Could someone explain how the graph of y=cos x can be obtained by translating the graph of y=sinx? Thank you!
horizontally phase shift the graph of y=sin(x) by pi/4 units to the left and reflect the graph over the x axis to acquire the graph y=cos(x)
sorry - no reflection was indicated. just translation.
sinx = sin(180-x)
cosx = sin(90-x) = sin(180-(90-x)) = sin(90+x)
check:
sin(90+x) = sin90 cosx + cos90 sinx = cosx
Well, imagine you have a group of sin(x) graphs having a wild party. Now, these graph folks are all dancing and having a grand old time, but one of them -- let's call him Mr. Sin -- decides that he wants to get a bit frisky and change his dance moves.
So, Mr. Sin decides to shift his dance routine, which involves a horizontal translation, because he's just fancy like that. He wants to go to the left, to negativeville, if you will. So he drags himself a few units to the left, saying, "Hey, now I'm gonna shake my booty over here!" (Very scientific terms, of course.)
And just like that, Mr. Sin has transformed into Mr. Cos! The graph of y = cos(x) is essentially the result of this horizontal shift of the sine wave. The whole crew will still recognize him because his shape remains the same, but now he's just hanging out in a different location on the dance floor. So it's like they're twins, but one is just a bit moodier than the other.
Hope that helps you picture it! Now if you'll excuse me, I need to go work on my own signature dance move. It's called the Sine-boogie-woogie. Trust me, it's going to be a hit!
Certainly! The graph of y = cos x can be obtained by translating the graph of y = sin x. To understand how this translation occurs, let's break it down step by step.
1. Start with the graph of y = sin x. This is the familiar "wave-like" shape that oscillates between -1 and 1.
2. The cosine function, y = cos x, is simply a horizontal shift (translation) of the sine function. To be more specific, the cosine function is the same shape as the sine function, but it is shifted to the left or right.
3. The shift in the horizontal direction is determined by the value of x. For example, if you have y = sin(x - a), where a is a positive value, the graph will shift a units to the right. Conversely, if a is negative, the graph will shift a units to the left.
4. In the case of y = cos x, we can see that there is no horizontal shift (a = 0). Therefore, the graph of y = cos x remains centered on the y-axis and retains the same shape as y = sin x.
In summary, to obtain the graph of y = cos x by translating the graph of y = sin x, all you need to do is make sure there is no horizontal shift (a = 0) in the equation. This will result in the cosine function being centered on the y-axis, with the same wave-like shape as the sine function.