Name: trigonometric limits for derivatives
Uploaded: 2026-01-12T12:24:48.000Z
Duration: 1 min 25 s
Channel: JoVE

一个基本的三角函数极限是当 θ 趋近于零时，sin θ 除以 θ 的极限，其值等于 1。为了直观理解，考虑一个单位圆，以及从正 x 轴测量的一个小角度 θ。绘制出对应于该角度的圆弧。对于单位圆，圆弧长度在数值上等于以弧度表示的角度。从圆弧的终点向 x 轴作一条垂直线。该线段表示 theta 的正弦值。由于线段 BC 始终小于圆弧 AB，因此可推导出 sin θ / θ < 1 的关系。半径从圆心 O 经过点 B 延长，直至与点 A 处的切线相交于点 D。线段 AD 的长度表示 tan θ。从几何上看，弧长 θ 小于切线段 AD 的长度，后者可用 sin θ 和 cos θ 表示。利用这两个关系可推导出该不等式。当 theta 趋近于零时，外侧边界值均收敛于 1。由于 sine 与 theta 的比值介于两者之间，因此它也趋近于 1。

三角函数的极限

&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x7684;&#x6781;&#x9650;&#x5728;&#x5FAE;&#x79EF;&#x5206;&#x4E2D;&#x5177;&#x6709;&#x57FA;&#x7840;&#x6027;&#x4F5C;&#x7528;&#xFF0C;&#x5C24;&#x5176;&#x662F;&#x5728;&#x5BFC;&#x6570;&#x7684;&#x5B9A;&#x4E49;&#x4E2D;&#x3002;&#x5176;&#x4E2D;&#x6700;&#x91CD;&#x8981;&#x7684;&#x7ED3;&#x679C;&#x4E4B;&#x4E00;&#x662F;&#xFF1A;<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19797/19797_PT_Equation_1_editor_19797.svg" alt="19797_PT_Equation_1.svg" /></figure>&#x8BE5;&#x7ED3;&#x679C;&#x5728;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6C42;&#x5BFC;&#x4E2D;&#x81F3;&#x5173;&#x91CD;&#x8981;&#xFF0C;&#x5E76;&#x88AB;&#x5E7F;&#x6CDB;&#x5E94;&#x7528;&#x4E8E;&#x6570;&#x5B66;&#x5206;&#x6790;&#x4E0E;&#x7269;&#x7406;&#x5B66;&#x7814;&#x7A76;&#x4E4B;&#x4E2D;&#x3002;&#x51E0;&#x4F55;&#x610F;&#x4E49;&#x8BC1;&#x660E;&#x4E0A;&#x8FF0;&#x7ED3;&#x679C;&#x7684;&#x4E00;&#x79CD;&#x5E38;&#x89C1;&#x65B9;&#x6CD5;&#x662F;&#x5206;&#x6790;&#x5355;&#x4F4D;&#x5706;&#x4E2D;&#x7684;&#x4E00;&#x4E2A;&#x6247;&#x5F62;&#xFF0C;&#x5176;&#x5706;&#x5FC3;&#x6240;&#x5BF9;&#x89D2;&#x4E3A;&#x67D0;&#x4E00;&#x7ED9;&#x5B9A;&#x89D2;&#x3002;&#x7531;&#x4E8E;&#x5F27;&#x957F;&#x5728;&#x6570;&#x503C;&#x4E0A;&#x7B49;&#x4E8E;&#x4EE5;&#x5F27;&#x5EA6;&#x8868;&#x793A;&#x7684;&#x5706;&#x5FC3;&#x89D2;&#x5927;&#x5C0F;&#xFF0C;&#x8FD9;&#x4E3A;&#x7406;&#x89E3;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6781;&#x9650;&#x63D0;&#x4F9B;&#x4E86;&#x91CD;&#x8981;&#x7684;&#x51E0;&#x4F55;&#x57FA;&#x7840;&#x3002;&#x6B64;&#x5916;&#xFF0C;&#x8BE5;&#x89D2;&#x5EA6;&#x8FD8;&#x6EE1;&#x8DB3;&#x5982;&#x4E0B;&#x4E0D;&#x7B49;&#x5F0F;&#xFF1A;<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/19797/19797_PT_Equation_2_editor_19797.svg" alt="19797_PT_Equation_2.svg" /></figure>&#x8FD9;&#x4E00;&#x4E0D;&#x7B49;&#x5F0F;&#x5728;&#x5229;&#x7528;&#x5939;&#x903C;&#x5B9A;&#x7406;&#x5EFA;&#x7ACB;&#x6781;&#x9650;&#x8FC7;&#x7A0B;&#x4E2D;&#x8D77;&#x7740;&#x5173;&#x952E;&#x4F5C;&#x7528;&#x3002;&#x5939;&#x903C;&#x5B9A;&#x7406;&#x7684;&#x5E94;&#x7528;&#x5939;&#x903C;&#x5B9A;&#x7406;&#x662F;&#x4E00;&#x79CD;&#x91CD;&#x8981;&#x7684;&#x6570;&#x5B66;&#x5DE5;&#x5177;&#xFF0C;&#x5B83;&#x901A;&#x8FC7;&#x5C06;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x4E0E;&#x4E24;&#x4E2A;&#x6536