How can I evaluate an expression like $\sin(3\pi/2)$ on a calculator and get an answer in terms of $\pi$? I have an expression like this that I need to evaluate:
$$16\sin(2\pi/3)$$
According to my book the answer is $8\sqrt{3}$. However, when I'm using my calculator to get this I get an answer like $13.86$. What I want to know, is it possible to make a calculator give the answer without evaluating $\pi$, so that $\pi$ is kept separate in the answer? And the same for in this case, $\sqrt{3}$. If the answer involves a square root, I want my calculator to say that, I don't want it to be evaluated.
I am using the TI-83 Plus if that makes a difference.
 A: Well we don't need a calculator here. There is no need to write anything in terms of $\pi$.
If you look closely at the sin function you will see that sin$(x) = $ sin$(\pi - x)$.
So sin$(\frac{2\pi}{3}) = $ sin$(\frac{\pi}{3})$.
But what is sin$(\frac{\pi}{3})$?
Well the answer is $\frac{\sqrt{3}}{2}$. You can get this by drawing an equilateral triangle of side length $2$ and splitting in half down the middle (to get a right angled triangle with one angle $\frac{\pi}{3}$ and side lengths $1$, $2$ and $\sqrt{3}$).
So the answer to your question is $16(\frac{\sqrt{3}}{2}) = 8\sqrt{3}$.
A: You may have learned some value of $\sin \theta$ . For example,
$\sin 0= 0$, $\sin \frac{\pi}{6}= \frac{1}{2}$ ,$\sin \frac{\pi}{4}= \frac{1}{\sqrt{2}}$,
$\sin \frac{\pi}{3}= \frac{\sqrt{3}}{2}$ and $\sin \frac{\pi}{2}= 1$, Other most useful value can be evaluated using these.
For example, in your case $$\sin\frac{2\pi}{3}= \sin{(\pi-\frac{\pi}{3}})$$
$$=\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$$
We know that $\sqrt{3}= 1.732..$, this gives
$$16.\sin\frac{2\pi}{3}=\frac{1.732..}{2}\times 16= 8\times 1.732.. = 13.856$$
A: Here’s something I used to tell students that might help. Among the angles that you’re typically expected to know the trig. values for ($30,$ $45,$ $60$ degrees and their cousins in the other quadrants), the only irrational values for the sine, cosine, tangent have the following magnitudes:
$$\frac{\sqrt{2}}{2}, \;\; \frac{\sqrt{3}}{2}, \;\; \sqrt{3}, \;\; \frac{\sqrt{3}}{3}$$
Note that if you square each of these, you get:
$$\frac{1}{2}, \;\; \frac{3}{4}, \;\; 3, \;\; \frac{1}{3}$$
Now consider the decimal expansions of these fractions:
$$0.5, \;\; 0.75, \;\; 3, \;\; 0.3333…$$
The important thing to notice is that if you saw any of these decimal expansions, you’d immediately know its fractional equivalent. (O-K, most people would know it!)
Now you can see how to use a relatively basic calculator to determine the exact value of $\sin\left(2\pi / 3 \right).$ First, use your calculator to find a decimal for $\sin\left(2\pi / 3 \right).$ Using a basic calculator (mine is a TI-32), I get $0.866025403.$ Now square the result. Doing this, I get $0.75.$ Therefore, I know that the square of $\sin\left(2\pi / 3 \right)$ is equal to $\frac{3}{4},$ and hence $\sin\left(2\pi / 3 \right)$ is equal to $\sqrt{\frac{3}{4}}.$ The positive square root is chosen because I got a positive value for $\sin\left(2\pi / 3 \right)$ when I used my calculator. Finally, we can rewrite this as $\frac{\sqrt{3}}{\sqrt{4}}=\frac{\sqrt{3}}{2}.$
What follows are some comments I posted in sci.math (22 June 2006) about this method.
By the way, I used to be very concerned in the early days of calculators that students could obtain all the exact values of the $30,$ $45,$ and $60$ degree angles by simply squaring the calculator output, recognizing the equivalent fraction of the resulting decimal [note that the squares of the sine, cosine, tangent of these angles all come out to fractions that any student would recognize (well, they used to be able to recognize) from its decimal expansion], and then taking the square root of the fraction. As the years rolled by, I got to where I didn't worry about this at all, because even when I taught this method in class (to help students on standardized tests and to help them for other teachers who were even more insistent about using exact values than I was), the majority of my students had more trouble doing this than just memorizing the values!
A: It's quite simple; simply enter your expression exactly as it's written into your TI-36X Pro ($18 at OfficeMax) while you're in RAD mode. 
The TI-36X Pro returns 8√3 quite nicely, or, it can return any answer in terms of pi.
A: Use a calculator
with a symbolic or CAS mode.
For example,
on the HP Prime,
in CAS mode,
when I entered
$\sin(2\pi/3)$,
I got $\frac12\sqrt{3}$.
A: No need for a calculator here.
Using polynomial division (or general knowledge), the polynomial $p=X^3-1\in \mathbb{C}[X]$ can be factored as $X^3-1=(X-1)(X^2+X+1)$. By employing the quadratic formula, we then see that $1$, $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $-\frac{1}{2}-\frac{\sqrt{3}}{2}i$ are the zeroes of $p$.
Using the complex exponential function, these third roots of unity can alternatively be calculated as $e^{2\pi i}$, $e^{\frac{2\pi}{3}i}$ and $e^{\frac{4\pi}{3}i}$. With Euler's formula, the equality $\operatorname{sin}\big(\frac{2\pi}{3}\big)=\frac{\sqrt{3}}{2}$ follows.
