MindMap Gallery The concept of trigonometric functions
This is a mind map about the concept of trigonometric functions. The main contents include: history of trigonometric functions, expansion of trigonometric functions, applications of trigonometric functions, trigonometric identities, images of trigonometric functions, properties of trigonometric functions, unit circle, definition .
Edited at 2024-12-13 17:37:24Find a streamlined guide created using EdrawMind, showcasing the Lemon 8 registration and login flow chart. This visual tool facilitates an effortless journey for American users to switch from TikTok to Lemon 8, making the transition both intuitive and rapid. Ideal for those looking for a user-centric route to Lemon 8's offerings, our flow chart demystifies the registration procedure and emphasizes crucial steps for a hassle-free login.
これは稲盛和夫に関するマインドマップです。私のこれまでの人生のすべての経験は、ビジネスの明確な目的と意味、強い意志、売上の最大化、業務の最小化、そして運営は強い意志に依存することを主な内容としています。
かんばんボードのデザインはシンプルかつ明確で、計画が一目で明確になります。毎日の進捗状況を簡単に記録し、月末に要約を作成して成長と成果を確認することができます。 実用性が高い:読書、早起き、運動など、さまざまなプランをカバーします。 操作簡単:シンプルなデザイン、便利な記録、いつでも進捗状況を確認できます。 明確な概要: 毎月の概要により、成長を明確に確認できます。 小さい まとめ、今月の振り返り掲示板、今月の習慣掲示板、今月のまとめ掲示板。
Find a streamlined guide created using EdrawMind, showcasing the Lemon 8 registration and login flow chart. This visual tool facilitates an effortless journey for American users to switch from TikTok to Lemon 8, making the transition both intuitive and rapid. Ideal for those looking for a user-centric route to Lemon 8's offerings, our flow chart demystifies the registration procedure and emphasizes crucial steps for a hassle-free login.
これは稲盛和夫に関するマインドマップです。私のこれまでの人生のすべての経験は、ビジネスの明確な目的と意味、強い意志、売上の最大化、業務の最小化、そして運営は強い意志に依存することを主な内容としています。
かんばんボードのデザインはシンプルかつ明確で、計画が一目で明確になります。毎日の進捗状況を簡単に記録し、月末に要約を作成して成長と成果を確認することができます。 実用性が高い:読書、早起き、運動など、さまざまなプランをカバーします。 操作簡単:シンプルなデザイン、便利な記録、いつでも進捗状況を確認できます。 明確な概要: 毎月の概要により、成長を明確に確認できます。 小さい まとめ、今月の振り返り掲示板、今月の習慣掲示板、今月のまとめ掲示板。
The concept of trigonometric functions
definition
angle
Measuring the size of an angle
Degree system
Range from 0 to 360 degrees
Each circle is divided into 360 degrees
radians
Based on the radius of the circle
The circumference of a circle is equal to 2π radians
right triangle
An angle is 90 degrees
The sum of the other two angles is 90 degrees
trigonometric ratio
sine
Ratio of opposite side to hypotenuse
in a right triangle
sin(θ) = length of opposite side / length of hypotenuse
cosine
Ratio of adjacent side to hypotenuse
in a right triangle
cos(θ) = length of adjacent side / length of hypotenuse
tangent
The ratio of the opposite side to the adjacent side
in a right triangle
tan(θ) = length of opposite side / length of adjacent side
unit circle
The center of the circle is at the origin
radius is 1
Define trigonometric functions
The terminal side of the angle intersects the unit circle
coordinates of intersection
The x coordinate corresponds to the cosine value
The y coordinate corresponds to the sine value
tangent
The ratio of y coordinate to x coordinate
Properties of Trigonometric Functions
cyclical
Sine and cosine functions
The period is 360 degrees or 2π radians
sin(θ) = sin(θ 360°)
cos(θ) = cos(θ 360°)
tangent function
Period is 180 degrees or π radians
tan(θ) = tan(θ 180°)
parity
sine function
odd function
sin(-θ) =sin(θ)
cosine function
even function
cos(-θ) = cos(θ)
tangent function
odd function
tan(-θ) =tan(θ)
range
Sine and cosine functions
1, 1
tangent function
all real numbers
Graphics of Trigonometric Functions
sine function image
wavy curve
cyclical fluctuations
Amplitude is 1
cosine function image
wavy curve
cyclical fluctuations
Amplitude is 1
The phase ratio sine function image is shifted 90 degrees to the left
Tangent function graph
periodic curve
Increases from negative infinity to positive infinity in each period
Period is π radians
Trigonometric Identities
Basic Identities
sin²θ cos²θ = 1
sum and difference formula
sin(α ± β) = sinαcosβ ± cosαsinβ
cos(α ± β) = cosαcosβ ∓ sinαsinβ
Double angle formula
sin2θ = 2sinθcosθ
cos2θ = cos²θ sin²θ = 2cos²θ 1 = 1 2sin²θ
half angle formula
sin²(θ/2) = (1 cosθ)/2
cos²(θ/2) = (1 cosθ)/2
Applications of Trigonometric Functions
geometry
Solve problems related to angles and distances
physics
Fluctuation analysis
Sound waves, light waves, etc.
Vibration analysis
Simple harmonic motion etc.
engineering
signal processing
Filter design, etc.
structural analysis
Calculation of stress and strain, etc.
astronomy
Calculation of celestial body positions
Prediction of star motion, etc.
electronics
AC circuit analysis
Phase difference between voltage and current, etc.
computer Science
Graphics
3D modeling and rendering, etc.
Algorithm design
Fast Fourier Transform, etc.
Extensions of Trigonometric Functions
inverse trigonometric function
arcsine
Inverse function of sine function
Arc cosine (arccos)
Inverse function of cosine function
arctan
inverse function of tangent function
hyperbolic function
Hyperbolic sine (sinh)
A type of hyperbolic trigonometric function
Hyperbolic cosine (cosh)
A type of hyperbolic trigonometric function
Hyperbolic tangent (tanh)
A type of hyperbolic trigonometric function
Complex Trigonometric Functions
Trigonometric functions over the field of complex numbers
Euler's formula
e^(iθ) = cosθ i*sinθ
Conversion between exponential and trigonometric forms of complex numbers
Convert using Euler's formula
History of Trigonometric Functions
ancient greek period
hippasos
Irrational numbers were discovered
Euclid
Discussion of angles in "Elements"
indian mathematician
Ayepodo
Early studies of trigonometric functions
islamic golden age
al batani
Compilation of trigonometric function tables
European Renaissance
Regmontanus
Systematic study of trigonometry
René Descartes
The creation of analytic geometry
The combination of trigonometric functions and coordinate systems
modern mathematics
Strict definition of trigonometric functions
Based on the concepts of limits and infinitesimal quantities
Applications of trigonometric functions in advanced mathematics
Calculus
complex analysis
Fourier analysis