Question

How do you solve `sin x = 0.25`?

Answer 1

A first approximation, given only two significant figures is to convert the value to degrees from radians (explained below). This gives a value of 14.3o.

Explanation:

“sin x = a” means “the angle ‘x’ whose sine is ‘a’ “. Sine is defined as the ratio of the opposite side and hypotenuse of a right triangle for the desired angle. Thus, you can construct a right triangle from the ratio and then measure the angle with a protractor.

Further refinement can be made, if desired, by working out the derivation with the Taylor Series, starting with this initial value. If the result does not come out as an equality, increment the value by 0.1 as demonstrated below.

Sin 14.4 = 0.25?
14.4 = 0.2513 radians
sin (0.2513 radians) = (0.2513) - (0.2513)3/3! + (0.2513)5/5!
sin (0.2513) = (0.2513) - 0.002645 + 0.000008 = 0.2487

Just a bit low, so lets try another 0.1 degree increment.

14.5 = 0.2531 radians
sin (0.2531 radians) = (0.2531) - (0.2531)3/3! + (0.2531)5/5!
sin (0.2531) = (0.2531) - 0.002702 + 0.0000087 = 0.2504
Just a TAD over the desired 0.25, and well within the accuracy of two significant figures.
NOW convert the radian value (0.2531) back into a degree:
0.2531 *180/pi = 14.5 degrees.

Arithmetically you can use the Taylor Series:
`sin x = x − x^3/3! + x^5/5! - x^7/7! +… = ∑_(n=0)^∞[(−1)n/(2n+1)!]x2n+1`

`cos x = 1 − x2/2! + x4/4! - x6/6! +… = ∑_(n=0)^∞[(−1)n/(2n)!]x2n`

The formulas for sine and cosine are the ones to focus on first. They converge very quickly, but you have to realize that the angles are measured in radians, where 2π radians =360∘. If you do the conversion, you'll be able to calculate quite quickly for yourself.

The radian value for a given degree is r = deg * π/180o
The degree value for a given radian value is d = r * 180o/π

EXAMPLE:
Assume that the angle is 25 degrees. Taylor series works in radians so convert 25 degrees into radians:

25 * (3.1416)/180 = 0.4363.

Use 0.436332 as x in the formula and plug this value into each part of the equation: `sin (0.4363) = (0.4363) - (0.4363)^3/3! + (0.4363)^5/5! - (0.4363)^7/7!`

Calculate the first value using a calculator: `(0.4363)^3/3! = (0.083071)/(1x2x3) = 0.013845` - Note: 3! Is a factorial, shorthand for 1x2x3.

Calculate the second value in the series: `(0.4363)^5/5!= (0.015815)/(1x2x3x4x5)`= 0.015815/120 = 0.000132. Note: 5! Is factorial shorthand for 1x2x3x4x5.
Calculate the last value of the series:
`(0.4363)^7/7!` = 0.003011/5040 = 0.000006.
Notice that the value of the factor diminishes rapidly, so only the first few need to be used in most approximations.

Plug in all the values:
sin 0.4363 = 0.4363 – 0.013845 + 0.000132 – 0.000006 = 0.422613

Again, you can see that within the accuracy of our original values, the first three terms are sufficient, with a value of 0.4226.
Therefore, the sine of 25 degrees is 0.4226.

http://www.ehow.com/how_5804771_sine-angle-calculator.html