)
Since cosine is an even function, both 12x and -12x will yield the same cosine value. Therefore, we can express B as:
B = ±12x
Conclusion for Part (b)
In conclusion, by applying the cosine double-angle identity, we simplified the given equation and found the expression for B. The general solution for B is B = ±12x, which accounts for the periodic and even nature of the cosine function. This problem further illustrates the power of using trigonometric identities to simplify and solve trigonometric equations.
General Conclusion
In this article, we tackled two trigonometric problems involving the difference of squares of cosine and sine functions. We successfully found the value of A in the equation cos²(24°) - sin²(24°) = cos(A°) and the expression for B in the equation cos²(6x) - sin²(6x) = cos(B). The key to solving these problems was recognizing and applying the cosine double-angle identity:
cos(2θ) = cos²(θ) - sin²(θ)
By using this identity, we were able to simplify the equations and isolate the variables we needed to solve for. We also discussed the importance of considering the periodic and even properties of trigonometric functions when finding general solutions to trigonometric equations. These concepts are crucial for mastering trigonometry and solving more complex problems in related fields.
Understanding and applying trigonometric identities is a fundamental skill in mathematics, physics, and engineering. By practicing and familiarizing ourselves with these identities, we can efficiently solve a wide range of problems. The examples we discussed in this article serve as a solid foundation for tackling more challenging trigonometric equations in the future.
For further exploration of trigonometric identities and equations, you can visit trusted resources such as Khan Academy's Trigonometry Section.
