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Sagot :
To solve the given equations analytically, we begin with the equations:
1. [tex]\(2 \cos x \cos y = 1\)[/tex]
2. [tex]\(\tan x + \tan y = 2\)[/tex]
Let's break down the steps to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
### Step 1: Simplify the First Equation
We start with the first equation:
[tex]\[ 2 \cos x \cos y = 1 \][/tex]
This can be simplified to:
[tex]\[ \cos x \cos y = \frac{1}{2} \][/tex]
### Step 2: Consider Possible [tex]\(x\)[/tex] and [tex]\(y\)[/tex] Pairs
We now need to find pairs [tex]\((x, y)\)[/tex] such that [tex]\(\cos x \cos y = \frac{1}{2}\)[/tex].
### Step 3: Simplify the Second Equation
Consider the second equation:
[tex]\[ \tan x + \tan y = 2 \][/tex]
### Step 4: Test Possible Pairs
We will test each of the given pairs to see if they satisfy both equations.
#### Option A: [tex]\(x = 45^\circ, y = 45^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(45^\circ = \frac{\pi}{4}\)[/tex]
2. Calculate [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 45^\circ \cos 45^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{1}{2} \][/tex]
[tex]\[ \cos 45^\circ \cos 45^\circ = \frac{1}{2} \quad \text{(correct)} \][/tex]
4. Calculate [tex]\(\tan 45^\circ = 1\)[/tex]
5. Verify the second equation:
[tex]\[ \tan 45^\circ + \tan 45^\circ = 1 + 1 = 2 \][/tex]
[tex]\[ \tan 45^\circ + \tan 45^\circ = 2 \quad \text{(correct)} \][/tex]
Since both conditions are satisfied, [tex]\((x, y) = (45^\circ, 45^\circ)\)[/tex] is a valid solution.
#### Option B: [tex]\(x = 45^\circ, y = 60^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(45^\circ = \frac{\pi}{4}, 60^\circ = \frac{\pi}{3}\)[/tex]
2. Calculate [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 45^\circ \cos 60^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) = \frac{\sqrt{2}}{4} \][/tex]
[tex]\[ \frac{\sqrt{2}}{4} \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (45^\circ, 60^\circ)\)[/tex] is not correct.
#### Option C: [tex]\(x = 60^\circ, y = 90^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(60^\circ = \frac{\pi}{3}, 90^\circ = \frac{\pi}{2}\)[/tex]
2. Calculate [tex]\(\cos 90^\circ = 0\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 60^\circ \cos 90^\circ = \left(\frac{1}{2}\right) (0) = 0 \][/tex]
[tex]\[ 0 \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (60^\circ, 90^\circ)\)[/tex] is not correct.
#### Option D: [tex]\(x = 90^\circ, y = 120^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(90^\circ = \frac{\pi}{2}, 120^\circ = \frac{2\pi}{3}\)[/tex]
2. Calculate [tex]\(\cos 90^\circ = 0\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 90^\circ \cos 120^\circ = (0) \left(-\frac{1}{2}\right) = 0 \][/tex]
[tex]\[ 0 \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (90^\circ, 120^\circ)\)[/tex] is not correct.
Therefore, the correct answer is:
A. [tex]\(45^\circ, 45^\circ\)[/tex]
1. [tex]\(2 \cos x \cos y = 1\)[/tex]
2. [tex]\(\tan x + \tan y = 2\)[/tex]
Let's break down the steps to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
### Step 1: Simplify the First Equation
We start with the first equation:
[tex]\[ 2 \cos x \cos y = 1 \][/tex]
This can be simplified to:
[tex]\[ \cos x \cos y = \frac{1}{2} \][/tex]
### Step 2: Consider Possible [tex]\(x\)[/tex] and [tex]\(y\)[/tex] Pairs
We now need to find pairs [tex]\((x, y)\)[/tex] such that [tex]\(\cos x \cos y = \frac{1}{2}\)[/tex].
### Step 3: Simplify the Second Equation
Consider the second equation:
[tex]\[ \tan x + \tan y = 2 \][/tex]
### Step 4: Test Possible Pairs
We will test each of the given pairs to see if they satisfy both equations.
#### Option A: [tex]\(x = 45^\circ, y = 45^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(45^\circ = \frac{\pi}{4}\)[/tex]
2. Calculate [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 45^\circ \cos 45^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{1}{2} \][/tex]
[tex]\[ \cos 45^\circ \cos 45^\circ = \frac{1}{2} \quad \text{(correct)} \][/tex]
4. Calculate [tex]\(\tan 45^\circ = 1\)[/tex]
5. Verify the second equation:
[tex]\[ \tan 45^\circ + \tan 45^\circ = 1 + 1 = 2 \][/tex]
[tex]\[ \tan 45^\circ + \tan 45^\circ = 2 \quad \text{(correct)} \][/tex]
Since both conditions are satisfied, [tex]\((x, y) = (45^\circ, 45^\circ)\)[/tex] is a valid solution.
#### Option B: [tex]\(x = 45^\circ, y = 60^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(45^\circ = \frac{\pi}{4}, 60^\circ = \frac{\pi}{3}\)[/tex]
2. Calculate [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 45^\circ \cos 60^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) = \frac{\sqrt{2}}{4} \][/tex]
[tex]\[ \frac{\sqrt{2}}{4} \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (45^\circ, 60^\circ)\)[/tex] is not correct.
#### Option C: [tex]\(x = 60^\circ, y = 90^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(60^\circ = \frac{\pi}{3}, 90^\circ = \frac{\pi}{2}\)[/tex]
2. Calculate [tex]\(\cos 90^\circ = 0\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 60^\circ \cos 90^\circ = \left(\frac{1}{2}\right) (0) = 0 \][/tex]
[tex]\[ 0 \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (60^\circ, 90^\circ)\)[/tex] is not correct.
#### Option D: [tex]\(x = 90^\circ, y = 120^\circ\)[/tex]
1. Convert degrees to radians: [tex]\(90^\circ = \frac{\pi}{2}, 120^\circ = \frac{2\pi}{3}\)[/tex]
2. Calculate [tex]\(\cos 90^\circ = 0\)[/tex]
3. Verify the first equation:
[tex]\[ \cos 90^\circ \cos 120^\circ = (0) \left(-\frac{1}{2}\right) = 0 \][/tex]
[tex]\[ 0 \neq \frac{1}{2} \quad \text{(incorrect)} \][/tex]
Thus, [tex]\((x, y) = (90^\circ, 120^\circ)\)[/tex] is not correct.
Therefore, the correct answer is:
A. [tex]\(45^\circ, 45^\circ\)[/tex]
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