[Tutor] Trigonometric Functions in Python

[boB Stepp]

Greetings April!

On Sun, May 24, 2020 at 08:09:32PM -0400, April Morone wrote:
>I signed up for the ITP100 Software Design (Python Programming) class for
>this Summer. I have been working ahead through the book, but came across an
>issue with understanding something within Chapter 4 of the book "Python for
>Everybody: Exploring Data in Python 3" that is by the author Charles
>Severance on page 45. Please explain for me how the following checks the
>result of the conversion from degrees to radius to get  0.7071067811865476
>to see if the result is correct:
>
>math.sqrt(2) / 2.
>
>of the following:
>
>>>> degrees = 45
>>>> radians = degrees / 360.0 * 2 * math.pi
>>>> math.sin(radians)
>0.7071067811865475

It is not totally clear to me exactly where you are not following things.
It appears that it is the math, not the Python, which is unclear.

By definition there will be 2 * pi radians in a full 360 degree angle, or
pi radians for a half-circle angle.  So 2 * pi radians = 360 degrees.  So
the line above might be rewritten:

2 * pi radians = 360 degrees
1 radian = 360/(2 * pi) degrees, or
1 degree = (2 * pi)/360 radians

Does that make sense?  So if 1 degree is the above, then 45 degrees would
be:

45 * 1 degrees = (45 * 2 * pi)/360 radians, which simplifies to
45 degrees = pi/4 radians

For a right triangle (implying a 90 degree angle) if one of the other two
angles is 45 degrees then the other one will be, too, as the sum of all
angles in a triangle must add up to 180 degrees.  And the two 45 degree
angles imply that the height of the right triangle will equal the width of
the triangle.

Using the Pythagorean Theorem which states that the sum of the squares of
the sides must equal the square of the hypotenuse, or, if the width of the
side is x, the height y and the hypotenuse length r, then

x**2 + y**2 = r**2

If we arbitrarily choose x = y = 1 for our 45 degree angle we would have:

r**2 = 1**2 + 1**2
r**2 = 2, and taking the square root of both sides:
r = sqrt(2)

Sine of an angle = height / hypotenuse, so

sine(45 degrees) = 1/sqrt(2)

If you multiply the top and bottom by sqrt(2) you get

sine(45 degrees) = (sqrt(2) * 1) / (sqrt(2) * sqrt(2)) = sqrt(2) / 2

If you divide that out you will get the result you stated.


>How does the following check the result of the above conversion from
>degrees to radius to see if the result is correct?
>
>>>> math.sqrt(2) / 2.0

See above.  Does any of this help?  Note that what I wrote above is not actual
Python code statements.  I am just trying to help you see the math.

-- 
Wishing you only the best,

boB Stepp