The problem asks us to find the number of values of \(x\) that satisfy the given trigonometric equation within the specified domain.

The given equation is: \(1 + 2(\sin x + \cos x)(\sin x - \cos x) = 0\), where \(0 < x < 360^\circ\).

Solving the Trigonometric Equation

Let's simplify the equation step by step:

1. Notice the product of the two binomials: \((\sin x + \cos x)(\sin x - \cos x)\). This is in the form of \((a+b)(a-b) = a^2 - b^2\).
2. So, \((\sin x + \cos x)(\sin x - \cos x) = \sin^2 x - \cos^2 x\).
3. Substitute this back into the original equation: \(1 + 2(\sin^2 x - \cos^2 x) = 0\).
4. We know the double angle identity for cosine: \(\cos(2x) = \cos^2 x - \sin^2 x\).
5. Therefore, \(\sin^2 x - \cos^2 x = -(\cos^2 x - \sin^2 x) = -\cos(2x)\).
6. Substitute this identity into the equation: \(1 + 2(-\cos(2x)) = 0\).
7. This simplifies to: \(1 - 2\cos(2x) = 0\).
8. Now, isolate \(\cos(2x)\): \(2\cos(2x) = 1\).
9. So, \(\cos(2x) = \frac{1}{2}\).

Finding the Values of 2x

We need to find the values of \(2x\) for which the cosine is \(\frac{1}{2}\). The general solution for \(\cos \theta = \frac{1}{2}\) is \(\theta = n \cdot 360^\circ \pm 60^\circ\), where \(n\) is an integer, because the principal value where cosine is \(\frac{1}{2}\) is \(60^\circ\).

In our case, \(\theta\) is \(2x\), so \(2x = n \cdot 360^\circ \pm 60^\circ\).

The given domain for \(x\) is \(0 < x < 360^\circ\). This means the domain for \(2x\) is \(0 < 2x < 720^\circ\).

Let's find the values of \(2x\) in the range \(0 < 2x < 720^\circ\) by trying different integer values for \(n\):

• For \(n = 0\): \(2x = 0 \cdot 360^\circ + 60^\circ = 60^\circ\). This is in the range. \(2x = 0 \cdot 360^\circ - 60^\circ = -60^\circ\). This is not in the range.
• For \(n = 1\): \(2x = 1 \cdot 360^\circ + 60^\circ = 360^\circ + 60^\circ = 420^\circ\). This is in the range. \(2x = 1 \cdot 360^\circ - 60^\circ = 360^\circ - 60^\circ = 300^\circ\). This is in the range.
• For \(n = 2\): \(2x = 2 \cdot 360^\circ + 60^\circ = 720^\circ + 60^\circ = 780^\circ\). This is not in the range. \(2x = 2 \cdot 360^\circ - 60^\circ = 720^\circ - 60^\circ = 660^\circ\). This is in the range.
• For \(n = -1\): \(2x = -1 \cdot 360^\circ + 60^\circ = -360^\circ + 60^\circ = -300^\circ\). This is not in the range. \(2x = -1 \cdot 360^\circ - 60^\circ = -360^\circ - 60^\circ = -420^\circ\). This is not in the range.

The values for \(2x\) that are within the range \(0 < 2x < 720^\circ\) are \(60^\circ, 300^\circ, 420^\circ, 660^\circ\).

Finding the Values of x

Now, we divide each value of \(2x\) by 2 to find the values of \(x\):

• \(2x = 60^\circ \implies x = \frac{60^\circ}{2} = 30^\circ\).
• \(2x = 300^\circ \implies x = \frac{300^\circ}{2} = 150^\circ\).
• \(2x = 420^\circ \implies x = \frac{420^\circ}{2} = 210^\circ\).
• \(2x = 660^\circ \implies x = \frac{660^\circ}{2} = 330^\circ\).

Let's check if these values of \(x\) are within the original domain \(0 < x < 360^\circ\):

• \(30^\circ\) is between \(0^\circ\) and \(360^\circ\).
• \(150^\circ\) is between \(0^\circ\) and \(360^\circ\).
• \(210^\circ\) is between \(0^\circ\) and \(360^\circ\).
• \(330^\circ\) is between \(0^\circ\) and \(360^\circ\).

All four values are within the domain. Therefore, there are four values of \(x\) that satisfy the equation.

Conclusion on the Number of Values

We found exactly four distinct values for \(x\) in the given domain \(0 < x < 360^\circ\) that satisfy the trigonometric equation \(1 + 2(\sin x + \cos x)(\sin x - \cos x) = 0\).

Revision Table: Key Trigonometric Identities

Additional Information: Solving Trigonometric Equations Steps

When solving trigonometric equations, it's useful to follow these steps:

• Simplify the equation: Use trigonometric identities to simplify the equation to a basic form involving a single trigonometric function of a single angle (like \(\sin x = k\) or \(\cos(2x) = k\)).
• Determine the general solution: Find the general formula for all possible angles that satisfy the basic equation using inverse trigonometric functions and periodicity.
• Consider the domain: Adjust the domain of the angle if the equation involves a multiple angle (like \(2x\), \(3x\), etc.). If the domain is \(0 < x < 360^\circ\), the domain for \(2x\) is \(0 < 2x < 720^\circ\).
• Find specific solutions within the domain: Use the general solution formula and test integer values (like \(n=0, 1, 2, \dots\)) to find all solutions that fall within the required domain for the modified angle (e.g., \(2x\)).
• Solve for the variable: Divide the solutions for the modified angle (e.g., \(2x\)) by the coefficient (e.g., 2) to get the values for the original variable (e.g., \(x\)).
• Verify solutions: Check if the final values of the variable fall within the original given domain. Count the number of valid solutions.