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2.3 Properties of Trigonometric Functions

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The important properties are:
The Pythagorean theorem (which is really our definition of distance as discussed below).
The addition theorems  which are expressions for  sin(a + b) and cos (a + b).
The half angle theorem (a consequence of the previous two.)
All trigonometric functions depend only on the angle mod 2.
The law of sines: In the triangle ABC, the ratio of the length AB and AC is the ratio of the sines of the opposite angles:. This is just the fact that both ABsin B and ACsin C are equal to AH.

The law of cosines.(see next chapter)
The coordinates of  the point on the unit circle at angle t with center at the origin are(cos, sin) which means y = sin, x = cos.
The tangent line to a circle with center at the origin through (x, y) is perpendicular to the line from the center to (x, y) i and points toward the y axis in the first quadrant.
Its direction is given by (-sin, cos)
All properties follow from the differential properties of the sine.
For the moment we assume:


Then using

,

We get


And we have cos 0 = 1
And so we have sin 0 = 0, (sin 0)’ = 1, (sin  0 )” = 0, (sin 0)”’ = -1 and further derivatives repeat as (from the start 0 1 0 -1 0 1 0 -1 0 1 0 -1, etc, at argument 0.    This information determines a power series formula for the sine of x.
What polynomial has derivative 1 at x = 0, and value 0 there?
When you realize that the monomial xk has all its derivatives 0 at x = 0 except the kth which is k!, we can read off the power series for sine from the sequence of derivative values at 0.
We get

and similarly

which implies

exp(ix) = cos x + i sin x.     (A)

We can therefore use the properties of the exponential function to deduce the addition theorem for sines and cosines:

exp i(a+b) = cos (a + b) + i sin( a + b) = exp ia * exp ib

                                = cos a cos b – sin a sin b + i(cos a sin b + cos b sin a)

Exercises:

2.10 Derive the formulae for sine x and cosine x in terms of exp(ix) and exp(–ix), that follows from  equation (A) above. Solution

2.11 Find expressions for (sin t/2)2 and (cos t/2)2  from the Pythagorean theorem and the cosine addition theorem. Solution

2.12 Set up a spreadsheet to compute the sin x for any inputted x. How many terms in the sin x power series expansion do you need to evaluate sin .5 to 8 decimal places? Solution