;&#x655B;&#x4E8E;&#x540C;&#x4E00;&#x6781;&#x9650;&#x503C;&#x7684;&#x51FD;&#x6570;&#x8FDB;&#x884C;&#x6BD4;&#x8F83;&#xFF0C;&#x4ECE;&#x800C;&#x6C42;&#x5F97;&#x5176;&#x6781;&#x9650;&#x3002;&#x5728;&#x672C;&#x4F8B;&#x4E2D;&#xFF0C;&#x901A;&#x8FC7;&#x5C06;&#x6B63;&#x5F26;&#x51FD;&#x6570;&#x5939;&#x903C;&#x4E8E;&#x4E24;&#x4E2A;&#x51E0;&#x4F55;&#x8868;&#x8FBE;&#x5F0F;&#x4E4B;&#x95F4;&#xFF0C;&#x4ECE;&#x800C;&#x53EF;&#x76F4;&#x63A5;&#x5F97;&#x51FA;&#x7ED3;&#x8BBA;&#xFF1A;19797_Equation_1&#x8BE5;&#x7ED3;&#x8BBA;&#x662F;&#x5B9A;&#x4E49;&#x6B63;&#x5F26;&#x51FD;&#x6570;&#x5BFC;&#x6570;&#x7684;&#x7406;&#x8BBA;&#x57FA;&#x7840;&#xFF0C;&#x5E76;&#x5728;&#x5FAE;&#x79EF;&#x5206;&#x7406;&#x8BBA;&#x4E2D;&#x5177;&#x6709;&#x6DF1;&#x8FDC;&#x5F71;&#x54CD;&#x3002;&#x53E6;&#x4E00;&#x4E2A;&#x91CD;&#x8981;&#x7684;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6781;&#x9650;&#x7C7B;&#x4F3C;&#x5730;&#xFF0C;&#x6781;&#x9650;&#xFF1A;<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19797/19797_PT_Equation_3_editor_19797.svg" alt="19797_PT_Equation_3.svg" /></figure>&#x53EF;&#x901A;&#x8FC7;&#x4EE3;&#x6570;&#x53D8;&#x5F62;&#x4EE5;&#x53CA;&#x5DF2;&#x77E5;&#x7684;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6781;&#x9650;&#x52A0;&#x4EE5;&#x63A8;&#x5BFC;&#x3002;&#x8BE5;&#x7ED3;&#x679C;&#x5728;&#x4F59;&#x5F26;&#x51FD;&#x6570;&#x5BFC;&#x6570;&#x7684;&#x5B9A;&#x4E49;&#x4E2D;&#x53D1;&#x6325;&#x7740;&#x6838;&#x5FC3;&#x4F5C;&#x7528;&#xFF0C;&#x5E76;&#x5728;&#x6570;&#x5B66;&#x7269;&#x7406;&#x4E0E;&#x5DE5;&#x7A0B;&#x5B66;&#x7684;&#x8BF8;&#x591A;&#x9886;&#x57DF;&#x4E2D;&#x5177;&#x6709;&#x91CD;&#x8981;&#x610F;&#x4E49;&#x3002;&#x5728;&#x6570;&#x5B66;&#x4E0E;&#x79D1;&#x5B66;&#x4E2D;&#x7684;&#x5E94;&#x7528;&#x8FD9;&#x4E9B;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6781;&#x9650;&#x7684;&#x7ED3;&#x8BBA;&#x4E0D;&#x4EC5;&#x5C40;&#x9650;&#x4E8E;&#x7EAF;&#x6570;&#x5B66;&#x7814;&#x7A76;&#xFF0C;&#x5728;&#x7269;&#x7406;&#x5B66;&#x4E2D;&#x540C;&#x6837;&#x5177;&#x6709;&#x5E7F;&#x6CDB;&#x5E94;&#x7528;&#xFF0C;&#x5C24;&#x5176;&#x4F53;&#x73B0;&#x5728;&#x6CE2;&#x52A8;&#x3001;&#x632F;&#x8361;&#x7CFB;&#x7EDF;&#x4EE5;&#x53CA;&#x5085;&#x91CC;&#x53F6;&#x5206;&#x6790;&#x7B49;&#x65B9;&#x9762;&#x3002;&#x5DE5;&#x7A0B;&#x5E08;&#x548C;&#x79D1;&#x5B66;&#x5BB6;&#x501F;&#x52A9;&#x8FD9;&#x4E9B;&#x6781;&#x9650;&#x6765;&#x7B80;&#x5316;&#x529B;&#x5B66;&#x3001;&#x4FE1;&#x53F7;&#x5904;&#x7406;&#x4EE5;&#x53CA;&#x7535;&#x78C1;&#x5B66;&#x4E2D;&#x7684;&#x76F8;&#x5173;&#x8868;&#x8FBE;&#x5F0F;&#x3002;&#x8FD9;&#x4E9B;&#x57FA;&#x672C;&#x6027;&#x8D28;&#x5145;&#x5206;&#x4F53;&#x73B0;&#x4E86;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x6781;&#x9650;&#x5728;&#x5404;&#x7C7B;&#x6570;&#x5B66;&#x53CA;&#x5E94;&#x7528;&#x5B66;&#x79D1;&#x4E2D;&#x7684;&#x91CD;&#x8981;&#x5730;&#x4F4D;&#x3002;

