Finding Angles A And B Given Trigonometric Equations

by ADMIN 53 views

In trigonometry, solving for unknown angles given trigonometric equations is a common and important task. This article aims to provide a comprehensive solution to the problem where we are given $\sin (A-B)=\frac{1}{2}$ and $\cos (A+B)=\frac{1}{2}$, with the constraints $0^{\circ} < A+B < 90^{\circ}$ and $A > B$. We will systematically break down the problem, leveraging our knowledge of trigonometric values for standard angles and algebraic manipulation to find the values of angles $A$ and $B$. This exploration will enhance your understanding of trigonometric equations and problem-solving strategies within trigonometry. The ability to manipulate and solve trigonometric equations is crucial in various fields such as physics, engineering, and navigation, where angles and their relationships are fundamental. This detailed solution not only addresses the specific problem but also reinforces the underlying principles of trigonometry, making it easier to tackle similar problems in the future. Throughout the solution, we will emphasize the importance of considering the given constraints, as they play a crucial role in narrowing down the possible solutions and ensuring the uniqueness of the answer. By carefully analyzing the trigonometric relationships and applying algebraic techniques, we will arrive at the precise values of angles $A$ and $B$ that satisfy the given conditions. This methodical approach will serve as a valuable guide for solving trigonometric problems and developing a deeper appreciation for the elegance and power of trigonometric principles.

Understanding the Problem

To effectively tackle this problem, we must first understand the given information and what we need to find. We are given two trigonometric equations: $\sin (A-B)=\frac1}{2}$ and $\cos (A+B)=\frac{1}{2}$. These equations relate the sine and cosine of the angles $(A-B)$ and $(A+B)$, respectively, to the value $\frac{1}{2}$. Additionally, we have two constraints $0^{\circ < A+B < 90^{\circ}$ and $A > B$. The first constraint limits the range of the angle $(A+B)$ to be between $0^{\circ}$ and $90^{\circ}$, which means it lies in the first quadrant. This is crucial because trigonometric functions have specific values within different quadrants. The second constraint, $A > B$, tells us that angle $A$ is greater than angle $B$, which is important for determining the signs and relative magnitudes of the angles. Our goal is to find the values of the angles $A$ and $B$ that satisfy both trigonometric equations and the given constraints. This involves using our knowledge of trigonometric values for standard angles, understanding the behavior of sine and cosine functions, and applying algebraic techniques to solve the equations. By systematically analyzing the given information and constraints, we can develop a clear strategy to find the unknown angles. Understanding the problem thoroughly is the first step towards a successful solution, as it allows us to approach the problem with a clear direction and avoid potential pitfalls. The constraints, in particular, play a vital role in ensuring that our solution is unique and consistent with the given conditions. This careful analysis will enable us to apply the appropriate trigonometric identities and algebraic manipulations to arrive at the correct answer.

Solving the Equations

To begin solving for angles $A$ and $B$, we need to utilize our knowledge of trigonometric values for standard angles. For the equation $\sin (A-B)=\frac1}{2}$, we recall that $\sin 30^{\circ} = \frac{1}{2}$. Therefore, one possible solution for $(A-B)$ is $30^{\circ}$. So, we can write our first equation as $A - B = 30^{\circ$. Next, we consider the equation $\cos (A+B)=\frac{1}{2}$. We know that $\cos 60^{\circ} = \frac{1}{2}$. Given the constraint $0^{\circ} < A+B < 90^{\circ}$, we can confidently say that $A + B = 60^{\circ}$. This gives us our second equation. Now we have a system of two linear equations with two variables, $A$ and $B$:

  1. A−B=30∘A - B = 30^{\circ}

  2. A+B=60∘A + B = 60^{\circ}

To solve this system, we can use the method of substitution or elimination. Here, the elimination method is straightforward. Adding the two equations, we get:

(A−B)+(A+B)=30∘+60∘(A - B) + (A + B) = 30^{\circ} + 60^{\circ}

2A=90∘2A = 90^{\circ}

Dividing both sides by 2, we find:

A=45∘A = 45^{\circ}

Now that we have the value of $A$, we can substitute it into either equation to find $B$. Let's use equation (2):

45∘+B=60∘45^{\circ} + B = 60^{\circ}

Subtracting $45^{\circ}$ from both sides, we get:

B=15∘B = 15^{\circ}

Thus, we have found that $A = 45^{\circ}$ and $B = 15^{\circ}$. It's important to verify that these values satisfy the given conditions and constraints. We will do this in the next section to ensure our solution is correct.

