Basic trigonometric equations
Example
Solve the equation \(4\sin x^\circ - 3 = 0\), where \(0 \le x \textless 360\).
Solution
First rearrange the equation.
\(4\sin x^\circ - 3 = 0\)
\(4\sin x^\circ = 0 + 3\)
\(4\sin x^\circ = 3\)
\(\sin x^\circ = \frac{3}{4}\)
The graph of this function looks like this:
From the graph of the function, we can see that we should be expecting 2 solutions: 1 solution between \(0^\circ\) and \(90^\circ\) and the other between \(90^\circ\) and \(180^\circ\).
\(\sin x^\circ = \frac{3}{4}\)
Since this is sin and is positive this means that we will be in the two quadrants where the sine function is positive - the first and second quadrants.
First quadrant
\(\sin x^\circ = \frac{3}{4}\)
\(x^\circ = {\sin ^{ - 1}}\left( {\frac{3}{4}} \right)\)
\(x^\circ = 48.59037...\)
\(x^\circ = 48.6^\circ\) (to 1 d.p.)
Second quadrant
\(x^\circ = 180^\circ - 48.6^\circ\)
\(x^\circ = 131.4^\circ\)
Therefore \(x^\circ = 48.6^\circ ,\,131.4^\circ\)