Glossary

4.2 Derivatives of Trigonometric Functions

3 min read•august 7, 2024

Trigonometric functions are essential in calculus, and knowing their derivatives is crucial. This section covers the derivatives of basic trig functions like sine, cosine, and tangent, as well as their reciprocals.

Understanding these derivatives allows us to solve more complex problems involving trigonometric functions. We'll also learn how to use trig identities to simplify expressions and verify our work when differentiating.

Derivatives of Basic Trigonometric Functions

Sine and Cosine Function Derivatives

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  • The derivative of the sine function sin⁡(x)\sin(x) is cos⁡(x)\cos(x)
    • Intuitively, the slope of the sine function at any point is equal to the value of the cosine function at that point
    • Example: ddx(sin⁡(x))=cos⁡(x)\frac{d}{dx}(\sin(x)) = \cos(x)
  • The derivative of the cosine function cos⁡(x)\cos(x) is −sin⁡(x)-\sin(x)
    • The slope of the cosine function at any point is the negative value of the sine function at that point
    • Example: ddx(cos⁡(x))=−sin⁡(x)\frac{d}{dx}(\cos(x)) = -\sin(x)
  • These derivatives can be derived using the limit definition of the derivative and trigonometric identities
    • For sin⁡(x)\sin(x): lim⁡h→0sin⁡(x+h)−sin⁡(x)h=cos⁡(x)\lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} = \cos(x)
    • For cos⁡(x)\cos(x): lim⁡h→0cos⁡(x+h)−cos⁡(x)h=−sin⁡(x)\lim_{h \to 0} \frac{\cos(x+h) - \cos(x)}{h} = -\sin(x)

Tangent Function Derivative

  • The derivative of the tangent function tan⁡(x)\tan(x) is sec⁡2(x)\sec^2(x)
    • This can be derived using the quotient rule, since tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}
    • ddx(tan⁡(x))=cos⁡2(x)+sin⁡2(x)cos⁡2(x)=sec⁡2(x)\frac{d}{dx}(\tan(x)) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \sec^2(x)
  • The can also be expressed as 1+tan⁡2(x)1 + \tan^2(x) using the sec⁡2(x)=1+tan⁡2(x)\sec^2(x) = 1 + \tan^2(x)
    • Example: ddx(tan⁡(x))=1+tan⁡2(x)\frac{d}{dx}(\tan(x)) = 1 + \tan^2(x)

Derivatives of Reciprocal Trigonometric Functions

Cotangent, Secant, and Cosecant Function Derivatives

  • The derivative of the cotangent function cot⁡(x)\cot(x) is −csc⁡2(x)-\csc^2(x)
    • Derived using the quotient rule, since cot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)}
    • Example: ddx(cot⁡(x))=−csc⁡2(x)\frac{d}{dx}(\cot(x)) = -\csc^2(x)
  • The derivative of the secant function sec⁡(x)\sec(x) is sec⁡(x)tan⁡(x)\sec(x)\tan(x)
    • Derived using the quotient rule, since sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}
    • Example: ddx(sec⁡(x))=sec⁡(x)tan⁡(x)\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)
  • The derivative of the cosecant function csc⁡(x)\csc(x) is −csc⁡(x)cot⁡(x)-\csc(x)\cot(x)
    • Derived using the quotient rule, since csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}
    • Example: ddx(csc⁡(x))=−csc⁡(x)cot⁡(x)\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)

Alternative Expressions using Pythagorean Identities

  • The derivatives of the reciprocal trigonometric functions can also be expressed using Pythagorean identities
    • ddx(cot⁡(x))=−csc⁡2(x)=−(1+cot⁡2(x))\frac{d}{dx}(\cot(x)) = -\csc^2(x) = -\left(1 + \cot^2(x)\right)
    • ddx(sec⁡(x))=sec⁡(x)tan⁡(x)=sec⁡(x)sec⁡2(x)−1\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x) = \sec(x)\sqrt{\sec^2(x) - 1}
    • ddx(csc⁡(x))=−csc⁡(x)cot⁡(x)=−csc⁡(x)csc⁡2(x)−1\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x) = -\csc(x)\sqrt{\csc^2(x) - 1}

