# Difference between Cos and sin

• tiffney
In summary, the difference between cos and sin is that cosine is a phase shift of sine (and vice versa), with one being a translation to the right by pi/2. On the unit circle, cos is the x-coordinate and sin is the y-coordinate, with both being the shadows of the radius on the x- and y-axes respectively. However, due to the symmetry of the circle, as one goes around the circle, the shadows look essentially the same but just out of phase. When creating a waveform, the sin function creates a sawtooth appearance while the cos function does not, due to the properties of the functions as an even and odd function, respectively.
tiffney
Can anyone tell me the difference btween cos and sin, and tell me when creating waveform why the sin function creates a sawtooth appearance and the cos function does. Any help would be appreiciated. :!) :!)

Cosine is a phase shift of sine (and visa-versa). In other words,

$$\sin \theta = \cos \left(\theta - \frac{\pi}{2}\right)$$

just a translation to the right by $\pi / 2$.

Given a right triangle and a non-right angle, the sine of that angle is equal to the length of the side facing the angle over the length of the hypotenuse. The cosine of the said angle is equal to the length of the side adjacent to the angle over the length of the hypotenuse.

A cosine wave is simply a sine wave that is shifted to the right by pi/2.

It would be much easier to explain if I can draw pictures. Your best chance is to google it or to check out MathWorld (http://mathworld.wolfram.com).

for a point on the unit circle, cos is the x coordinate and sin is the y coordinate. I.e. one is the shadow of the radius on the x-axis and the other the shadow of the radius on the y axis. Due to the symmetry of the circle as you go around the length of the shadow of the radius on the x-axis or on the y-axis look essentially the same, just out of phase.

mathwonk said:
for a point on the unit circle, cos is the x coordinate and sin is the y coordinate. I.e. one is the shadow of the radius on the x-axis and the other the shadow of the radius on the y axis. Due to the symmetry of the circle as you go around the length of the shadow of the radius on the x-axis or on the y-axis look essentially the same, just out of phase.

And the y-axis is perpendicular (90 degrees; pi/2 radians) to the x-axis. Which is why:

Data said:
Cosine is a phase shift of sine (and visa-versa). In other words,

$$\sin \theta = \cos \left(\theta - \frac{\pi}{2}\right)$$

just a translation to the right by .

The "difference" between cos(x) and sin(x) is |cos(x) - sin(x)|.

That was a joke.

Really.

tiffney said:
when creating waveform why the sin function creates a sawtooth appearance and the cos function does.

Eh, what?

I suggest google, it's not the easiest thing to explain

several people have explained it here already :tongue:

Not his sawtooth conjecture, which beats me.

The only thing that connects sawteeth & sin/cos is Fourier series/transform...

Daniel.

An important property of sin(x) and cos(x) is that
cos(x) is an "even function of x" [that is, cos(-x)=cos(x)] and
sin(x) is an "odd function of x" [that is, sin(-x)=-sin(x)].

I'm thinkin' something's missing here.

No cos or sin function, at least involving perhaps your basic constant coefficients, generates a sawtooth wave, does it? It should generate a sine wave and another wave 90 degrees out of phase. Perhaps his problem involves some Fourier or other transfoms as Dexter already suggested?

## 1. What is the difference between cos and sin?

Both cos (cosine) and sin (sine) are trigonometric functions used in mathematics to calculate the relationship between the sides and angles of a triangle. The main difference between them is that cos calculates the ratio of the adjacent side to the hypotenuse, while sin calculates the ratio of the opposite side to the hypotenuse.

## 2. How are cos and sin related?

Cos and sin are related through the Pythagorean identity, which states that cos²θ + sin²θ = 1. This means that the square of the cosine of an angle plus the square of the sine of the same angle will always equal 1.

## 3. Can cos and sin be negative?

Yes, both cos and sin can be negative depending on the quadrant of the angle in which they are being evaluated. In the first and fourth quadrants, cos is positive and sin is negative. In the second and third quadrants, both cos and sin are negative.

## 4. What is the difference between the graphs of cos and sin?

The graph of cos is a curve that starts at 1 and decreases to -1 as the angle increases from 0 to 180 degrees. The graph of sin is a curve that starts at 0, increases to 1 at 90 degrees, decreases to 0 at 180 degrees, and then becomes negative as the angle increases further.

## 5. In what real-life situations are cos and sin used?

Cos and sin have numerous applications in real life, including in architecture, engineering, navigation, and physics. For example, cos is used in calculating the stability of structures, while sin is used in determining the trajectory of a projectile. They are also used in fields such as music, sound engineering, and animation for creating smooth curves and movements.

• General Math
Replies
3
Views
3K
• General Math
Replies
1
Views
743
• General Math
Replies
17
Views
4K
• General Math
Replies
9
Views
1K
• General Math
Replies
2
Views
1K
• General Math
Replies
3
Views
808
• General Math
Replies
1
Views
1K
• General Math
Replies
13
Views
2K
• General Math
Replies
6
Views
2K
• General Math
Replies
3
Views
2K