双语教学课件：Sine and cosine
三角函数歌英文英译文
三角函数歌英文英译文Title: The Song of Trigonometric Functions.In the vast realm of mathematics, trigonometric functions hold a unique and fascinating place. They are the melodies that resonate through the geometric landscapes, painting the canvas of angles, sides, and circles withtheir harmonious patterns. Let us embark on a journey through the English translation of "The Song of Trigonometric Functions," a celebration of their beauty and significance.Verse One: The Introduction.Oh, trigonometric functions, you are the dancers of the mathematical ball.Your steps are precise, your rhythms are exact, and your melodies are divine.From sine to cosine, tangent to secant, you spin and twirl in perfect harmony.Your grace and elegance are unmatched, as you traverse the plane with ease.Verse Two: The Sine Function.Sine, you are the queen of the waves, your crests and troughs defining the sea.Your cycle is infinite, your peaks and valleys repeating with precision.You are the voice of the pendulum, swinging back and forth with regularity.Your graph is a sine curve, a wave that flows through time and space.Verse Three: The Cosine Function.Cosine, you are the twin of sine, your steps mirroring.。
英国A-level数学教材内容汇总大全-推荐下载
英国A-LEVEL教材汇总Core Mathematics1（AS/A2）——核心数学11.Algebra and functions——代数和函数2.Quadratic functions——二次函数3.Equations and inequalities——等式和不等式4.Sketching curves——画图（草图）5.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何6.Sequences and series——数列7.Differentiation——微分8.Integration——积分Core Mathematics2（AS/A2）——核心数学21.Algebra and functions——代数和函数2.The sine and cosine rule——正弦和余弦定理3.Exponentials and logarithm——指数和对数4.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何5.The binomial expansion——二项展开式6.Radian measure and its application——弧度制及其应用7.Geometric sequences and series——等比数列8.Graphs of trigonometric functions——三角函数的图形9.Differentiation——微分10.Trigonometric identities and simple equations——三角恒等式和简单的三角等式11.Integration——积分Core Mathematics3（AS/A2）——核心数学31.Algebra fractions——分式代数2.Functions——函数3.The exponential and log functions——指数函数和对数函数4.Numerical method——数值法5.Transforming graph of functions——函数的图形变换6.Trigonometry——三角7.Further trigonometric and their applications——高级三角恒等式及其应用8.Differentiation——微分Core Mathematics4（AS/A2）——核心数学41.Partial fractions——部分分式2.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何3.The binomial expansion——二项展开式4.Differentiation——微分5.Vectors——向量6.Integration——积分A-Level：核心数学Core Maths，力学数学，统计数学，决策数学Core Mathematics1（AS/A2）——核心数学11.Algebra and functions——代数和函数2.Quadratic functions——二次函数3.Equations and inequalities——等式和不等式4.Sketching curves——画图（草图）5.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何6.Sequences and series——数列7.Differentiation——微分8.Integration——积分每章内容：Core Mathematics2（AS/A2）——核心数学21.Algebra and functions——代数和函数2.The sine and cosine rule——正弦和余弦定理3.Exponentials and logarithm——指数和对数4.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何5.The binomial expansion——二项展开式6.Radian measure and its application——弧度制及其应用7.Geometric sequences and series——等比数列8.Graphs of trigonometric functions——三角函数的图形9.Differentiation——微分10.Trigonometric identities and simple equations——三角恒等式和简单的三角等式11.Integration——积分每章内容：1.Algebra fractions——分式代数2.Functions——函数3.The exponential and log functions——指数函数和对数函数4.Numerical method——数值法5.Transforming graph of functions——函数的图形变换6.Trigonometry——三角7.Further trigonometric and their applications——高级三角恒等式及其应用8.Differentiation——微分每章内容：1.Partial fractions——部分分式2.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何3.The binomial expansion——二项展开式4.Differentiation——微分5.Vectors——向量6.Integration——积分每章内容：。
高中数学 三角函数
