Trigonometric functions for acute angles 銳角的三角函數
There are three basic trigonometric functions for acute angles: Sine (Sin), Cosine (Cos), and Tangent (Tan).
銳角有三個基本的三角函數。正弦（Sin），餘弦（Cos），和正切（Tan）。
When using a right-angled triangle we get 當使用一個直角三角形時，我們得到:
These functions have a unique value for an acute angle that can be obtained from a scientific calculator.
這些函數對一個銳角有一個獨特的數值，可以從科學電腦中得到。
These formulae are only applicable for an acute angle in a right-angled triangle, and so the next stage is to extend to work with any angle in radians and degrees.
這些公式隻适用于直角三角形中的銳角，是以下一階段要擴充到以弧度和度數計算的任何角度。
Trigonometric functions for positive and negative angles 正和負的角度的三角函數
On a coordinate grid a general angle is measured from the positive x-axis and is represented by the angle through which a line OM rotates about the origin.
在坐标網格上，一般的角度是從正X軸開始測量的，并由線OM圍繞原點旋轉的角度來表示。
When we rotate anti-clockwise, the angle is positive while a clockwise rotation gives a negative angle.
當我們逆時針旋轉時，角度是正的，而順時針旋轉則是負的。
Trigonometric functions for any angle in radians and degrees 以弧度和度為機關的任何角度的三角函數
The four quadrants of the Cartesian axes are as follows 直角坐标軸的四個象限如下:
As the line OM rotates, the point M moves to the first quadrant where its coordinates are both positive, and into the second quadrant, where the x-coordinate becomes negative.
随着直線OM的旋轉，點M移動到第一象限，其坐标都是正的，并進入第二象限，其x坐标變成了負的。
In the third quadrant, both coordinates are negative and finally, in the fourth quadrant, the point has a positive x- and negative y-coordinate. (See below.)
在第三象限，兩個坐标都是負的，最後，在第四象限，該點的X坐标是正的，Y坐标是負的。(見下文）。
You can see that the angle MON, called a, is always acute, and measured from the x-axis.
你可以看到角度MON，稱為a，總是銳角，并從X軸開始測量。
For example:
The signs of the trigonometric functions depend on which quadrant the point M lies in and represent the signs of the x- and y-coordinates of M.
三角函數的符号取決于點M位于哪個象限，代表M的x坐标和y坐标的符号。
Learn the information in the following diagrams to help you understand the signs.
學習以下圖表中的資訊，以幫助你了解這些符号。
First quadrant 第一象限
All the functions are positive. 所有的函數都是正數。
Second quadrant 第二象限
By looking at the signs of the coordinates of M, we see that the trigonometric functions of are 通過觀察M的坐标的符号，我們看到M的三角函數是:
Third quadrant 第三象限
The signs of the coordinates of M show us that the trigonometric functions are M的坐标的符号告訴我們，三角函數是:
Fourth quadrant 第四象限
The signs of the coordinates of M show us that the trigonometric functions of are M的坐标的符号告訴我們，其三角函數是:
This can be summarised as 這可以歸納為:
These sign rules and the value of the acute angle a allow you to find the value of any trigonometric function of any angle.
這些符号規則和銳角a的值允許你找到任何角度的三角函數的值。
Example:
Find the values of sin 150, sin 210 and sin 690 if sin 30 = 0.5. 如果sin 30=0.5，求sin 150、sin 210和sin 690的值。
sin 150 = sin 30 = 0.5
sin 210 = - sin 30 = - 0.5
sin 690 = sin 330 = - sin 30 = -0.5
You also need to be aware of negative angles created from the rotation of M in a clockwise direction.
你還需要注意M沿順時針方向旋轉所産生的負角。
i.e. each position of line OM gives us two different values of theta, one that is positive and one that is negative.
即OM線的每個位置都給我們兩個不同的theta值，一個是正值，一個是負值。
Example:
Here a = 200 so both angles have the same trigonometric functions.
這裡a=200，是以兩個角的三角函數相同。
Therefore:
sin 160° = sin (-200° ) = + sin 20°
cos 160° = cos (-200° ) = - cos 20°
tan 160° = tan (-200° ) = - tan 20°