The vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ satisfy $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{c}\| = 2,$ and \[\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c},$ then find all possible values of $\theta,$ in degrees.
What I tried:
I began with subtracting both sides with b, and taking the magnitude.
I get \[|\mathbf{a} \times (\mathbf{a} \times \mathbf{c})|=1\]
\(|a|*|c|*\sin(\theta)*\sin(\theta)=1\)
\(\sin^2(\theta)=1/2, \sin(\theta)=\dfrac{1}{\sqrt2}\)
\(\theta=45, 135, 225, 315\)
The answer registered it incorrect, can somebody help?
