Certainly! To determine all the solutions to the equation:
[tex]\[ (\cos 3\theta)(\cos \theta) + 1 = (\sin 3\theta)(\sin \theta) \][/tex]
we can follow these steps:
1. Utilize Trigonometric Identities:
To simplify the equation, we start by using some well-known trigonometric identities. Specifically, we use the triple angle formulas for cosine and sine:
[tex]\[ \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \][/tex]
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]
2. Substitute these identities into the given equation:
[tex]\[ (4\cos^3(\theta) - 3\cos(\theta))\cos(\theta) + 1 = (3\sin(\theta) - 4\sin^3(\theta))\sin(\theta) \][/tex]
3. Expand and Simplify the resulting equation:
[tex]\[ 4\cos^4(\theta) - 3\cos^2(\theta) + 1 = 3\sin^2(\theta) - 4\sin^4(\theta) \][/tex]
4. Express in terms of a single trigonometric function:
Since [tex]\(\sin^2(\theta) = 1 - \cos^2(\theta)\)[/tex], substitute into the equation:
[tex]\[ 4\cos^4(\theta) - 3\cos^2(\theta) + 1 = 3(1 - \cos^2(\theta)) - 4(1 - \cos^2(\theta))^2 \][/tex]
Simplifying this yields a polynomial in [tex]\(\cos(\theta)\)[/tex]. After solving this polynomial, we find the specific solutions for [tex]\(\theta\)[/tex] within the range [tex]\(0 \leq \theta < 2\pi\)[/tex].
5. Solve for [tex]\(\theta\)[/tex]:
This simplifies to obtaining multiple angles for [tex]\(\theta\)[/tex] that satisfy the equation. After solving the polynomial, we get the critical points.
Given our result, the solutions to the equation are:
[tex]\[ \theta = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
These values correspond to the angles in radians where the given equation holds true. Notice that these solutions fit within the principal range for [tex]\(\theta\)[/tex]; we can also express them in an extended form including cyclic repetitions due to the periodic nature of trigonometric functions.
Therefore, all the solutions to the equation [tex]\((\cos 3\theta)(\cos \theta) + 1 = (\sin 3\theta)(\sin \theta)\)[/tex] are:
[tex]\[ \theta = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
