Graphs, symmetries and periodicities of Sin, Cos and Tan sin、cos和tan的图形、对称性和周期性

The graphs of the three major functions are very important and you need to learn the characteristics of each.

三个主要函数的图形是非常重要的，你需要学习每个函数的特点。

The sine function 正弦函数

1. This graph is continuous (there are no breaks). 这个图形是连续的（没有中断）
2. The range is -1 ≤ sin θ ≤ 1. 范围是-1 ≤ sin θ ≤ 1
3. The shape of the graph from θ = 0 to θ = 2π is repeated every 2π radians. 从θ=0到θ=2π的图形形状每2π弧度重复一次。
4. This is called a periodic or cyclic function and the width of the repeating pattern that is measured on the horizontal axis, is called the period. The sine wave has a period of 2π, a maximum value of 1, and a minimum value of -1. 这被称为周期性或循环性函数，在横轴上测量的重复图案的宽度被称为周期。正弦波的周期为2π，最大值为 1，最小值为-1。
5. The greatest value of the sine wave is called the amplitude. 正弦波的最大值称为振幅。

The cosine function 余弦函数

1. This graph is continuous. 这个图形是连续的
2. The range is -1 ≤ cos θ ≤ 1. 其范围是 -1 ≤ cos θ ≤ 1
3. It has a period of 2π. 它的周期为2π
4. The shape is the same as the sine wave but displaced a distance of π ⁄ 2 to the left on the horizontal axis. This is called a phase shift. 其形状与正弦波相同，但在横轴上向左移了π ⁄ 2的距离。这被称为相移。

The tan function 正切函数

The tan function is found using 正切函数的求法是用:

It therefore follows that tan θ = 0, when sin θ = 0, and tan θ is undefined when cos θ = 0. 因此可以看出，当sin θ=0时，tan θ=0，当cos θ=0时，tan θ未定义。

1. This graph is continuous, but is undefined when 这个图形是连续的，但在以下情况下是未定义的

2. The range of values for tan θ is unlimited. tan θ的取值范围是无限的。

3. It has a period of π. 它的周期为π

All of the three functions periodical** repeat their values and the simplest way to learn this, is to make sure that you understand the general rules below which use 'n' to represent any integer (i.e. any whole number, both positive and negative).

所有这三个函数都会定期重复它们的值，学习这个的最简单方法是确保你理解下面的一般规则，用'n'来代表任何整数（即任何整数，包括正数和负数）。

(Remember: nπ means "every 180 degrees", and 2nπ means "every 360 degrees".)

(记住：nπ表示 "每180度"，而2nπ表示 "每360度"）。

Solutions of trigonometric equations 三角形方程的解决方案

A trigonometric equation contains at least one trigonometric function, and when asked to solve the equation we must find the angle(s) for which it is valid.

一个三角方程至少包含一个三角函数，当被要求解决这个方程时，我们必须找到方程有效的角度（s）。

We are normal** required to find particular values of θ in a given interval.

我们通常被要求在一个给定的区间内找到θ的特定值。

Example:

Solve the equation cos θ = 0, for −π ≤ θ ≤ π. 求解方程cos θ = 0, -π ≤ θ ≤ π.

The finite solution set is θ = − π ⁄ 2 and π ⁄ 2. 有限的解集是θ=-π ⁄2，π ⁄2。

There are two methods to find the solution of a trigonometric equation 有两种方法可以找到三角方程的解:

1. Use the graph of the trigonometric functions. 使用三角函数的图形
2. Use the four quadrants of the coordinate grid. 使用坐标网格的四个象限

The first step in both cases is to find the principal value, (or PV of θ which is the value you get from the calculator).

在这两种情况下，第一步是找到主值，（或者说是θ的PV，这是你从计算器得到的值）。

Principal values for sin, cos and tan sin、cos和tan的主要数值

Principal values for sin θ sin θ的主要数值

Any equation for sin θ = S for the domain 任何领域的sin θ = S的方程

has one solution in this interval called the principal value of θ. 在这个区间有一个解，叫做θ的主值。

It is in the first or fourth quadrant. 它位于第一象限或第四象限。

The range is shown in the diagram. 其范围如图所示。

Principal values for cos θ cos θ的主要数值

Any equation cos θ = C for the domain [0, π], has one solution in this interval called the principal value of θ.对于域[0, π]的任何方程cos θ = C，在这个区间有一个解，称为θ的主值。

It is in the first or second quadrant. 它位于第一象限或第二象限。

The range is shown in the diagram. 其范围如图所示。

Principal values for tan θ tan θ的主要数值

All the possible values for tan θ = T occur in the interval tan θ = T的所有可能值都出现在区间内

The one solution in this interval called the principal value of θ. 这个区间内的一个解决方案称为θ的主值。

It is in the first or fourth quadrant. 它位于第一象限或第四象限。

The range is shown in the diagram. 其范围如图所示。

Secondary values of sin, cos and tan sin、cos和tan的次要数值

Each trig. function has two solutions in a 360° or 2π interval. The first solution is the principal value, the other solution is called the secondary value, (SV), and lies in a different quadrant.

