sine and cosine have a period 2pi
tangent and cotangent have period pi
Can someone explain why? thanks a lot.
well tangent is sine/cosine and there a place where the tangent function is undefined and that is where the asymptotes occur. the same is true with the cotangent since it is cosine/sine.
Sure! I'd be happy to explain why the sine and cosine functions have a period of 2π, while the tangent and cotangent functions have a period of π.
The period of a function represents the distance between two consecutive repetitions of the function's values. In the case of trigonometric functions, the period determines how often the values of the function repeat.
Let's start with the sine and cosine functions, which are fundamental trigonometric functions. The sine function represents the y-coordinate of a point on the unit circle, while the cosine function represents the x-coordinate of that same point.
When we examine the unit circle, we can see that as we move along the circumference, the x and y coordinates of the points on the unit circle are periodic. After completing one full revolution around the circle, the coordinates of the point repeat, resulting in the same values for sine and cosine. This full revolution corresponds to 2π radians or 360 degrees in the trigonometric circle.
So, both the sine and cosine functions have a period of 2π because their values repeat every 2π radians or 360 degrees.
Now let's move on to the tangent and cotangent functions. Tangent is defined as the ratio of sine to cosine, while cotangent is defined as the ratio of cosine to sine.
To understand the period of tangent and cotangent, we need to consider the points where the values of sine and cosine are zero or undefined. These points are called the asymptotes.
At π/2 and 3π/2 (or 90 degrees and 270 degrees), the cosine function equals zero. As a result, the tangent function becomes undefined at these points because division by zero is not defined. Similarly, at 0 and π, the sine function equals zero, making the cotangent function undefined at these points.
Considering the periodic nature of the sine and cosine functions, we can observe that the values of sine and cosine repeat every 2π radians or 360 degrees. However, the values of tangent and cotangent become undefined every time the cosine or sine becomes zero.
Therefore, the tangent and cotangent functions have a period of π because their values repeat every π radians or 180 degrees, excluding the points where the tangent and cotangent are undefined.
In summary, the sine and cosine functions have a period of 2π because their values repeat every 2π radians or 360 degrees. On the other hand, the tangent and cotangent functions have a period of π because their values repeat every π radians or 180 degrees, excluding the points where they are undefined.