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Differentiation Rules

微分法则

KEY TERMS AND CONCEPTS

关键术语与概念

Limits on Trigonometric Functions

trigonometric limits for derivatives

Trigonometric limits are key tools in calculus, especially when derivatives are being defined. One of the most important results is the limit of sine over its angle, and this result is used often in math and physics. It helps connect trigonometry with rates of change.

A common way to understand this limit uses a sector of a unit circle. In radians, the arc length matches the angle, so the circle gives a clear geometric setting for the proof. The angle also fits inside an inequality that is needed to apply the squeeze theorem.

The squeeze theorem is a method for finding a limit by placing one expression between two others that approach the same value. Here, the sine function is bounded between two geometric expressions. That comparison leads directly to the limit result for sine.

A second important trigonometric limit is found with algebraic manipulation and the first limit. It is the limit used with cosine, and it also helps define the derivative of the cosine function. Together, these results are central in calculus.

These limits matter beyond pure mathematics. They are used in physics for wave motion and oscillating systems. Engineers and scientists also use them in Fourier analysis, mechanics, signal processing, and electromagnetism.

Trigonometric Limits for Derivatives

derivatives of simple functions

Slope Rules for Simple Functions

Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

the product rule

Derivative of a Changing Rectangle

The Product Rule helps find the derivative of a product of two changing functions. In calculus, it states that the derivative of one function times another equals the first function times the rate of change of the second, plus the second function times the rate of change of the first. This rule shows how to account for both functions when they vary at the same time. A useful way to see the rule is with a rectangle whose width and height both change over time. If the width and height represent...

the quotient rule

Differentiating Ratios with the Quotient Rule

The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

Trig Derivatives in Ferris Wheel Motion

Trig derivatives can be understood through the motion of a Ferris wheel. As the wheel turns at a constant speed, a rider’s height above the ground changes in a smooth, repeating pattern over time. That vertical motion can be modeled with a sine function, because it matches the rise and fall of circular motion. The derivative of that height tells how fast the rider’s position is changing at each moment. This rate of change is not constant. It is greatest when the rider is moving fastest upward...

limits on trigonometric functions

Trigonometric limits are key tools in calculus, especially when derivatives are being defined. One of the most important results is the limit of sine over its angle, and this result is used often in math and physics. It helps connect trigonometry with rates of change. A common way to understand this limit uses a sector of a unit circle. In radians, the arc length matches the angle, so the circle gives a clear geometric setting for the proof. The angle also fits inside an inequality that is...