每个三角函数在3600或2π的区间内有两个解。第一个解是主值，另一个解被称为次值（SV），位于不同的象限内。

This can be found by drawing the graph or using the four quadrants of the coordinate grid as follows.

这可以通过画图或使用坐标网格的四个象限来找到，方法如下

Solve tan θ = 2, −π ≤ θ ≤ π 解决tan θ = 2，-π ≤ θ ≤ π

The first solution, (or principal value) is found using the calculator.

第一个解决方案，（或主值）是用计算器找到的。

θ = 1.11 (3 sf)

The second solution, (or secondary value), is found using the fact that tan is also positive in the third quadrant (it repeats every θ radians).

第二个解决方案（或次要值）是利用tan在第三象限也是正值的事实（它每隔θ弧度重复一次）找到的。

θ = π 1.11 = 4.25 (3 sf)

general Solutions of Trigonometric Equations 三角形方程的一般解

A general solution refers to all angles that satisfy the equation. So, it is an infinite set of angles. This is the same as before but we have to remember the period of the graph to list the rest of the solutions.

一般解决方案是指满足方程的所有角度。因此，它是一个无限的角度**。这和以前一样，但我们必须记住图形的周期来列出其余的解。

As sin and cos repeat every 360° or 2π radians we 由于sin和cos每隔360°或2π弧度重复一次，我们:

1. Find the two solutions in the initial range, (e.g. −π ≤ θ ≤ π) 找到初始范围内的两个解，（例如-π≤θ≤ π）
2. Add 360n or 2nπ to both of these 将这两个解加上360n或2nπ

As tan repeats every 180° or π radians we 由于tan每隔180°或π弧度就会重复一次，所以我们:

1. Find the first solution (PV) initial range, (e.g. −π ≤ θ ≤ π) 找到第一个解决方案（PV）的初始范围，（例如：-π ≤ θ ≤ π）
2. Add 180n or nπ

This can be summarised as 这可以归纳为:

For sin θ = S (where |S| ≤ 1), the general solution is 对于sin θ = S（其中|S| ≤ 1），一般的解是

For cos θ = C, the general solution is 对于cos θ = C，一般解决方案是

For tan θ = T, the general solution is 对于tan θ = T，一般解决方案是

The graphs of trigonometric functions of compound angles 复角的三角函数的图形

The graph of the function sin cθ where c is a constant, is a sine wave with a period of 2π ⁄ c. The frequency is c times that of sin θ. This is shown in the diagram below 函数sin cθ的图形，其中c是一个常数，是一个周期为2π ⁄ c的正弦波，其频率是sin θ的c倍:

This rule is also true for cos θ, and tan θ. 这一规则对于cos θ和tan θ也是如此

This means that when solving trigonometric equations with a multiple of θ, there will be a different number of solutions in a 360° range. In these situations find the two initial solutions, make the general set of solutions, and then rearrange to find θ.

这意味着，当求解θ的倍数的三角方程时，在360°范围内会有不同数量的解。在这些情况下，找到两个初始解，做出一般的解集，然后重新排列，找到θ。

Example:

Solve cos (3θ 45) = −0.5 解出cos (3θ 45) = -0.5

(3θ 45) = −120 (from calculator) and,

(3θ 45) = 120 (cos is negative in the second and third quadrants) (cos在第二和第三象限是负的)

Therefore, (3θ 45) = −120 ± 360n and (3θ 45) = 120 ± 360n.

Therefore, θ = −55 ± 120n , and θ = 25 ± 120n

A final hint. Watch out for trigonometric equations that are quadratics.

最后一个提示。留意那些四边形的三角方程。Example:

2 sin2 θ sin θ − 1 = 0

This has to be factorised and then solved. 这必须先进行因式分解，然后再求解。

(2 sin θ − 1) (sin θ 1) = 0, where 0 ≤ θ ≤ 360

sin θ = 0.5 or sin θ = -1 and solve as before to get,

θ = 30, 150, or 270. (See if you can get these solutions.)