# \ #

# \mbox{The equation can be simplified using the addition formula for} \ \ \mbox{the sine function. See below.} #

# \mbox{Given equation:} \qquad sin \theta cos 3\theta + cos \theta sin 3\theta \ = \ 0. #

# \mbox{Addition formula for sine:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad sin(x + y) \ = \ sin x cos y + cos x sin y. #

# \mbox{In reverse:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad sin x cos y + cos x sin y \ = \ sin(x + y). #

# \mbox{So we can rewrite the LHS of the given equation as:} #

# \qquad sin \theta cos 3\theta + cos \theta sin 3\theta \ = \ sin( \theta + 3\theta ) \ = \ sin( 4\theta ). #

# \mbox{Substituting back into the LHS of the given equation:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad sin( 4\theta ) \ = \ 0. #

# \mbox{This is easily solved now:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad sin( 4\theta ) \ = \ 0. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 4\theta \ = \ k \pi, \qquad k \ \ \mbox{any integer}. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \theta \ = \ k \pi/4, \qquad k \ \ \mbox{any integer}. \qquad \qquad \qquad \qquad (1) #

# \mbox{We want (restriction on} \ \theta \mbox{:)} \qquad \qquad 0 <= \theta < 2 \pi. #

# \mbox{Thus:} \qquad \qquad \qquad \qquad \qquad \qquad 0 <= k \pi/4 < 2 \pi, \qquad \qquad k \ \ \mbox{any integer}. #

# \mbox{As} \ \ 4/\pi > 0: \qquad \quad 4/\pi \cdot 0 <= 4/\pi \cdot k \pi/4 < 4/\pi \cdot 2 \pi, \qquad \qquad k \ \ \mbox{any integer}. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad 0 <= \color(red){cancel{4/\pi} } \cdot k\color(red){cancel{\pi/4} } < 4/\color(red){cancel{\pi} } \cdot 2 \color(red){cancel{\pi} }, \qquad \quad \ k \ \ \mbox{any integer}. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad 0 <= k < 8, \qquad \qquad \qquad \qquad \qquad \qquad \qquad k \ \ \mbox{any integer}. #

# \qquad \quad \ :. \qquad \qquad \qquad \quad k = 0, 1, 2, 3, 4, 5, 6, 7. #

# \quad \ :. \quad \mbox{(by eqn. (1))} \qquad \quad \theta \ = \ k \pi/4; \qquad k = 0, 1, 2, 3, 4, 5, 6, 7. #

# \quad \ :. \quad \qquad \quad \theta \ = 0, \pi/4, 2\pi/4, 3\pi/4, 4\pi/4, 5\pi/4, 6\pi/4, 7\pi/4. #

# \mbox{And finally ...} #

# \quad \ :. \quad \qquad \quad \theta \ = 0, \pi/4, \pi/2, {3\pi}/4, \pi, {5\pi}/4, {3\pi}/2, {7\pi}/4. #

# \ #

# \mbox{Summary:} #

# \mbox{Equation:} \qquad \quad sin \theta cos 3\theta + cos \theta sin 3\theta \ = \ 0; \quad \theta \in [0, 2\pi]. #

# \mbox{Solutions:} \quad \quad \qquad \quad \theta \ = 0, \pi/4, \pi/2, {3\pi}/4, \pi, {5\pi}/4, {3\pi}/2, {7\pi}/4. #

Describe your changes (optional) 200