the chain rule

Chain Rule in a Gear System

The Chain Rule appears in a gear system with three connected gears. The first gear is represented by x, the second by z, and the third by y. Each gear’s motion depends on the gear before it, so the system creates a linked change from one stage to the next. This setup shows a composite function, which is a function built from another function. In the first stage, the second gear’s rotation depends on the first gear, so the intermediate variable is written as z = z(x). In the second stage, the...

the chain rule: problem solving

Chain Rule in Thermal Expansion

The Chain Rule appears in thermal expansion when a rod’s length changes because its temperature changes over time. In this example, a metal rod is heated, and its length depends on temperature. The temperature itself changes with time in a quadratic way. For linear thermal expansion, the rod’s length L depends on temperature T with a constant rate of change, using the initial length L0 = 2 m and the coefficient of thermal expansion α. The temperature follows a quadratic relationship in time,...

implicit differentiation

Tangent Lines in Circular Motion

Tangent lines in circular motion show how implicit differentiation helps describe an object’s direction of motion. In classical mechanics, motion is often written as a relationship between position and time. A car moving along a straight highway with constant acceleration is a simple example where velocity is an explicit function of time, so basic differentiation works directly. A satellite in circular orbit is different because its path is given by an implicit function. The circle equation...

second derivatives of implicit functions

Slope and Curvature of Elliptical Arches

Elliptical arches connect slope and curvature to the shape of an ellipse. In this geometry, the height y and the horizontal position x are related implicitly, so y cannot be written as an explicit function of x. That is why implicit differentiation is needed to study the arch. The ellipse is centered at the origin and has a semi-major axis a and a semi-minor axis b. Differentiating the ellipse equation with respect to x uses the chain rule. After solving for dy / dx, the first derivative shows...

implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

Implicit differentiation is used to find the slope of a curve when x and y cannot be separated easily. The conchoid of Nicomedes is one example of such a curve. Its equation links x and y in a way that makes solving for one variable difficult, so a special method is needed to study the curve. To work with the conchoid, y is treated as a function of x. The chain rule is then applied to any term that contains y. The derivative is taken on both sides of the equation, which introduces dy/dx terms.<br...

derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

Logarithmic functions and exponential functions are inverses of each other. If y = log b x, it can also be written as b^y = x. That inverse relationship makes implicit differentiation useful for finding derivatives of logarithmic functions. Logarithmic scales are also important for measuring data that cover many orders of magnitude. The Richter scale for earthquake magnitude and decibels for sound intensity are both examples. These scales help compare very small and very large values on the...

logarithmic differentiation

Differentiating Tire Stress with Logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain. The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material...

the number e as a limit

e from Natural Log Derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships. The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That...

application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial Doubling in Exponential Growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly. During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The...

related rates

Balloon Inflation and Changing Rates

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

Tangent Line Estimates Near a Point

Linearization uses a tangent line to estimate a nonlinear function near a chosen reference point. It replaces a difficult function with a simpler linear model when the input changes only a little. This makes it useful for small deviations from a known value. A square root example shows how the method works. The value at 4 is known exactly, so 4 is a good reference point. The function value and the rate of change are both easy to find there. Estimating the value at a nearby input, such as 4.1,...

application of linearization and approximation

application-of-linearization-and-approximation

Estimating Altitude Changes with Linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

inverse-hyperbolic-functions-and-their-derivatives

Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

hyperbolic and inverse hyperbolic functions: problem solving

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

Tangent Lines on a Hyperbolic Arch

Hyperbolic cosine functions can model an arched gate with a smooth, symmetric shape. When the arch is centered at the origin, its highest point is at the center. That symmetry means any lower height appears at two horizontal points on opposite sides of the centerline. The arch is checked at a height of five meters by adjusting the function to match that level. Solving for the horizontal positions shows that there are exactly two points where the gate reaches five meters. These points are equal...

key terms and concepts

Learn how key trigonometric limits are proved with the unit circle and squeeze theorem, and why they matter for derivatives, waves, and physics.

Trigonometric Limits for Derivatives | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Trigonometric Limits for Derivatives

Description

Transcript