Finding the range of $3+\frac{3}{2}\sin 2\theta + 2\cos 2\theta$

Question

How to find range of this function? $$3+\frac{3}{2}\sin 2\theta + 2\cos 2\theta$$

The original equation I had was $$\sin^2\theta+ 3\sin \theta \cos \theta +5\cos^2 \theta$$ If possible, can I also find range from original directly?

You can look here too: https://math.stackexchange.com/questions/2855431/how-do-i-find-the-range-of-y-a-sin-x-b-cos-x — bigbang, Jun 27 '21 at 18:17

score 2 · Accepted Answer · answered Jun 27 '21 at 18:59

I believe that simplifying the original formula as you have done was the right approach. Continuing, it can be written as

$3+\frac 3 2 \sin 2\theta+2 \cos 2\theta$

$=3+A\cos(\phi+2\theta)$ for some constants A and $\phi$

But since $A=\sqrt {\left(\frac 3 2\right)^2+ 2^2}=\frac 5 2$ we have,

$=3+ \frac 5 2 \cos(\phi+2\theta)$

and cos ranges from $-1$ to $1$ so our formula ranges from $\frac 1 2$ to $\frac {11} 2$

score 1 · Answer 2 · answered Jun 27 '21 at 18:30

1

Another approach to this could be finding extrema of your function.

That is, take the first derivative of the function, put it to zero and find roots. I won't go further, because this is well known schema, read about it here.

answered Jun 27 '21 at 18:30

valuemis

156

Finding the range of $3+\frac{3}{2}\sin 2\theta + 2\cos 2\theta$

2 Answers2