How to Determine Where a Sine/Cosine Graph Intersects the X-Axis

[biologystu]

Hi... I'm learning how to graph sine and cosine functions, and I can't seem to figure out how to determine where the line intersects the x-axis? I think I understand how to shift up/down and to the right/left, as well as amplitude and period, but can't understand where to intersect. Can anyone help me?

I did notice that the sine of pi= 0, and this is where a sine graph intersects, but what if the x-axis is not labeled in pi's but in integers instead?

Thanks for any help!

 

[tiny-tim]

welcome to pf!

hi biologystu! welcome to pf!

Hi... I'm learning how to graph sine and cosine functions, and I can't seem to figure out how to determine where the line intersects the x-axis?

sin intersects the x-axis at multiples of π (ie 0, ±π, ±2π, ±3π, …)

cos intersects the x-axis at odd multiples of π/2 (ie ±π/2, ±3π/2, ±5π/2, …)

I did notice that the sine of pi= 0, and this is where a sine graph intersects, but what if the x-axis is not labeled in pi's but in integers instead?

tough!

you'll just have to write the multiples of π in
​

 

[cepheid]

Remember that a sine function is periodic (it repeats itself in a regular way). You also know that sin(0) = 0 and sin(π) = 0. You can make this argument just from looking at the unit circle.

Since sine is periodic, this means that after the function goes through one full cycle (which is 2pi radians) it should come back to value that it started with. Therefore, not only does sin(0) = 0, and sin(π) = 0, BUT also

sin(0 + 2nπ) = 0
sin(π + 2nπ) = 0

where n is an integer. I.e. n = 1, 2, 3,... etc. So, the first equation above tells you that sin(2π), sin(4π), sin(6π) etc. are ALL equal to zero, because these angles are spaced apart by 2π radians (which is one period). Similarly, the second equation says that sin(3π), sin(5π), sin(7π) etc. are all equal to 0. So, basically, for the sine function, at every integer multiple of π radians, the function is equal to 0.

You can apply this argument to find the zeros of the cosine function as well, but I'll let YOU do that, just to make sure that you have understood what I'm saying.

If the axis isn't labelled in multiples of pi, that's not a big deal. You can still draw the intercepts approximately where they are supposed to be. I.e. π ≈ 3.14, 2π ≈ 6.28, etc. It's just a sketch, right? It doesn't have to be perfect.