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：。
常用角度的三角函数值
常用角度的三角函数值
三角函数是一种常用角度处理运算，它被应用于许多科学和工程
技术中，通过利用三角函数可以解决各种解决问题。
三角函数主要包
括正弦Sine，余弦Cosine和正切Tangent三个角度处理运算。
正弦是一种三角函数，它以形如y = A sin (Bx + C)的方式表示，它的定义是半径的角的正弦值，也就是弧度/radians的值为A，B为周期，一般情况下B都为2π或π，C表示平移量，即x轴的.
一般常用角度的三角函数的值如下：0度的sin值为0，90度的
sin值为1，180度的sin值为0，270度的sin值为-1；0度的cos值
为1，90度的cos值为0，180度的cos值为-1，270度的cos值为0；0度的tan值为0，90度的tan值为无穷，180度的tan值为0，270
度的tan值为无穷。
由此可见，三角函数对于解决特定的角度问题非常有用，而在很
多日常生活中也有它的应用，比如在构成一个几何形状时，需要计算
三角函数来构成各个角落的位置。
在科学研究方面，三角函数的用处也非常大，比如研究正弦波的
容积计算，可以用到B=2π的正弦函数，来辅助计算正弦波的容积。
此外，三角函数还被广泛的用于电气工程，比如在相位分析，将大小
供比较的电流和电压分解成正弦和余弦函数，来计算相位差。
总之，三角函数是当今非常重要的角度处理运算，它不仅能够派
上用场解决日常生活中几何问题，还可以在科学研究及工程方面拥有
巨大的潜力，将来也会更加重要的，使世界变得更加智能。
【数学课件】双语教学课件：Sine and cosine
THANKS！
1、做老师的只要有一次向学生撒谎撒漏了底，就可能使他的全部教育成果从பைடு நூலகம்为之毁灭。——卢梭
好好学习，天天向上。 2、教育人就是要形成人的性格。——欧文
cos A b 2 c 2 a 2
2bc
cos B a 2 c2 b2
cos C a 2 b 2 c 2
2ac
2ab
Question1: Q is 25 km from P on a
bearing（方位角） of 75 and 40 km
from R ．P is due North（正北方） of R.
Calculate the bearing of Q from R.
N
75
Q
P
R
Question2: Find the length BC and the angle B.
B
30
C
A
Question3: Find the length BC and the angle B.
B
30
A 22mm
C
b
Sine and Cosine Rules 正弦(余弦)定理 (二)
Review: 1. Sine Rule
a b c sin A sin B sin C
2R
2、Cosine Rule
a2 b2 c2 2bc cos A
b2 a2 c2 2bc cosB
c2 a2 b2 2ab cosC
3、自我教育需要有非常重要而强有力的促进因素——自尊心、自我尊重感、上进心。——苏霍姆林斯基 4、追求理想是一个人进行自我教育的最初的动力，而没有自我教育就不能想象会有完美的精神生活。我认为，教会学生自己教育自己，这是一种
北师大版八上Unit3《Countries and Cities》(Lesson10)课件
The Beach Hotel
Built : 2000 Price :420 ￥ (each) Room : single,two-three
The pink ruler is longer than the blue one . The blue ruler is shorter than the pink one .
The picture is nicer than that one . The white T-shirt is larger than the yellow one .
rooms Close to the beach Good views
The Garden Hotel
Built : 1980 Price :150 ￥ (each) Room : single,two rooms Clean ,clear air, Nice surounding : park
The Mississippi River The Nile River
Байду номын сангаас iver
ountain
ake
esert
6,300 km (long)
5,464 km (long)
The Yangtze River
The Yellow River
The Yangtze River is longer than The Yellow River .
32.4 million square (large) 1 million square (large) The Takelamagan Desert The Maowusu Desert
数字信号处理ppt课件
4
Signal-----
• An indicator, such as a gesture or colored light, that serves as a means of communication.See Synonyms at gesture
• 信号：一种用作通讯交流手段的指示，比如一种手势或有色 的光参见 gesture
• the center is too bright, while the border is too dark
13
•a great example of how DSP can improve the
understandability of data.
• Digital filtering was able to convert the raw image (on the left) into a processed image (on the right).
• Suppose we attach an analog-to-digital converter to a computer, and then use it to acquire a chunk of real world data. DSP answers the question: What next?
• A message communicated by such means. • 信号：用这种手段传达的信息 • Electronics An impulse or a fluctuating electric quantity, such as
voltage, current, or electric field strength, whose variations represent coded information. • 【电子学】 电波：电脉冲或变化的电量，比如电压、电流或 电场强度，它们的变化表示着编码后的信息 • The sound, image, or message transmitted or received in telegraphy, telephony, radio, television, or radar. • 信号：由电报、电话、收音机、电视机或雷达传播或收到的 声音、影像或信息
学校观摩课课件Unit-6-Lesson-2-It-is-nice-and-not-expensiv
want to buy five.
Shop assistant: Here you are .
Tom: Here is the money. Thanks.
2 A: How much is the dress? B: It is 150 yuan.
3
A:What is the price of the dress? B: It is 150 yuan.
Complete the dialogues
Dialogue 1 Talking about the price
A: You are welcome. Here is your
change.
Talking about color
1 A: What color do you like best?
B: I like
best.
blue
red
orange
green
white
black
purple yellow
2
A: What is your favorite color?
