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

• biologystu
In summary, you will need to find the zeros of the cosine function using the same argument as for the sine function. Remember that cos(0) = 0 and cos(π) = 0.
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!

welcome to pf!

hi biologystu! welcome to pf!
biologystu said:
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

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.

## 1. How do I determine the x and y values for a sine/cosine graph?

The x values for a sine/cosine graph are typically the angle values, measured in radians or degrees. The y values are the corresponding values of the sine or cosine function at those angles. You can use a table of values or a calculator to determine these values.

## 2. How do I label the x and y axes for a sine/cosine graph?

The x axis should be labeled with the angle values in either radians or degrees, depending on the units used for the x values. The y axis should be labeled with the values of the sine or cosine function.

## 3. What is the amplitude of a sine/cosine graph?

The amplitude of a sine/cosine graph is the distance from the center line to the highest or lowest point on the graph. It is equal to half the difference between the maximum and minimum values of the function.

## 4. How do I find the period of a sine/cosine graph?

The period of a sine/cosine graph is the length of one complete cycle. It can be calculated by dividing the full range of x values (usually 2π or 360 degrees) by the number of cycles in the graph.

## 5. How do I plot a point on a sine/cosine graph?

To plot a point on a sine/cosine graph, first determine the x value (angle) for the point. Then, use the function to find the corresponding y value. Plot the point at the intersection of the x and y values on the graph.

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