Cosine or Sine of (angle+angle) always equal to 1?

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opus
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Main Question or Discussion Point

I'll start off with a given problem.

Find ##cos\left(α-β\right)## given that
##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III.
##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.


Solution:
##cos\left(α-β\right)=1##

This had we wonder if this continued for other angles of the same measure so I tried different inputs such as ##cos\left(30°-30°\right)## and again the solution was 1 after using the difference of cosines formula.

My question is, when we take the difference of cosines with the same angle, is it always equal to 1?
And following this train of thought, could it be true that in taking the difference of sines with the same angle, this would be equal to 0?
And would this extend to addition of the angles as well?
 

Answers and Replies

  • #2
tnich
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I'll start off with a given problem.

Find ##cos\left(α-β\right)## given that
##cos\left(α\right)=\frac{-12}{13}## and α is in quadrant III.
##sin\left(β\right)=\frac{-5}{13}## and β is in quadrant III.


Solution:
##cos\left(α-β\right)=1##

This had we wonder if this continued for other angles of the same measure so I tried different inputs such as ##cos\left(30°-30°\right)## and again the solution was 1 after using the difference of cosines formula.

My question is, when we take the difference of cosines with the same angle, is it always equal to 1?
And following this train of thought, could it be true that in taking the difference of sines with the same angle, this would be equal to 0?
And would this extend to addition of the angles as well?
You can answer this if you figure out what 30°-30° is, or in general, given angle θ, what θ - θ is.
 
  • #3
opus
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Dumb question then! What made me unsure is that when first learning these formulas, I would see something such as ##sin\left(240°\right)## and write it as ##sin\left(180°\right)+sin\left(60°\right)## which at first glance seems like a fair statement but it isn't. So with these, some of the statements that might seem obvious are actually not true.
 
  • #4
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Dumb question then! What made me unsure is that when first learning these formulas, I would see something such as ##sin\left(240°\right)## and write it as ##sin\left(180°\right)+sin\left(60°\right)## which at first glance seems like a fair statement but it isn't. So with these, some of the statements that might seem obvious are actually not true.
Sine and cosine are ratios. The ratios between two lengths of a right triangle. Now ratios aren't additive, since they usually have two different denominators. A denominator is a kind of scale: the nominator says how many units in terms of the denominator we have. So different denominators means different scales means not comparable. E.g. you cannot add distances of a street map from NYC with the distances of a map of the US. It's the same here, and that's why the ratios sine or cosine cannot be added. Fortunately there is a formula for ##\sin(\alpha \pm \beta)## and ##\cos(\alpha \pm \beta)\,.## You can look up all those details e.g. on Wikipedia, where the ratios are explained as well as the addition theorems noted.
 
  • #5
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Dumb question then! What made me unsure is that when first learning these formulas, I would see something such as ##sin\left(240°\right)## and write it as ##sin\left(180°\right)+sin\left(60°\right)## which at first glance seems like a fair statement but it isn't.
This goes back to a misconception you had before in another thread. For an arbitrary function f, it is not true in general that f(a + b) = f(a) + f(b). So you should not expect sin(240°) to be equal to sin(180°) + sin(60°). For the same reason you shouldn't expect ##\sqrt{9 + 16}## to be equal to ##\sqrt 9 + \sqrt{ 16}## or ##(2 + 3)^2## to be equal to ##2^2 + 3^2##.

There's a concept of "linear function" or "linear transformation" a little further along in mathematics. A function or transformation f is linear if it satisfies two properties:
  • f(a + b) = f(a) + f(b)
  • f(ka) = kf(a)

An example of such a linear function, as defined above, is y = f(x) = 3x. A nonexample would be y = g(x) = 3x + 2. The terminology is somewhat confusing, because the graphs of both functions are straight lines, but since the graph of the latter function doesn't pass through the origin, it isn't considered a "linear function." Instead it is called an affine linear function.

Obviously, none of the trig functions are linear functions, nor are functions involving radicals, rational functions, most polynomials, exponential functions, or log functions, just to name a few kinds of functions.
opus said:
So with these, some of the statements that might seem obvious are actually not true.
They only seem obvious until you know better.
 

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