# Using the R method for finding all solutions to $\sin(2a) - \cos(2a) = \frac{\sqrt{6}}{2}$. My solution differs from official answer.

How many solutions does $$\sin(2a) - \cos(2a) = \frac{\sqrt{6}}{2}$$ have between $$-90^\circ$$ and $$90^\circ$$?

I used the R method and got $$2a-45^\circ = \arcsin\left(\frac{\sqrt{3}}{2}\right).$$ Since $$a$$ is between $$-90^\circ$$ and $$90^\circ$$, then $$2a$$ is between $$-180^\circ$$ and $$180^\circ$$. The RHS can be $$60^\circ$$, $$120^\circ$$, $$-240^\circ$$, and $$-300^\circ$$. Only $$60^\circ$$ and $$120^\circ$$ fit the criteria, but the answer is 4 solutions.

Where did I go wrong?

You are right. Here is the image of the function using google.

• Thanks, but i want to know whats wrong with this solution that got 4 as the answer. Sin 2a - cos 2a = root 6 /2, then he squared it, getting 1-2sin2acos2a= 3/2 which is sin 4a = -1/2 which does have 4 solutions. – SuperMage1 Jan 14 at 2:54
• Since I don't see how they got 4 solutions, there is no way to explain what they did wrong. Do you know which solutions they got? – Andrei Jan 14 at 2:56
• No solutions were given – SuperMage1 Jan 14 at 2:58
• @SuperMage1 when we square we add solutions $x=2$ square $x^2 = 4$ two solutions -/+2. If you square you need to return to the original equation and check the answers – AmerYR Jan 14 at 3:25
• @Ameryr That's why you have a plot of the original equation, so you can confirm that there are only two solutions, not 4 – Andrei Jan 14 at 3:28

Hint:

$$2a-45^\circ=180^\circ n+(-1)^n\arcsin\dfrac{\sqrt3}2$$

If $$n$$ is odd$$=2m+1$$(say)

$$2a-45=360m+180-60$$

But $$-180-45\le2a-45\le180-45$$

$$-225\le360m+120\le135$$

$$?\le m\le?$$

What if $$n$$ is even $$=2m$$(say)

• Sorry, but wat do i do after gettong the bounds in this solution, and it does seem a bit more tedious. – SuperMage1 Jan 14 at 3:03
• @SuperMage1, Please let me know if u find one better – lab bhattacharjee Jan 14 at 3:24

$$\sin(2a-45) = \frac{\sqrt{3}}{2}$$

$$2a-45 = 60 + 360n \Rightarrow a = 52.5 + 180n$$

$$2a-45 = 120+ 360n \Rightarrow a = 82.5 + 180n$$

Either $$a = 52.5$$ or $$a = 82.5$$. If you square

$$\sin(4a) = \frac{-1}{2}$$

$$4a = 210 + 360n \Rightarrow a = 52.5 + 90n$$

$$4a = 330 + 360n \Rightarrow a = 82.5 + 90n$$ First two solution $$a = 52.5$$

$$a= 52.5 - 90 = -38.5$$

The other two $$a = 82.5 , 82.5 - 90 = -7.5$$

In both cases the second solution is rejected see Checking $$\sin(-15) - \cos(-15) = \frac{-\sqrt{6}}{2}$$ which is wrong same for the other