Trigonometric functions can have several solutions. Sine, cosine and tangent all have different positive or negative values depending on what quadrant they are in.

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Many students think they've solved a trig equation when they get one answer (one size of angle \(x^\circ\)). However there's often more than one answer expected so be aware of this.

Solve the equation \(\sin x^\circ = 0.5\), where 0 ≤ x < 360.

Let's remind ourselves of what the sine graph looks like so that we can see how many solutions we should be expecting:

Therefore, from the graph of the function, we can see that we should be expecting two solutions: one solution being between 0˚ and 90˚ and the other solution between 90˚ and 180˚.

\[\sin x^\circ = 0.5\]

\[x^\circ = {\sin ^{ - 1}}(0.5)\]

\[x^\circ = 30^\circ\]

So we know that the first solution is 30˚ as previously predicted from the graph.

To get the other solution, we need to go back to our quadrants and use the appropriate rule:

Therefore since the trig equation we are solving is sin and it is positive (0.5), then we are in the first and second quadrants. We have already found the first solution which is the acute angle from the first quadrant, so to find the second solution, we need to use the rule in the second quadrant.

\[x^\circ = 180^\circ - 30^\circ\]

\[x^\circ = 150^\circ\]

Therefore \(x^\circ = 30^\circ ,150^\circ\)

Now try the example questions below.

- Question
Solve the equation \(\sin x^\circ = -0.349\), where 0 ≤ x < 360.

From the graph of the function, we can see that we should be expecting two solutions: one solution between 180° and 270° and the other between 270° and 360°.

\[\sin x^\circ = - 0.349\]

Since this is sin, but negative, this means that we will be in the two quadrants where the sine function is negative, ie the third and fourth quadrants. We need to firstly find the acute angle to use with the rules in these quadrants.

When calculating the acute angle, we ignore the negative.

\[x^\circ = {\sin ^{ - 1}}(0.349)\]

\[x^\circ = 20.4^\circ \,(to\,1d.p.)\]

**Third quadrant**\[x^\circ = 180^\circ + 20.4^\circ\]

\[x^\circ = 200.4^\circ\]

**Fourth quadrant**\[x^\circ = 360^\circ - 20.4^\circ\]

\[x^\circ = 339.6^\circ\]

Therefore: \(x^{\circ}= 200.4^{\circ}, 339.6^{\circ}\)

- Question
Solve the equation \(4\sin x^\circ - 3 = 0\), where \(0 \le x \textless 360\).

When solving a question like this one, we need to rearrange it first.

\[4\sin x^\circ - 3 = 0\]

\[4\sin x^\circ = 0 + 3\]

\[4\sin x^\circ = 3\]

\[\sin x^\circ = \frac{3}{4}\]

The graph of this function looks like this:

From the graph of the function, we can see that we should be expecting two solutions: one solution between 0° and 90° and the other between 90° and 180°.

\[\sin x^\circ = \frac{3}{4}\]

Since this is sin and is positive this means that we will be in the two quadrants in which the sine function is positive, ie the first and second quadrants.

**First quadrant**\[\sin x^\circ = \frac{3}{4}\]

\[x^\circ = {\sin ^{ - 1}}(\frac{3}{4})\]

\[x=sin^{-1}0.75\]

\[x^\circ = 48.6^\circ (to\,1\,d.p.)\]

**Second quadrant**\[x^\circ = 180^\circ - 48.6^\circ\]

\[x^\circ = 131.4^\circ\]

Therefore \(x^\circ = 48.6^\circ ,131.4^\circ\)