高中数学：三角函数一、概述三角函数是高中数学的一个重要组成部分，是解决许多数学问题的关键工具。
它涉及的角度、边长、面积等，都是几何和代数的核心元素。
通过学习三角函数，我们可以更好地理解图形的关系，掌握数学的基本概念。
二、三角函数的定义三角函数是以角度为自变量，角度对应的边长为因变量的函数。
常用的三角函数包括正弦函数（sine）、余弦函数（cosine）和正切函数（tangent）。
这些函数的定义如下：1、正弦函数：sine（θ） = y边长 / r （其中，θ是角度，r是从原点到点的距离）2、余弦函数：cosine（θ） = x边长 / r3、正切函数：tangent（θ） = y边长 / x边长三、三角函数的基本性质1、周期性：正弦函数和余弦函数都具有周期性，周期为 2π。
正切函数的周期性稍有不同，为π。
2、振幅：三角函数的振幅随着角度的变化而变化。
例如，当角度增加时，正弦函数的值也会增加。
3、相位：不同的三角函数具有不同的相位。
例如，正弦函数的相位落后余弦函数相位π/2。
4、奇偶性：正弦函数和正切函数是奇函数，余弦函数是偶函数。
5、导数：三角函数的导数与其自身函数有关。
例如，正弦函数的导数是余弦函数，余弦函数的导数是负的正弦函数。
四、三角函数的实际应用三角函数在现实生活中有着广泛的应用，包括但不限于以下几个方面：1、物理：在物理学中，三角函数被广泛应用于描述波动、振动、电磁场等物理现象。
例如，简谐振动可以用正弦或余弦函数来描述。
2、工程：在土木工程和机械工程中，三角函数被用于计算角度、长度等物理量。
例如，在桥梁设计、建筑设计等过程中，需要使用三角函数来计算最佳的角度和长度。
3、计算机科学：在计算机图形学中，三角函数被用于生成二维和三维图形。
例如，使用正弦和余弦函数可以生成平滑的渐变效果。
4、金融：在金融学中，三角函数被用于衍生品定价和风险管理。
例如，Black-Scholes定价模型就使用了正态分布（一种特殊的三角函数）。
拉密定理和正余弦定
拉密定理和正余弦定英文回答：The Lami's theorem, also known as the Lami's theorem of equilibrium, states that if three forces acting on a particle are in equilibrium, then each force isproportional to the sine of the angle between the other two forces. This theorem is named after the Italian mathematician and physicist Giovanni Battista Lami.To understand the Lami's theorem, let's consider a simple example. Imagine a particle at rest, with three forces acting on it. The magnitudes of these forces are F1, F2, and F3, and the angles between them are α, β, and γ, respectively. According to Lami's theorem, the forces can be represented as:F1/sin(α) = F2/sin(β) = F3/sin(γ)。
This means that the ratio of each force to the sine ofthe angle between the other two forces is constant. Inother words, the forces are in equilibrium when this ratiois maintained.Now let's move on to the cosine and sine laws. These laws are used to find the lengths of sides and measures of angles in triangles.The cosine law, also known as the law of cosines,relates the lengths of the sides of a triangle to thecosine of one of its angles. It states that in a triangle with sides of lengths a, b, and c, and opposite angles A, B, and C respectively, the following equation holds:c^2 = a^2 + b^2 2abcos(C)。
正余弦公式推论
正余弦公式推论The sine and cosine formulas, also known as the trigonometric addition formulas, are essential in mathematics and other fields of study. These formulas enable us to find the sine and cosine values of the sum or difference of two given angles, which is crucial in various applications such as physics, engineering, and architecture. The derived formulas for sine and cosine can be quite complex but are incredibly useful in solving problems related to triangles and circular motion.正弦和余弦公式，也称为三角函数加法公式，在数学和其他学科中至关重要。
这些公式使我们能够找到两个给定角的和或差的正弦和余弦值，这在物理、工程和建筑等各种应用中至关重要。
派生的正弦和余弦公式可能相当复杂，但在解决与三角形和圆周运动相关的问题时非常有用。
The cosine addition formula states that the cosine of the sum of two angles is equal to the product of the cosines of the angles minus the product of the sines of the angles. This formula is expressed as cos(A + B) = cosA cosB - sinA sinB. It is a fundamental concept in trigonometry and is used extensively in calculus, physics, andengineering. Understanding and applying this formula can help solve problems involving circular motion, harmonic motion, and wave propagation.余弦加法公式规定了两个角之和的余弦等于两个角的余弦乘积减去两个角的正弦乘积。
函数英文ppt课件ppt课件ppt课件
Definition: The composite function f (g (x)) is formed by replacing the input of function g with the output of function f. For example, if f (x)=x ^ 2 and g (x)=3x+2, then the composite function f (g (x))=(3x+2) ^ 2 for all x
The characteristics of linear functions include: when a>0, the function is an increasing function; When a<0, the function is a subtraction function.
