Solving trigonometric equations in degrees


Solve the equation \sin x^\circ  = 0.5, where 0 \le x \textless 360.


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

Sine graph displaying two solutions

Therefore, from the graph of the function, we can see that we should be expecting 2 solutions: 1 solution being between 0^\circ and 90^\circ and the other solution between 90^\circ and 180^\circ.

\sin x^\circ  = 0.5

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

x^\circ  = 30^\circ

So we know that the first solution is 30^\circ as previously predicted from the graph.

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

Top r: 1st quadrant, x degrees, all positive. Top l: 2nd quadrant, 180-x degrees, Sin positive. Bottom l: 3rd quadrant, 180+x degrees, Tan positive. Bottom r: 4th quadrant, 360-x degrees, Cos positive

Therefore since the trig equation we are solving is sin and it is positive (0.5), then we are in the 1st and 2nd quadrants.

We have already found the first solution which is the acute angle from the 1st quadrant, so to find the second solution, we need to use the rule in the 2nd quadrant.

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

x^\circ  = 150^\circ

x^\circ  = 30^\circ ,\,150^\circ


Solve the equation \sin x^\circ  =  - 0.349, where 0 \le x \textless 360.

Sine graph with two solutions when y=-0.349

From the graph of the function, we can see that we should be expecting 2 solutions: 1 solution between 180^\circ and 270^\circ and the other between 270^\circ and 360^\circ.

\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 - the third and fourth quadrants.

We need to firstly find the acute angle to use with the rules in these quadrants.

\sin x^\circ  =  - 0.349

When calculating the acute angle, we ignore the negative.

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

x^\circ  = 20.4261...

x^\circ  = 20.4^\circ (to 1 d.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