Verifying the Solution

After obtaining potential values for $A$ and $B$, it is crucial to verify that these values satisfy both the original trigonometric equations and the given constraints. This step ensures the correctness and validity of our solution. We found that $A = 45^{\circ}$ and $B = 15^{\circ}$. Let's first check the trigonometric equations. For the first equation, $\(sin (A-B)=\frac{1}{2}$, we substitute the values of $A$ and $B$:

sin⁡(45∘−15∘)=sin⁡(30∘)=12\sin (45^{\circ} - 15^{\circ}) = \sin (30^{\circ}) = \frac{1}{2}

This satisfies the first equation. Next, we check the second equation, $\cos (A+B)=\frac{1}{2}$:

cos⁡(45∘+15∘)=cos⁡(60∘)=12\cos (45^{\circ} + 15^{\circ}) = \cos (60^{\circ}) = \frac{1}{2}

This also satisfies the second equation. Now, we need to verify the constraints. The first constraint is $0^{\circ} < A+B < 90^{\circ}$. We have:

A+B=45∘+15∘=60∘A + B = 45^{\circ} + 15^{\circ} = 60^{\circ}

Since $0^{\circ} < 60^{\circ} < 90^{\circ}$, this constraint is satisfied. The second constraint is $A > B$. We have $A = 45^{\circ}$ and $B = 15^{\circ}$, so $45^{\circ} > 15^{\circ}$, which means this constraint is also satisfied. Since our values for $A$ and $B$ satisfy both trigonometric equations and the given constraints, we can confidently conclude that our solution is correct. Verification is a critical step in problem-solving, especially in mathematics, as it helps to identify and correct any potential errors. By ensuring that our solution aligns with all the given conditions, we can have confidence in the accuracy of our results. This methodical approach reinforces the importance of precision and attention to detail in solving mathematical problems.

Final Answer

In conclusion, by solving the given trigonometric equations and adhering to the specified constraints, we have successfully determined the values of angles $A$ and $B$. Through a systematic approach involving the use of standard trigonometric values and algebraic manipulation, we found that $A = 45^{\circ}$ and $B = 15^{\circ}$. These values were then rigorously verified against both the trigonometric equations and the constraints, ensuring the accuracy and validity of our solution. The problem required a careful understanding of trigonometric relationships and the ability to solve a system of linear equations. The constraints $0^{\circ} < A+B < 90^{\circ}$ and $A > B$ played a crucial role in narrowing down the possible solutions and ensuring the uniqueness of the answer. By checking these constraints, we confirmed that our solution was consistent with the given conditions. This exercise highlights the importance of a methodical approach in problem-solving, particularly in trigonometry. By breaking down the problem into smaller, manageable steps and applying relevant trigonometric identities and algebraic techniques, we were able to arrive at the correct answer. Furthermore, the verification step underscores the necessity of ensuring that the solution aligns with all the given conditions. This thorough approach not only provides the correct answer but also enhances our understanding of trigonometric principles and problem-solving strategies. Therefore, the final answer to the problem is:

A=45∘,B=15∘\boxed{A = 45^{\circ}, B = 15^{\circ}}

This result demonstrates the power of combining trigonometric knowledge with algebraic techniques to solve for unknown angles, a skill that is valuable in various mathematical and scientific contexts. The process of solving this problem reinforces the importance of precision, attention to detail, and a systematic approach in tackling complex mathematical challenges.