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

• B
• opus
In summary, the conversation discusses finding the value of ##cos\left(α-β\right)## given the values of ##cos\left(α\right)## and ##sin\left(β\right)## in quadrant III. The conversation also explores the idea of taking the difference of sines and cosines with the same angle, and whether it is always equal to 1 and 0 respectively. The experts clarify that this is not always true and provide examples and explanations of why. They also mention the concept of linear functions and how it relates to this topic.
opus
Gold Member
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?

opus said:
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.

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.

baldbrain
opus said:
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.

baldbrain
opus said:
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.

## 1. What is the formula for calculating the cosine or sine of (angle+angle)?

The formula for calculating the cosine of (angle+angle) is: cos(angle+angle) = cos(angle) * cos(angle) - sin(angle) * sin(angle)

The formula for calculating the sine of (angle+angle) is: sin(angle+angle) = 2 * sin(angle) * cos(angle)

## 2. Why is the result always equal to 1?

The result is always equal to 1 because of the trigonometric identity known as the double angle formula. This formula states that the cosine or sine of (angle+angle) is equal to the cosine or sine of the individual angles multiplied together and subtracted or added, respectively.

## 3. Can this formula be applied to any angle?

Yes, this formula can be applied to any angle because the trigonometric functions of cosine and sine are continuous and defined for all real numbers.

## 4. How is this formula used in real-world applications?

The formula for calculating the cosine and sine of (angle+angle) is used in many fields, including engineering, physics, and astronomy. It is used to solve problems involving periodic motion, such as the motion of a pendulum or a wave.

## 5. Are there any other identities related to this formula?

Yes, there are several other identities related to this formula, such as the half-angle formula, which allows for the calculation of the cosine and sine of an angle divided by 2. Additionally, there is the sum and difference formula, which allows for the calculation of the cosine and sine of the sum or difference of two angles.

Replies
8
Views
1K
Replies
3
Views
1K
Replies
2
Views
1K
Replies
3
Views
1K
Replies
6
Views
1K
Replies
8
Views
5K
Replies
16
Views
2K
Replies
7
Views
2K
Replies
3
Views
1K
Replies
2
Views
6K