Trigonometric Identities in Differentiation

Applying Trigonometric Identities

  • Trigonometric identities can be used to simplify expressions before differentiation or to simplify the resulting derivative
    • Example: ddx(sin⁡2(x))=2sin⁡(x)cos⁡(x)=sin⁡(2x)\frac{d}{dx}(\sin^2(x)) = 2\sin(x)\cos(x) = \sin(2x) using the double angle formula
    • Example: ddx(cos⁡2(x))=−2sin⁡(x)cos⁡(x)=−sin⁡(2x)\frac{d}{dx}(\cos^2(x)) = -2\sin(x)\cos(x) = -\sin(2x) using the double angle formula
  • Common trigonometric identities used in differentiation include:
    • Pythagorean identities: sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1, 1+tan⁡2(x)=sec⁡2(x)1 + \tan^2(x) = \sec^2(x), 1+cot⁡2(x)=csc⁡2(x)1 + \cot^2(x) = \csc^2(x)
    • : sin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x), cos⁡(2x)=cos⁡2(x)−sin⁡2(x)\cos(2x) = \cos^2(x) - \sin^2(x)
    • Half angle formulas: sin⁡2(x)=1−cos⁡(2x)2\sin^2(x) = \frac{1 - \cos(2x)}{2}, cos⁡2(x)=1+cos⁡(2x)2\cos^2(x) = \frac{1 + \cos(2x)}{2}

Verifying Derivatives using Identities

  • Trigonometric identities can be used to verify the correctness of derivatives
    • Example: Verifying ddx(tan⁡(x))=sec⁡2(x)\frac{d}{dx}(\tan(x)) = \sec^2(x) using the quotient rule and Pythagorean identity:
      • ddx(tan⁡(x))=cos⁡2(x)+sin⁡2(x)cos⁡2(x)=1cos⁡2(x)=sec⁡2(x)\frac{d}{dx}(\tan(x)) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} = \sec^2(x)
    • Example: Verifying ddx(sec⁡(x))=sec⁡(x)tan⁡(x)\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x) using the quotient rule and Pythagorean identity:
      • ddx(sec⁡(x))=sin⁡(x)cos⁡2(x)=1cos⁡(x)⋅sin⁡(x)cos⁡(x)=sec⁡(x)tan⁡(x)\frac{d}{dx}(\sec(x)) = \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x)

Key Terms to Review (21)

