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Sagot :
To solve the trigonometric equation \(\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} = 2\), we need to proceed systematically to find all possible values of \(A\).
### Step 1: Use Trigonometric Identities
We start by utilizing trigonometric identities to simplify the given expressions.
Recall the trigonometric identities for \(\sin 3A\) and \(\cos 3A\):
[tex]\[ \sin 3A = 3 \sin A - 4 \sin^3 A \][/tex]
[tex]\[ \cos 3A = 4 \cos^3 A - 3 \cos A \][/tex]
### Step 2: Rewrite the Equation
Substitute the trigonometric identities into the given equation:
[tex]\[ \frac{3 \sin A - 4 \sin^3 A}{\sin A} - \frac{4 \cos^3 A - 3 \cos A}{\cos A} = 2 \][/tex]
This simplifies to:
[tex]\[ 3 - 4 \sin^2 A - 4 \cos^2 A + 3 = 2 \][/tex]
### Step 3: Simplify the Expression
We know from the Pythagorean identity that:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
Using this identity, we can rewrite:
[tex]\[ 3 - 4 \sin^2 A - 4 \cos^2 A + 3 = 2 \][/tex]
### Step 4: Combine Like Terms
Combine the constants and use the Pythagorean identity:
[tex]\[ 6 - 4 (\sin^2 A + \cos^2 A) = 2 \][/tex]
Since \(\sin^2 A + \cos^2 A = 1\), we have:
[tex]\[ 6 - 4 \cdot 1 = 2 \][/tex]
[tex]\[ 6 - 4 = 2 \][/tex]
[tex]\[ 2 = 2 \][/tex]
Thus, the original equation simplifies correctly, confirming our approach is consistent.
### Step 5: Solve For \( A \)
The remaining task is to identify the values of \(A\) that satisfy the original equation within the range of trigonometric functions:
By examination and known trigonometric solutions:
1. \(\sin 3A\) implies solutions at multiples of \(\pi\).
2. \(\cos 3A\) implies additional restrictions valid for angles such that the equation is held true.
### Solution Conclusion:
The set of solutions that satisfy the given trigonometric equation are:
[tex]\[ A = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
These solutions accommodate the periodic nature of sine and cosine functions ensuring the trigonometric identity holds true for the given angle solutions.
Thus, the solutions to the given equation \(\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} = 2\) are:
[tex]\[ A = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
### Step 1: Use Trigonometric Identities
We start by utilizing trigonometric identities to simplify the given expressions.
Recall the trigonometric identities for \(\sin 3A\) and \(\cos 3A\):
[tex]\[ \sin 3A = 3 \sin A - 4 \sin^3 A \][/tex]
[tex]\[ \cos 3A = 4 \cos^3 A - 3 \cos A \][/tex]
### Step 2: Rewrite the Equation
Substitute the trigonometric identities into the given equation:
[tex]\[ \frac{3 \sin A - 4 \sin^3 A}{\sin A} - \frac{4 \cos^3 A - 3 \cos A}{\cos A} = 2 \][/tex]
This simplifies to:
[tex]\[ 3 - 4 \sin^2 A - 4 \cos^2 A + 3 = 2 \][/tex]
### Step 3: Simplify the Expression
We know from the Pythagorean identity that:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
Using this identity, we can rewrite:
[tex]\[ 3 - 4 \sin^2 A - 4 \cos^2 A + 3 = 2 \][/tex]
### Step 4: Combine Like Terms
Combine the constants and use the Pythagorean identity:
[tex]\[ 6 - 4 (\sin^2 A + \cos^2 A) = 2 \][/tex]
Since \(\sin^2 A + \cos^2 A = 1\), we have:
[tex]\[ 6 - 4 \cdot 1 = 2 \][/tex]
[tex]\[ 6 - 4 = 2 \][/tex]
[tex]\[ 2 = 2 \][/tex]
Thus, the original equation simplifies correctly, confirming our approach is consistent.
### Step 5: Solve For \( A \)
The remaining task is to identify the values of \(A\) that satisfy the original equation within the range of trigonometric functions:
By examination and known trigonometric solutions:
1. \(\sin 3A\) implies solutions at multiples of \(\pi\).
2. \(\cos 3A\) implies additional restrictions valid for angles such that the equation is held true.
### Solution Conclusion:
The set of solutions that satisfy the given trigonometric equation are:
[tex]\[ A = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
These solutions accommodate the periodic nature of sine and cosine functions ensuring the trigonometric identity holds true for the given angle solutions.
Thus, the solutions to the given equation \(\frac{\sin 3A}{\sin A} - \frac{\cos 3A}{\cos A} = 2\) are:
[tex]\[ A = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
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