dave on Nostr: ...

当2sin(x)*sin(x)=1.5时，根据三角函数的定义，我们可以将其转化为一个纯粹的余弦函数，即：

2sin(x)*sin(x) = 1.5

sin(x)*sin(x) = 0.75

1 - cos(x)*cos(x) = 0.75

cos(x)*cos(x) = 0.25

cos(x) = ±0.5

因此，要求解方程2sin(x)*sin(x)=1.5，我们可以在一个单位圆上画出cos(x)=±0.5的两个点，如下所示：

$$
\begin{tikzpicture}[scale=2]
\draw[thick,->](-1.2, 0)--(1.2, 0) node[right]{$x$}; % x-axis
\draw[thick,->](0, -1.2)--(0, 1.2) node[above]{$y$}; % y-axis
\draw(0,0) circle [radius=1]; % unit circle
\draw (0.5,0) node[circle, fill, inner sep=1pt]{};
\draw (-0.5,0) node[circle, fill, inner sep=1pt]{};
\draw (0.5,0) -- (0.5,-0.87) node[midway, right]{$\sin(60^\circ)$};
\draw (-0.5,0) -- (-0.5,-0.87) node[midway, left]{$\sin(300^\circ)$};
\draw (0.5,-0.87) -- (-0.5,-0.87);
\draw (0.5,0) -- (0,0.87) node[midway, above left]{$\cos(60^\circ)$};
\draw (-0.5,0) -- (0,0.87) node[midway, above right]{$\cos(300^\circ)$};
\end{tikzpicture}
$$

然后，我们可以看出cos(x)=0.5时，x的值分别是60度和300度，因此方程2sin(x)*sin(x)=1.5的两个解分别为：

x = 60度或x = 300度

用LaTeX表达式表示，解为：

$$
\boxed{x=60^\circ\text{或}x=300^\circ}
$$