# Solve for $x: \sin 2x = - \frac 12$

I am trying to solve the following trigonometric equation algebraically, where $0\leq \displaystyle x \leq2 \pi$

$$\sin2x = -\frac{1}{2}.$$

My answer must be an exact solution.

Here is what I have tried: If $\displaystyle \sin(30°) = \frac{1}{2}$, then $\displaystyle \sin (2\times 15°) = \frac{1}{2}$.

Sin is negative in quadrants III and IV, so with a reference angle of $15°$ degree, $x$ can equal $180°+15°=\bf{195°}$, and $360°-30°=\bf{345°}$

Feeling confident about my answer, I checked the solution with my graphing calculator. The graph intersects the x-axis at $345°$ ($6.021$ rad), but it doesn't intersect the graph at $195°$.

Where did I make my mistake? I know I can work backwards, by taking the radian values from the graph and finding their "degree equivalents", but I need to be able to solve this algebraically.

Edit

210 degrees/2 = 105 degrees

330 degrees/2 = 165 degrees

210+360/2 = 285 degrees

330+360/2 = 345 degrees

Answer: x= 105˚, 165˚, 285˚, 345˚

• $360-30\not=345$ – teadawg1337 Dec 7 '14 at 21:51
• I think he means $360-15$. – Harto Saarinen Dec 7 '14 at 21:52
• The sine of what angle gives $-1/2$? This is one solution to $\sin \theta = -1/2$. Now solve $\theta = 2x$ for $x$. Also, draw a graph of $y = \sin 2x$. How many values between $0$ and $2\pi$ satisfy $\sin 2x = -1/2$? – Jon Dec 7 '14 at 21:52
• @teadawg1337 My mistake, I did mean 360-15=345. – McB Dec 7 '14 at 21:55
• Yes. sine of 2 times each of those 4 answers is -$\frac{1}{2}$ – turkeyhundt Dec 7 '14 at 22:10

Because the two values of sin that equal $-\frac{1}{2}$ are 210 and 330 (30 degrees below x axis). After getting these two answers, then you can divide the angle by 2.

You were correct that sin of 30 is $\frac{1}{2}$ and then looking to place the reference angle in the correct quadrants. You just need to think of the angle that when multiplied by 2 will give you the correct angles.

• 210/2 = 105 degrees, 330/2 = 165 degrees. Now how do I find the other two solutions? – McB Dec 7 '14 at 21:58
• There are only 2, he made a mistake. – Cyclohexanol. Dec 7 '14 at 21:58
• Since 210 and 330 were solutions, go around the circle one more time by adding 360. After dividing that by 2, you should still be withing the bounds... – turkeyhundt Dec 7 '14 at 21:58
• Never mind. you are correct. only 2 answers. – turkeyhundt Dec 7 '14 at 21:59
• @turkeyhundt: We need to find first the values of $y=2x$ between $0$ and $4\pi$ at which the sin of $y$ is $-1/2$. You were right in your first comment. – André Nicolas Dec 7 '14 at 22:02

There is a general formula:

$$\sin x = \sin \theta\implies x =\begin {cases} 2 k \pi + \theta,& k \in \mathbb Z \\\\ \text{or}\\\\ 2k\pi +\pi - \theta,&k \in \mathbb Z \end{cases}$$

It is sufficient to find a $\theta_0$ such that $\sin\theta_0=-\dfrac 12$ and then you can solve the inequalities

$$0\le 2k\pi +\theta_0\le2 \pi$$ and $$0\le 2k\pi +\pi -\theta_0\le 2 \pi$$ to find all suitable $k \in \mathbb Z$.

Notice that in your case you have $\sin(2x)$.

• And of course, if you want to calculate in degrees, you can replace each instance of $\pi$ by $180^\circ$. – Eff Dec 7 '14 at 23:07