Question

A solution \((x, y)\) of the system of equation \(x-y=1 / 3\) and \(\cos ^{2}(\pi x)-\sin ^{2}(\pi y)=1 / 2\) is given by (1) \((2 / 3,1 / 3)\) (2) \((5 / 3,4 / 3)\) (3) \((13 / 6,11 / 6)\) (4) \((5 / 12,1 / 12)\)

Short answer

(\( \frac{13}{6}, \frac{11}{6} \)) is the solution.

Step-by-step solution

Analyze the First Equation

The first equation is given by: \[ x - y = \frac{1}{3} \] We will need to check which pair \(x, y\) satisfies this equation.

Analyze the Second Equation

The second equation is given by: \[ \cos^{2}(\pi x) - \sin^{2}(\pi y) = \frac{1}{2} \] We will need to check which pair \(x, y\) satisfies this equation.

Check Each Given Pair

Let's check each given pair against both equations.**Option (1) (\(\frac{2}{3}, \frac{1}{3}\))**:1. For the first equation: \[ \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \] This holds true.2. For the second equation: \[ \cos^{2}(\pi \cdot \frac{2}{3}) - \sin^{2}(\pi \cdot \frac{1}{3}) = \cos^{2}(\frac{2\pi}{3}) - \sin^{2}(\frac{\pi}{3}) \] We know that \[ \cos(\frac{2\pi}{3}) = -\frac{1}{2} \] and \[ \sin(\frac{1\pi}{3}) = \frac{\sqrt{3}}{2} \] Therefore: \[ \left( -\frac{1}{2} \right)^{2} - \left( \frac{\sqrt{3}}{2} \right)^{2} = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \] This does not hold true.**Option (2) (\(\frac{5}{3}, \frac{4}{3}\))**:1. For the first equation: \[ \frac{5}{3} - \frac{4}{3} = \frac{1}{3} \] This holds true.2. For the second equation: \[ \cos^{2}(\pi \cdot \frac{5}{3}) - \sin^{2}(\pi \cdot \frac{4}{3}) = \cos^{2}(\frac{5\pi}{3}) - \sin^{2}(\frac{4\pi}{3}) \] We know that \[ \cos(\frac{5\pi}{3}) = \frac{1}{2} \] and \[ \sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2} \] Therefore: \[ \left( \frac{1}{2} \right)^{2} - \left( -\frac{\sqrt{3}}{2} \right)^{2} = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \] This does not hold true.**Option (3) (\( \frac{13}{6}, \frac{11}{6} \))**:1. For the first equation: \[ \frac{13}{6} - \frac{11}{6} = \frac{1}{3} \] This holds true.2. For the second equation: \[ \cos^{2}(\pi \cdot \frac{13}{6}) - \sin^{2}(\pi \cdot \frac{11}{6}) = \cos^{2}(\frac{13\pi}{6}) - \sin^{2}(\frac{11\pi}{6}) \] We know that \[ \cos(\frac{13\pi}{6}) = \frac{\sqrt{3}}{2} \] and \[ \sin(\frac{11\pi}{6}) = -\frac{1}{2} \] Therefore: \[ \left( \frac{\sqrt{3}}{2} \right)^{2} - \left( -\frac{1}{2} \right)^{2} = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \] This holds true. Therefore, \( \frac{13}{6}, \frac{11}{6} \) is a solution.**Option (4) (\(5/12, 1/12\))**:1. For the first equation: \[ \frac{5}{12} - \frac{1}{12} = \frac{1}{3} \] This does not hold true.

Conclusion

From the analysis, Option (3) (\( \frac{13}{6}, \frac{11}{6} \)) satisfies both equations.

Key concepts

Trigonometric Equations

In this problem, we encounter a trigonometric equation: \ \ \[\cos^{2}(\pi x) - \sin^{2}(\pi y) = \frac{1}{2}\] \ This is an important aspect of the problem and understanding how to manipulate trigonometric functions is crucial. Trigonometric identities often simplify complex expressions. In this case, we use: \ \ \[\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta\] \ So \ \ \[\cos 2(\pi x) = 2\cos^{2}(\pi x) - 1\] \ helps us understand the manipulation needed to solve our equation. Remember to also know your key trigonometric values: \ \ \[\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}, \text{ and } \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\] \ These are common values that are likely to appear in equations and can help quickly verify solutions. Calculators can sometimes be necessary too for more complex angles.

Linear Equations

Linear equations are fundamental and straightforward to solve. Here, we deal with a simple linear equation: \ \ \[x - y = \frac{1}{3}\] \ By solving for one variable in terms of the other (transposing), we simplify the evaluation. For example: \ \ \[x = y + \frac{1}{3}\] \ This allows us to substitute \(y\) with \(x - \frac{1}{3}\). Remember to always isolate one variable when possible:

• Check if you can simplify the form for easier evaluation.
• Keep equations balanced by performing the same operation on both sides.

Linear equations are the basis of more complex systems. Solve them first to simplify the remaining problem.

Solving Equations

Combining both trigonometric and linear equations requires strategy. First, isolate variables and substitute them into the trigonometric equation. Let's revisit: \ \ \[x - y = \frac{1}{3}\] \[ \cos^{2}(\pi x) - \sin^{2}(\pi y) = \frac{1}{2}\] \ Follow these steps:

• Solve the linear equation first to find relationships between variables.
• Use the relationships as substitutions in the trigonometric equation.
• Simplify the trigonometric equation using identities.
• Verify solutions against both original equations.

Through this, we decreased our options by validating each pair, ultimately finding that: \ \ \[ \left( \frac{13}{6}, \frac{11}{6} \right) \] met criteria for both equations, showing systematic approach in problem-solving.