三角函数值域和定义域

三角函数值域和定义域
英文回答：
The domain and range of trigonometric functions depend on the specific function being considered. Let's start with the sine function (sin(x)). The domain of sine is all real numbers, as it can take any angle as an input. However, the range of sine is limited to values between -1 and 1. This is because the sine function oscillates between these two values as the angle increases or decreases. For example, sin(0) = 0, sin(π/2) = 1, sin(π) = 0, sin(3π/2) = -1, and so on.
Moving on to the cosine function (cos(x)), its domain and range are also all real numbers. The cosine function also oscillates between -1 and 1, but it starts at 1 when the angle is 0. For example, cos(0) = 1, cos(π/2) = 0,
cos(π) = -1, cos(3π/2) = 0, and so on.
Next, we have the tangent function (tan(x)). The domain
of tangent is all real numbers except for the values where the cosine function equals zero. This is because tangent is defined as the ratio of sine to cosine, and division by zero is undefined. Therefore, the values where cosine is zero (e.g., π/2, 3π/2, etc.) are excluded from the domain of tangent. The range of tangent is all real numbers, as it can take any value depending on the angle. For example,
tan(0) = 0, tan(π/4) = 1, tan(π/2) is undefined, tan(π) = 0, and so on.
Moving on to the cosecant (csc(x)), secant (sec(x)), and cotangent (cot(x)) functions, their domains and ranges are similar to their reciprocal functions (sine, cosine, and tangent). The only difference is that their domains exclude the values where the sine, cosine, or tangent functions equal zero, respectively.
中文回答：
三角函数的定义域和值域取决于具体的函数。

让我们从正弦函数（sin(x)）开始。

正弦函数的定义域是所有实数，因为它可以接受任何角度作为输入。

然而，正弦函数的值域限制在-1到1之间。

这是因为正弦函数在角度增加或减少时在这两个值之间振荡。

例如，sin(0) = 0，sin(π/2) = 1，sin(π) = 0，sin(3π/2) = -1，依
此类推。

接下来是余弦函数（cos(x)），它的定义域和值域也是所有实数。

余弦函数也在-1到1之间振荡，但当角度为0时，它的值为1。

例如，cos(0) = 1，cos(π/2) = 0，cos(π) = -1，cos(3π/2) = 0，依此类推。

接下来是正切函数（tan(x)）。

正切函数的定义域是除了余弦
函数等于零的值之外的所有实数。

这是因为正切被定义为正弦除以
余弦，而除以零是未定义的。

因此，余弦为零的值（例如π/2，
3π/2等）被排除在正切的定义域之外。

正切函数的值域是所有实数，因为它可以取决于角度而取任何值。

例如，tan(0) = 0，
tan(π/4) = 1，tan(π/2)未定义，tan(π) = 0，依此类推。

接下来是余割函数（csc(x)）、正割函数（sec(x)）和余切函
数（cot(x)），它们的定义域和值域与它们的倒数函数（正弦、余
弦和正切）类似。

唯一的区别是它们的定义域分别排除了正弦、余
弦或正切函数等于零的值。