# Calculating trigonometry without calculator?

• B
• Xforce
In summary, the conversation discusses various methods for calculating trigonometric values without a calculator. These include memorizing certain values, using basic equations and polygons, estimating values for random degrees, and using Taylor series or CORDIC algorithms. It is also mentioned that ancient mathematicians used tables and sometimes hired savants to calculate values. Finally, there is a joke about log tables being invented for the benefit of snakes.
Xforce
TL;DR Summary
Trigonometry is vital for calculating angles and lengths, but unfortunately sometimes I forgot my calculator, what should I do?
Speaking of trigonometry without a calculator, I usually only memorizes the trig values of 30°, 45° and 60°. then by I can apply basic equations and applying to polygons or other geometry shapes I can get trig values for angles like 15° Or 75°. When people have enough time, people on Wikipedia even got the exact value of any degrees divisible by 3. However, constructing a 120-sided polygon takes way too long...
In some cases, for random degrees, it usually wants to get an estimated value, not an exact value. These manual calculations usually gives off a exact value... if we combine them using equations like sin(a+b)=sina*cosb+sinb*cosa and use a trial-and-error method, until we get the number of decimals we want, it will be too complicated. Despite that, estimating a surd is also hard.
So, is it possible to efficiently calculate the estimate of a trig ratio of random numbers? How does calculators do it? And how ancient people calculate them before the invention of computers?

Xforce said:
So, is it possible to efficiently calculate the estimate of a trig ratio of random numbers?
One way is to use their Taylor expansions.
How does calculators do it?
IIRC they use certain algorithms which are more efficient than Taylor, but I have forgotten which.
And how ancient people calculate them before the invention of computers?
They didn't. They only had certain values: the ratios in a right triangle with integer side lengths. Euclid's geometry is done by ruler and compass, not by numeric.

That depends on what you call "efficiently". The simplest concept is a Taylor series but it isn't very efficient in terms of number of calculations. A standard computer algorithm is CORDIC but that relies on tables of some pre-computed values stored somewhere.

fresh_42
Without a computer, there are half-angle formulas and angle sum formulas that can be used to get values. Early mathematicians used to hire idiot savants who could generate tables. The savants could not describe how they were doing the calculations in any way that a normal person could understand.

SLIDERULE, if you know how and if you have one

Keith_McClary, fresh_42 and FactChecker
fresh_42 said:
IIRC they use certain algorithms which are more efficient than Taylor, but I have forgotten which.
I believe they are called CORDIC (for COordinate Rotation DIgital Computer), aka Volder's algorithm. See https://en.wikipedia.org/wiki/CORDIC.

When computers were still many-millions of \$, there were tables. You looked up the value closest to what you wanted in the table. If you needed more accuracy you used various standard formulas and interpolated. For certain special values or special ranges you used other formulas. Such as for very small angles and the sin() or tan() functions, or angles close to pi/2 for the cos() function.

The same general process was used for logs. There are entire books of similar tables, various other special functions as well such as Bessel functions and so on.

In public school, mumble years ago, when personal computer power simply was not available, I started in memorizing the log tables. I had read a science fiction story about a "boy scout on the moon" or some such. He had managed to solve his problem in his head by using memorized log tables. If you recall that you can multiply two numbers by taking the log of each, adding them, then getting the inverse-log, a log table in your head let's you do a lot of fairly complicated arithmetic. Sadly, my memory was not really up to the task.

Which reminds me of a joke. The forest ranger noticed that a certain kind of snake in his park was not successfully breeding. The tourists visiting the park were disturbing them at just the wrong moment, and nobody was happy about that, since the snakes were likely to be aggressive in such situations. So he invented a particular type of picnic bench-and-table that would allow the tourists to sit on it, and the snakes could safely hide in a compartment under the table and be undisturbed by the tourists. And an attractive feature of these picnic surfaces was they could be easily constructed from local materials. Thus leading to the moral of the story: log tables were invented so that adders could multiply.

Keith_McClary, Nik_2213 and FactChecker
In my first year at secondary school we had Miss Rankine for maths. One of the few things I can remember from that class is that the part of the logarithm after the decimal point is called the "mantissa".

## 1. How do I calculate trigonometric functions without a calculator?

To calculate trigonometric functions without a calculator, you can use the unit circle and the basic trigonometric identities. This involves memorizing the values of sine, cosine, and tangent for common angles, and using the Pythagorean theorem to find the values for other angles. You can also use trigonometric tables or formulas to calculate the values.

## 2. What are the basic trigonometric identities?

The basic trigonometric identities are sine squared plus cosine squared equals one, tangent equals sine over cosine, and cotangent equals cosine over sine. These identities can be used to simplify trigonometric expressions and solve equations.

## 3. How do I use the unit circle to calculate trigonometric functions?

The unit circle is a circle with a radius of one and is used to calculate trigonometric functions for any angle. The x-coordinate of a point on the unit circle represents the cosine value, and the y-coordinate represents the sine value. You can also use the Pythagorean theorem to find the values for other angles on the unit circle.

## 4. Are there any tricks to help me memorize the values of trigonometric functions?

Yes, there are several tricks that can help you memorize the values of trigonometric functions. One trick is to use the acronym "SOH-CAH-TOA" to remember the ratios of sine, cosine, and tangent. Another trick is to create a mnemonic device, such as "Some Old Hippie Caught Another Hippie Tripping On Acid" to remember the values of sine, cosine, and tangent for the angles 0, 30, 45, 60, and 90 degrees.

## 5. Can I use trigonometric functions to solve real-world problems without a calculator?

Yes, trigonometric functions can be used to solve real-world problems without a calculator. For example, you can use the tangent function to calculate the height of a building or the distance between two objects. You can also use trigonometric functions to solve problems in physics, engineering, and navigation.

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