I’d like…
a dress
Some gifts a pair of shoes
a cake
a pack of gum
a mobile phonesome apples
a cup of coffee
Offering suggestions
What about this one? How about this one? This one fits you.
I want to buy five.
Shop assistant: Here you are .
Tom:
新汉英数学词汇 角平分线 英语
新汉英数学词汇角平分线英语全文共3篇示例，供读者参考篇1Title: New Chinese-English Mathematical Terminology: Angle BisectorIntroduction:As the world becomes more interconnected, the need for standardized mathematical terminology in multiple languages has become increasingly important. In recent years, there has been a push to create new Chinese-English mathematical terminology to better facilitate communication between Chinese and English-speaking mathematicians. One such term that has been recently standardized is "角平分线" or "angle bisector" in English.What is an Angle Bisector?An angle bisector is a line or ray that divides an angle into two congruent angles. In other words, it is a line that passes through the vertex of an angle and splits it into two equal parts. The angle bisector is an important concept in geometry and iscommonly used in various mathematical proofs and constructions.Importance of Standardized Terminology:Having standardized mathematical terminology is crucial for effective communication and collaboration between mathematicians from different linguistic backgrounds. By creating new Chinese-English mathematical terminology such as "angle bisector," we can ensure that mathematical concepts are accurately and clearly conveyed across different languages.Usage of the Term:The term "angle bisector" is widely used in geometry and trigonometry to describe a line that divides an angle into two equal parts. Mathematicians use angle bisectors to solve problems involving angles, triangles, and other geometric shapes. Understanding the concept of an angle bisector is essential for mastering various mathematical concepts and techniques.Conclusion:The standardization of new Chinese-English mathematical terminology, such as "angle bisector," is a positive step towards improving global mathematical communication andcollaboration. By creating clear and consistent terminology, mathematicians can more easily exchange ideas, share knowledge, and work together to advance the field of mathematics. The term "angle bisector" is just one example of how standardized mathematical terminology can enhance cross-cultural communication and cooperation in the mathematical community.篇2Title: New Chinese-English Mathematical Vocabulary: Angle BisectorIntroductionMathematics is a universal language that transcends cultural and linguistic barriers. To facilitate communication and collaboration among mathematicians around the world, it is essential to have a standardized set of mathematical vocabulary in various languages. In recent years, there has been a concerted effort to update and expand the Chinese-English mathematical vocabulary to better reflect the advancements in the field of mathematics. One important concept that has been included in the new vocabulary is the angle bisector.What is an Angle Bisector?An angle bisector is a line that divides an angle into two equal parts. In other words, it is a line that cuts an angle into two congruent angles. The point where the angle bisector intersects the angle is called the vertex of the angle.In geometric terms, the angle bisector is a line that passes through the vertex of the angle and divides the angle into two smaller angles of equal measure. The angle bisector is commonly denoted by the symbol "AD", where "A" is the vertex of the angle and "D" is the point where the angle bisector intersects the angle.Applications of Angle BisectorsAngle bisectors have numerous applications in geometry and trigonometry. They are used in various geometric constructions, such as constructing equilateral triangles, finding the incenter of a triangle, and bisecting a line segment. In trigonometry, angle bisectors are used to find the angle bisectors of a given angle, as well as to solve problems involving angles and triangles.ConclusionIn conclusion, the inclusion of the angle bisector in the new Chinese-English mathematical vocabulary is a significant steptowards promoting international collaboration and communication in the field of mathematics. By standardizing the mathematical vocabulary across different languages, mathematicians from different countries can more easily share ideas, collaborate on research projects, and contribute to the advancement of mathematics as a global discipline. The angle bisector is just one example of the many new terms that have been added to the Chinese-English mathematical vocabulary, and it serves as a reminder of the importance of language in the pursuit of knowledge and understanding.篇3Angle bisector, also known as angle partition or angle trisector in some contexts, is a geometric line that divides an angle into two equal parts. This concept plays an important role in geometry and trigonometry, helping us solve various problems related to angles and shapes. In this article, we will explore the term "angle bisector" in English and discuss its significance in mathematics.In English, the term "angle bisector" refers to a line or ray that divides an angle into two equal parts. The angle bisector can be either an internal or external line, depending on the position of the point where it intersects the angle. When an angle bisectoris drawn within the angle, it is called an internal angle bisector, while an angle bisector drawn outside the angle is referred to as an external angle bisector.The angle bisector theorem is a fundamental concept in geometry, stating that in a triangle, the angle bisector of an angle divides the opposite side in two segments that are proportional to the lengths of the other two sides. This theorem is often used to solve various problems involving angles and sides in triangles.Another important concept related to angle bisectors is the angle bisector construction. Given an angle, the angle bisector construction involves drawing a line that divides the angle into two equal parts. This construction is often used in geometric constructions and proofs to demonstrate the properties of angles and shapes.In trigonometry, the concept of angle bisectors is also important. When working with trigonometric functions like sine, cosine, and tangent, angle bisectors help us determine the values of these functions for different angles. Understanding the properties of angle bisectors can thus enhance our ability to solve trigonometric problems and equations.Overall, the concept of angle bisectors is a key element in geometry and trigonometry, providing us with a valuable tool for solving problems related to angles and shapes. By understanding the properties and applications of angle bisectors, we can enhance our mathematical skills and tackle a wide range of mathematical challenges.。