# Finding the Value of a Triginometric Function Without a Calculator

• DCircuit
In summary, calculators use Taylor series expansions or iterative algorithms to calculate trigonometric functions, such as sin(41.3°) and arcsin, with only basic operations like multiplication and addition. However, these methods can only give decimal approximations and may not be precise enough for certain calculations.
DCircuit
How can you find, say, sin(41.3°) without using a calculator? Or maybe a better question is: How does a calculator find that value when you punch it in? Also, what about arcsin, arccos, etc... How does a calculator find those values?

They use Taylor series (http://en.wikipedia.org/wiki/Taylor_series) expansions to get an answer that is correct up to a limited number of digits. That way only repeated additions and multiplications are needed.

They for sure don't use Taylor because that is much too slow in convergence.
For trigonometric functions in particular some simple systems might use
http://en.wikipedia.org/wiki/CORDIC
In general there are iterative algorithms for each elementary function which use only the basic operation (multiplication,...). I once heard a lecture but can't find the notes anymore :(
Maybe you find some information on
http://www.math.niu.edu/~rusin/known-math/index/65-XX.html
http://www.math.niu.edu/~rusin/known-math/98/special.func.comp

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Actually, Euler's Formula gives the exact values of the Trig functions and is very easy to use.

e^ix = cos(x) + isin(x)

:-D

How exactly would you calculate exp(i*41.3°) on paper? (with multiplication and addition only) ;-)

And furthermore, I believe that this formula applies only to real numbers (i.e., radian measure), not angles in degree measure, so you would need to convert 41.3° to radians.

41.3° = 41.3pi/180 That is easy.

$$sinx\ =\ \frac{1}{2i}\left(e^{ix}\ -\ e^{-ix}\right)$$

So go ahead and do the complex exponential :D
How would you think calculators do it?
(Hint: They use the trig functions!)

They can't use the trig functions to get the trig functions. I guess the only way to evaluate complex exponents is with the Taylor's power series, which will only give you a decimal approximation depending on how far you want to go.

Good point. And the conclusion is?
(Hint: Calculators don't use the complex exponential to find trig functions)

If you go back to Taylor series, then you start over at Post #2. And the answer is post #3 which says Taylor series converge too slowly to be useful.

## 1. How do I find the value of a trigonometric function without a calculator?

To find the value of a trigonometric function without a calculator, you can use trigonometric identities and special angles to simplify the expression. You can also use a table of values or a graph to estimate the value.

## 2. What are some common trigonometric identities that can help me find the value of a function without a calculator?

Some common trigonometric identities that can help you find the value of a function without a calculator include the Pythagorean identities, double angle identities, and half angle identities. These identities allow you to simplify the expression and find the value using basic trigonometric functions.

## 3. Can I use the unit circle to find the value of a trigonometric function without a calculator?

Yes, the unit circle can be a helpful tool in finding the value of a trigonometric function without a calculator. By understanding the relationship between the coordinates on the unit circle and the trigonometric functions, you can determine the value of a function for any given angle.

## 4. Are there any tricks or shortcuts for finding the value of a trigonometric function without a calculator?

Yes, there are some tricks and shortcuts that can help you find the value of a trigonometric function without a calculator. For example, you can use the symmetry of the unit circle to find the value of a function for angles greater than 90 degrees. You can also use special angles, such as 30-60-90 triangles, to quickly find the value of a function.

## 5. How can I check if my answer is correct when finding the value of a trigonometric function without a calculator?

You can check your answer by using a calculator to find the value of the function for the same angle. If the values match, then your answer is likely correct. You can also use a graphing calculator or online graphing tool to graph the function and visually confirm the value at a given angle.

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