Solving the Equation $\sin x = a$
📌 Why This Matters
One of the most fundamental equations in trigonometry is $\sin x = a$. Unlike algebraic equations that may have one or two solutions, trigonometric equations can have infinitely many solutions because the sine function is periodic. Learning to express all solutions using a general formula is a key skill.
🔑 Key Prerequisites
Range of Sine:
$-1 \leq \sin x \leq 1$
So $\sin x = a$ has solutions only if $|a| \leq 1$.
$-1 \leq \sin x \leq 1$
So $\sin x = a$ has solutions only if $|a| \leq 1$.
Period of Sine:
$\sin(x + 2\pi) = \sin x$
The function repeats every $2\pi$ radians ($360°$).
$\sin(x + 2\pi) = \sin x$
The function repeats every $2\pi$ radians ($360°$).
Symmetry Property:
$\sin(\pi - x) = \sin x$
This gives us the second family of solutions.
$\sin(\pi - x) = \sin x$
This gives us the second family of solutions.