How do you find the sine, cosine, or tangent of an angle without a calculator?

 
Relevant MM Lessons:This incredibly important skill requires a decent mastery of all of the underlying concepts. What follows will show the "how-to" steps, but will require your mastery of the prerequisites. Make sure you're up to snuff, and go through all of the related lessons listed above for in-depth coverage.

Before You Begin

The underlying reason that teachers expect us to be able to do this without a calculator is the strong presence of patterns. The unit circle, which has all the answers on it, demonstrates these patterns.Namely, if you find the point on the circle that lines up with the angle you want, the sine of the angle is the $y$ coordinate, and the cosine of the angle is the $x$ coordinate. Furthermore, all $30$° reference angles ($30$°, $150$°, $210$°, and $330$°) have the same numbers for their coordinates. Similarly, all $45$° reference angles ($45$°, $135$°, $225$°, and $315$°) and all $60$° reference angles ($60$°, $120$°, $240$°, and $300$°) have this commonality within their respective categories. The only difference is whether the numbers are positive or negative.First and foremost, memorize the first quadrant of the unit circle. This knowledge is what we'll use repeatedly when evaluating any other given angle. Check out my full lesson on evaluating sine and cosine with the unit circle » to see how I recommend going about this task.Second, make sure you know which trig functions are positive versus negative in which quadrants. Understanding why helps as well, to aide memory. Check out my full lesson on evaluating trig functions » to reinforce the foundation.Third, understand thoroughly what we mean when we say "reference angles". If you've read this far without knowing, you're probably already confused. Start here ».

Evaluating sin and cos

With the basics under your belt, here's the failsafe way to get the sine or cosine of any angle.Step 1 - ID the AngleWhat reference angle does it correspond with? This is the first question you need to answer, and if you're working in radians, you don't even have to think about it! When simplified properly, all radian angle measures with the same reference angle family have the same denominator. It's actually harder to do this whole process in degrees, for this reason. For example, both of the following lists are $30$° reference angles:$30$°, $150$°, $210$°, and $330$°$$\frac{\pi}{6}, \,\,\,\, \frac{5\pi}{6} \,\,\,\, \frac{7\pi}{6} \,\,\,\, \frac{11\pi}{6}$$Which one makes it clearer that they are all related? Also it's nice that you literally can't come up with a wrong answer if you use the denominator of $6$ - for angles between $0$ and $2\pi$, only these numerators work to give you the four angles in the family. If you pick any other numerator, you'll get something that can simplify to have a different denominator.Step 2 - ID the QuadrantNext, be aware what quadrant the angle you're using is in. If you're in degrees, use the $90$° quadrantal angle cutoffs to determine this. If you're working in radians, the best approach is to compare $\pi$ fractions to see where the angle lands. For example, if I'm working on $7\pi/6$, I know that it's bigger than $\pi$ because $\pi$ is $6\pi/6$. $7\pi/6$ is just a little bigger than $6\pi/6$, so I can be confident that $7\pi/6$ is in the third quadrant.Step 3 - Choose the SignWith the combination of which trig function you're looking at and which quadrant you're in, you can determine whether the answer is positive or negative. Remember All Students Take Calculus! » All trig functions are positive in quadrant 1, only the sine is positive in quadrant 2, only the tangent is positive in quadrant 3, and only the cosine is positive in quadrant 4.Step 4 - Choose the ValuePretend that your angle is really its quadrant 1 reference angle. Find the value of the trig function at that angle - that will be the number part of your answer. In conjunction with the plus or minus sign that you already determined in Step 3, you're done!

Let's Try Together

Let's try one example together. Check out the full lesson » for plenty of practice problems.Find the exact value of $\cos\left(\frac{3\pi}{4}\right)$.Step 1 - What angle is this? It's reference angle is $\pi/4$ (or $45$° for the degrees fans).Step 2 - What quadrant? $3/4$ is bigger than $2/4$ and smaller than $4/4$ so $3\pi/4$ is bigger than $\pi/2$ and smaller than $\pi$, and must lie in quadrant 2.Step 3 - Choose the sign. The cosine function yields negative answers in Q2, according to ASTC. Therefore our answer will be negative.Step 4 - Choose the value. This numeric value will match that of the cosine of $\pi/4$. That's a value we know (via memorization of the Q1 values). In this case we're looking for $\sqrt{2}/2$. Therefore, putting this all together, we have$$\cos\left(\frac{3\pi}{4}\right) = - \, \frac{\sqrt{2}}{2}$$

Evaluating Tangent

In my opinion, the only meaningful difference for tangent is that we do not have a unit circle that we could look at to get the answers. The points on the unit circle only describe the sines and cosines of angles.Instead of memorizing the Unit Circle's Q1 values like we did for sine and cosine, I would recommend memorizing the following table, though it can be derived quickly on a test should you forget by using $\tan(x) = \sin(x) / \cos(x)$:$$ \begin{array}{l|r} \theta & \tan(\theta) \\ \hline 0 & 0 \\ \frac{\pi}{6} & \frac{\sqrt{3}}{3} \\ \frac{\pi}{4} & 1 \\ \frac{\pi}{3} & \sqrt{3} \\ \frac{\pi}{2} & \mathrm{DNE} \end{array} $$Here's the way I go about remembering these facts: start with $0$, which has a tangent of $0$ because $\sin(0) = 0$. Also, once you become familiar with graphs of tangent functions », you'll also be able to recall that $(0,0)$ is a point on the graph of $\tan(x)$. Second, $\tan(\pi/4)$ is $1$ because that's the place that $\sin(x)$ and $\cos(x)$ have the same value. Then, $\pi/2$ doesn't have a defined tangent because $\cos$ is zero at that angle. Finally, we have the "root three" answers. I used to mix up which is which, but the anchor I hold on to is the fact that the tangent function gets larger as you go along. $\sqrt{3}$ is bigger than $1$, so it must belong to the $60$° angle, not the $30$° one.After that, the same four steps apply. Figure out your reference angles and quadrant locations to get the values and negative signs that you need for your answer.