Linear functions have a wide range of applications in real life, such as economics, statistics, and other fields.
Horizontal shift
Shifting a function Horizontally is equivalent to adding or subtracting a constant from all inputs For example, if f (x)=x ^ 2 is shifted left by 1 unit, it benefits (x-1) ^ 2
Exponential and logarithmic functions are also widely used in real life, such as in fields such as statistics and finance.
南开大学光学工程内部课件Fresnel diffraction
(6)
(7)
Fresnel approximation for diffraction integral 3/4
The aperture is small compared to both r´ and s´ :
2 2 x 0 y 0 2 x 0 y 0 r r´ 3 r´ 2 r´ 2 r´ 2 2 x y 2 x y s s´ 3 s´ 2 s´ 2 s´
13 13 5 7 Q w 1 2 2 2 4 w w P w 1 1 3 5 1 3 5 7 9 2 2 3 2 5 w w w
We combine (22) into a comblex integral (27)
Fresnel’s integrals 1/7
The integral equation (10) can be written in the form
(12)
U P B C jS ,
(intensity I (P )=|B |2(C 2+S 2))
where (13)
B A C S j
1 2 1 cos r' s' 1 1 r' s'
2
2
2
2
u ,
2
(17)
2
v .
(18)
Then
dd
dudv
1 2 1 cos r' s'
英国Alevel数学教材内容汇总
Core Mathematics1（AS/A2）——核心数学11.Algebra and functions——代数和函数2.Quadratic functions——二次函数3.Equations and inequalities——等式和不等式4.Sketching curves——画图（草图）5.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何6.Sequences and series——数列7.Differentiation——微分8.Integration——积分Core Mathematics2（AS/A2）——核心数学21.Algebra and functions——代数和函数2.The sine and cosine rule——正弦和余弦定理3.Exponentials and logarithm——指数和对数4.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何5.The binomial expansion——二项展开式6.Radian measure and its application——弧度制及其应用7.Geometric sequences and series——等比数列8.Graphs of trigonometric functions——三角函数的图形9.Differentiation——微分10.Trigonometric identities and simple equations——三角恒等式和简单的三角等式11.Integration——积分Core Mathematics3（AS/A2）——核心数学31.Algebra fractions——分式代数2.Functions——函数3.The exponential and log functions——指数函数和对数函数4.Numerical method——数值法5.Transforming graph of functions——函数的图形变换6.Trigonometry——三角7.Further trigonometric and their applications——高级三角恒等式及其应用8.Differentiation——微分Core Mathematics4（AS/A2）——核心数学41.Partial fractions——部分分式2.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何3.The binomial expansion——二项展开式4.Differentiation——微分5.Vectors——向量6.Integration——积分A-Level：核心数学Core Maths，力学数学，统计数学，决策数学Core Mathematics1（AS/A2）——核心数学11.Algebra and functions——代数和函数2.Quadratic functions——二次函数3.Equations and inequalities——等式和不等式4.Sketching curves——画图（草图）5.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何6.Sequences and series——数列7.Differentiation——微分8.Integration——积分每章内容：Core Mathematics2（AS/A2）——核心数学21.Algebra and functions——代数和函数2.The sine and cosine rule——正弦和余弦定理3.Exponentials and logarithm——指数和对数4.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何5.The binomial expansion——二项展开式6.Radian measure and its application——弧度制及其应用7.Geometric sequences and series——等比数列8.Graphs of trigonometric functions——三角函数的图形9.Differentiation——微分10.Trigonometric identities and simple equations——三角恒等式和简单的三角等式11.Integration——积分每章内容：1.Algebra fractions——分式代数2.Functions——函数3.The exponential and log functions——指数函数和对数函数4.Numerical method——数值法5.Transforming graph of functions——函数的图形变换6.Trigonometry——三角7.Further trigonometric and their applications——高级三角恒等式及其应用8.Differentiation——微分每章内容：1.Partial fractions——部分分式2.Coordinate geometry in the （x，y）plane——平面坐标系中的坐标几何3.The binomial expansion——二项展开式4.Differentiation——微分5.Vectors——向量6.Integration——积分每章内容：。