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Study Notes 032:
Trigonometry 6:
By William E. Steinman:
Okay, in study Notes 31 we dealt with the trigonometric functions of an angle. Of course, that was a single angle.
We also have trigonometric functions of two angles.
These can also be proved, but it takes a bit more effort.
Let us begin with the formulas for addition.
sin (a + b) = sin a cos b + cos a sin b
cos (a + b) = cos a cos b – sin a sin b
tan (a + b) = (tan a + tan b)/(1 – tan a tan b)
Let us see if we can prove the first formula, sin (a + b) = sin a cos b + cos a sin b
First, we can construct a baseline OX (below).
Then we can construct the line OY to give us the first angle. Call it angle a.
We can place the angle b on top of our first angle with the line OP.
We can see that the sum of these guys will be (a + b).
Great!
When we draw a perpendicular from point A on our baseline to point P it will give us a right triangle for the angle (a + b).
We can draw a perpendicular from point B on OY to P and we get another right triangle.
We can also draw BC perpendicular to our baseline and BD perpendicular to AP.
Since the corresponding sides ( OA and AP, and
Think this through.
It’s tedious, perhaps even mind numbing, but not particularly difficult.
Since we have some right triangles, this means our basic relationship can apply.
Sin (a + b) = AP/OP
cos (a + b) = OA/OP
sin a = CB/OB
cos a = DP/BP
sin b = BP/OP
cos b =OB/OP
Now, we can plug in the values and see what we get.
sin (a + b) = sin a cos b + cos a sin b
AP/OP = (CB/OB)(OB/OP) + (DP/BP)(BP/OP)
We can see that
Great! We can also see that BP will cancel out of the second term to give us DP/OP.
Well, what does that give us?
AP/OP = CB/OP + DP/OP
But, that is an equality.
Look at the drawing. AP/OP = (CB + DP)/OP = CB/OP + DP/OP
Shazam!
Ain’t that swell?
There are a few other trigonometric functions of two angles. Here they are.
Formulas for subtraction:
Sin (a - b) = sin a cos b – cos a sin b
cos (a – b) = cos a cos b + sin a sin b
tan (a – b) = (tan a – tan b)/1 + tan a tan b
Double angle formulas:
sin 2a = 2 sin a cos b
cos 2a = cos2 a – sin2 b
tan 2a = (2tan a)/ (1- tan2 a)
Half angle formulas:
You can accept all of these formulas as identities without proof.
However, depending on how interested you are in math you may want to go through the exercise of proving them to yourself. It will be handy later on to be comfortable in manipulating these buggers. In most cases, you will find that you will need to make a sketch to see what is necessary.
Enough for now!