Download Math 123 - College Trigonometry Euler`s Formula and Trigonometric

Math 123 - College Trigonometry
Euler’s Formula and Trigonometric Identities
Euler’s formula, named after Leonhard Euler, states that:
For any real number x,
eix = cos(x) + isin(x)
(1)
Where e is the base of the natural logarithm (e = 2.71828 . . .), and i is the imaginary unit.
Using this formula you can derive all the trigonometric formulas (Pythagorean formula, sum
and difference formulas, etc.).
Pythagorean identity:
cos2 (x) + sin2 (x) = 1
By the rule of exponent we know that
eix · e−ix = 1
However, using the Euler’s formula, we know
eix = cos(x) + i sin(x)
and
e−ix = cos(x) − i sin(x)
Hence,
1 = eix · e−ix
= cos(x) + i sin(x) · cos(x) − i sin(x)
= cos2 (x) − i cos(x) sin(x) + i cos(x) sin(x) − i2 sin2 (x)
= cos2 (x) + sin2 (x)
Therefore, cos2 (x) + sin2 (x) = 1.
1
Sum formulas:
Here we want to use the Euler’s formula to derive the formulas for:
cos(x + y), sin(x + y) and tan(x + y)
First, note that by the Euler’s formula we have:
ei(x+y) = cos(x + y) +i sin(x + y)
| {z }
| {z }
Real Part
(2)
Imaginary Part
Now, expanding ei(x+y) gives you:
ei(x+y) = eix · eiy
= cos(x) + i sin(x) · cos(y) + i sin(y)
= cos(x) cos(y) + i cos(x) sin(y) + i sin(x) cos(y) + i2 sin(x) sin(y)
= cos(x) cos(y) − sin(x) sin(y) +i sin(x) cos(y) + cos(x) sin(y)
{z
} |
{z
}
|
Imaginary Part
Real Part
Now, by equation (2), ei(x+y) = cos(x + y) + i sin(x + y), you can easily see that:
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
and
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
To get the formula for tan(x + y), all we need to do is applying the definition of tangent.
tan(x + y) =
=
=
=
=
sin(x + y)
cos(x + y)
sin(x) cos(y) + cos(x) sin(y)
cos(x) cos(y) − sin(x) sin(y)
sin(x) cos(y) + cos(x) sin(y)
cos(x) cos(y)
÷
(divide by 1 in a clever way)
cos(x) cos(y) − sin(x) sin(y)
cos(x) cos(y)
sin(x) cos(y) + cos(x) sin(y) ÷ cos(x) cos(y)
cos(x) cos(y) − sin(x) sin(y) ÷ cos(x) cos(y)
tan(x) + tan(y)
(using algebra and definition of tangent)
1 − tan(x) tan(y)
Thus,
tan(x + y) =
tan(x) + tan(y)
1 − tan(x) tan(y)
2
Difference formulas:
Here we want to use the Euler’s formula to derive the formulas for:
cos(x − y), sin(x − y) and tan(x − y)
This is similar to the sum formulas. Instead of expanding ei(x+y) , we will expand ei(x−y)
instead.
(3)
ei(x+y) = cos(x + y) +i sin(x + y)
| {z } | {z }
Real Part
Imaginary
Now, expanding ei(x+y) gives you:
ei(x−y) = eix · e−iy
= cos(x) + i sin(x) · cos(y) − i sin(y)
= cos(x) cos(y) − i cos(x) sin(y) + i sin(x) cos(y) − i2 sin(x) sin(y)
= cos(x) cos(y) + sin(x) sin(y) +i sin(x) cos(y) − cos(x) sin(y)
|
{z
} |
{z
}
Imaginary Part
Real Part
Using equation (3), we can equate the real and imaginary part. Hence,
cos(x − y) = cos(x) cos(y) + sin(x) sin(y)
and
sin(x − y) = sin(x) cos(y) − cos(x) sin(y)
To get tan(x − y), you compute
sin(x−y)
,
cos(x−y)
tan(x − y) =
similar to what we did for tan(x + y). Hence,
tan(x) − tan(y)
1 + tan(x) tan(y)
3