Chain Rule: The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative of that composite function can be found by multiplying the derivative of the outer function by the derivative of the inner function evaluated at the inner function. This concept is essential when dealing with differentiability and continuity, as well as in applying basic differentiation rules to more complex scenarios.
Continuity: Continuity refers to the property of a function that ensures it does not have any abrupt changes, jumps, or holes in its graph. A function is continuous if, intuitively, you can draw its graph without lifting your pencil from the paper. This concept is crucial in understanding limits, differentiability, and the behavior of functions across different scenarios.
D/dx: The notation 'd/dx' represents the derivative of a function with respect to the variable x, indicating how a function changes as its input x changes. This fundamental concept is crucial in calculus for understanding rates of change and slopes of curves. It connects directly to various rules and techniques for finding derivatives, which are essential for analyzing functions and solving real-world problems involving motion, growth, and optimization.
Derivative at 0 degrees: The derivative at 0 degrees refers to the slope of the tangent line to a trigonometric function, such as sine or cosine, at an angle of 0 degrees. This concept is crucial for understanding how trigonometric functions change at specific points, particularly at their peaks and troughs. It allows us to analyze the behavior of these functions and how they relate to rates of change in various applications.
Derivative at 90 degrees: The derivative at 90 degrees refers to the concept of finding a slope of a tangent line that is perpendicular to a given curve at a specific point. In the context of trigonometric functions, this concept is crucial for understanding how these functions behave and how their rates of change relate to angles and direction, particularly in applications involving circles and oscillations.
Derivative of cosecant: The derivative of cosecant refers to the rate of change of the cosecant function, which is defined as the reciprocal of the sine function. Specifically, the derivative of cosecant can be expressed mathematically as $$\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)$$. This function is crucial when dealing with rates of change in trigonometric applications, especially in calculus where understanding the behavior of trigonometric functions and their derivatives is key to solving various problems.
Derivative of cosine: The derivative of cosine is a fundamental concept in calculus, representing the rate at which the cosine function changes with respect to its input. Specifically, it is given by the formula $$\frac{d}{dx}(\cos(x)) = -\sin(x)$$, illustrating that the slope of the cosine curve at any point is equal to the negative sine of that angle. This relationship is essential for understanding the behavior of trigonometric functions and their applications in various fields such as physics and engineering.
Derivative of cotangent: The derivative of cotangent is the rate of change of the cotangent function with respect to its variable, typically represented as \(\frac{d}{dx} \cot(x)\). It is a crucial concept in calculus, particularly when dealing with trigonometric functions and their applications in various mathematical scenarios, like solving equations and optimization problems. Understanding this derivative also lays the groundwork for more complex calculus operations involving trigonometric identities and integrals.
Derivative of Secant: The derivative of secant refers to the rate at which the secant function changes with respect to its input variable. In calculus, understanding how to find the derivative of secant is crucial as it relates to the broader category of derivatives of trigonometric functions. The secant function, represented as $$ ext{sec}(x)$$, is defined as the reciprocal of cosine, and its derivative reveals important characteristics about the behavior and properties of trigonometric functions in calculus.
Derivative of sine: The derivative of sine refers to the rate of change of the sine function with respect to its angle in radians. This fundamental concept in calculus reveals how the sine function behaves and changes as its input varies, which is crucial for understanding the properties of trigonometric functions and their applications in various fields such as physics and engineering.
Derivative of Tangent: The derivative of tangent refers to the rate at which the tangent function changes with respect to its input variable, usually denoted as $$\frac{d}{dx} \tan(x)$$. It is a fundamental concept in calculus, particularly in understanding how trigonometric functions behave and interact with other mathematical expressions. This derivative plays a crucial role in applications such as optimization, motion analysis, and in understanding periodic functions.
Double angle formulas: Double angle formulas are trigonometric identities that express the sine, cosine, and tangent of an angle that is twice as large as a given angle. These formulas are essential for simplifying expressions and solving equations in trigonometry, especially when dealing with derivatives of trigonometric functions. They provide a way to relate double angles back to the original angles, allowing for easier manipulation and calculation in various mathematical contexts.
F'(x): The notation f'(x) represents the derivative of the function f(x) with respect to the variable x, indicating the rate at which the function's value changes as x varies. This concept is central to understanding how functions behave and provides insight into their continuity, differentiability, and the various rules for computing derivatives.
L'hôpital's rule: L'Hôpital's rule is a mathematical theorem used to evaluate limits that yield indeterminate forms, particularly when both the numerator and denominator approach zero or infinity. This rule states that if the limit of a function is in the form of '0/0' or '∞/∞', you can take the derivative of the numerator and the derivative of the denominator separately, then find the limit again. This process can simplify complex limit calculations, especially in calculus involving trigonometric, exponential, and logarithmic functions.
Limit as x approaches 0 of sin(x)/x: The limit as x approaches 0 of sin(x)/x is a fundamental concept in calculus, denoting that as the value of x gets closer to 0, the value of the expression sin(x)/x approaches 1. This limit is crucial for understanding the behavior of trigonometric functions near zero and serves as a basis for deriving the derivatives of trigonometric functions.
Limit of cos(x) as x approaches 0: The limit of cos(x) as x approaches 0 is the value that the function cos(x) approaches as the variable x gets infinitely close to 0. In this case, the limit evaluates to cos(0), which equals 1. This concept is fundamental in understanding how trigonometric functions behave and is crucial for finding derivatives of such functions.
Maxima and Minima in Wave Functions: Maxima and minima in wave functions refer to the highest and lowest points in the graph of a wave function, where the amplitude reaches its peak or drops to its lowest value. These points are crucial for understanding the behavior of waves, such as sound or light, and can help identify the frequency and wavelength characteristics of the wave. Analyzing these points involves using derivatives, particularly when examining trigonometric functions that describe wave behaviors.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative of the function is equal to the average rate of change over that interval. This theorem connects the concepts of continuity and differentiability, revealing crucial insights about the behavior of functions.
Periodicity: Periodicity refers to the characteristic of a function where it repeats its values at regular intervals. This concept is particularly important in understanding the behavior of functions that exhibit cyclical patterns, allowing for predictions and calculations based on these repeated behaviors. Recognizing periodicity is essential for analyzing trigonometric functions, their derivatives, and their representations in polar coordinates.
Pythagorean Identity: The Pythagorean Identity is a fundamental relationship in trigonometry that connects the squares of the sine and cosine functions, expressed as $$ ext{sin}^2(x) + ext{cos}^2(x) = 1$$. This identity is essential because it illustrates the interdependence of these trigonometric functions, allowing for transformations and simplifications in various mathematical contexts. It serves as a cornerstone for deriving other trigonometric identities and plays a crucial role in calculus, particularly when taking derivatives and evaluating integrals involving trigonometric functions.
Rate of change in oscillations: The rate of change in oscillations refers to the speed at which a function representing periodic motion, such as trigonometric functions, varies over time. This concept is closely tied to derivatives, as they measure how a quantity changes with respect to another variable. Understanding this rate of change is essential for analyzing the behavior of oscillating systems, particularly in physics and engineering, where sinusoidal functions model waves and other repetitive motions.
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Glossary
Glossary
4.3 Derivatives of Exponential and Logarithmic Functions