How to find the sine of 10 degrees?

Question

Mathematics

I know that there is a formula for the sine, this is the leg on the hypotenuse, but should this formula be used to calculate the value.
You can get radians of 10 degrees, this is pi / 18, but how to calculate it with a sine, I can’t figure out what to divide or multiply by.

Answer 1

If you just need to calculate, then the calculator will help.
If you yourself and without evaluation, then expand the sine into a series and count as many members as you need accuracy.

Answer 2

Approximate values ​​are usually looked up in tables or obtained in calculators.
There is also such a formula ( Taylor series ):
sin(x) ~= x ⁵ /5! + x ³ /3! + x
where x is in radians:
If the task is to find approximately sin (10 °) without technology, you can use paper, pencil, ruler and compass. sin() of an angle in a right triangle is the ratio of the opposite leg to the hypotenuse. If you draw a circle of radius 1, then sin() will be equal to the length of the opposite leg.
Draw a circle with a radius of, say, 10cm (let's say 1 = 10cm). Measure the angle in 1/18 of a semicircle (divide the right angle into three parts and again into three). Drop a perpendicular from a point on the circle to the horizontal axis. Its length divided by 10 cm will be the value of the sine.

Answer 3

Use the formulas of sines and cosines of the sum and bring sin3X to f (sinX), and then solve the resulting equation

Answer 4

There is a Bradis table for this.

Answer 5

Sine 30 gr \u003d 0.5 further, through the triple angle formula, it is easy to find the exact answer. No series are needed, the exact result in radicals is known. And yes, you get a 3rd degree equation that you have to solve. But nothing, it is solved, according to Cardano's formulas. I can’t tell which one now, I don’t remember the triple angle formula ... look it up in Google ... By the way, Taylor is a bad option. its basis is non-orthogonal in the func- tion space. Therefore, the errors begin to grow as the number of terms increases. The sine is actually a vector in an infinite-dimensional function space. We're looking for his coordinates. For Taylor they are known and for Chebyshev too. Everything has been calculated for a long time and not only waiting for these two bases. But for a sine, a periodic limited function, they are not suitable. But an increase in the number of terms in the Chebyshev expansion already steadily increases the accuracy, in contrast to Taylor.

Answer 6

According to the Cardano formula, an irreducible case is obtained when the real root is expressed in complex radicals, but is not expressed in real ones.