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Derivatives of Trigonometric Functions
Derivatives of trigonometric functions can be found using the standard derivative formula.
Learning Objective

Identify the derivatives of the most common trigonometric functions
Key Points
 The derivative of the sine function is the cosine function.
 The derivative of the cosine function is the negative of the sine function.
 The derivative of the tangent function is the squared secant function.
Term

secant
a straight line that intersects a curve at two or more points
Full Text
The trigonometric functions (also called the circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the
With this in mind, we can use the definition of a derivative to calculate the derivatives of different trigonometric functions:
For example, if
Sine and Cosine
In this image, one can see that where the line tangent to one curve has zero slope (the derivative of that curve is zero), the value of the other function is zero.
The same procedure can be applied to find other derivatives of trigonometric functions. The most common are the following:
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Key Term Reference
 definition
 Appears in these related concepts: Defining an Informative Speech, Infinite Limits, and Types of Informative Speeches
 derivative
 Appears in these related concepts: Integration By Parts, Separable Equations, and Overview of Derivatives
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Types of Muscle Tissue, and ThreeDimensional Coordinate Systems
 slope
 Appears in these related concepts: Equations of Lines and Planes, Tangent Planes and Linear Approximations, and Slope and Intercept
 tangent
 Appears in these related concepts: Derivatives of Exponential Functions, Direction Fields and Euler's Method, and Tangent Vectors and Normal Vectors
 trigonometric
 Appears in these related concepts: Trigonometric Integrals, Trigonometric Substitution, and Double Integrals in Polar Coordinates
 trigonometric function
 Appears in these related concepts: Inverse Trigonometric Functions: Differentiation and Integration, Trigonometric Limits, and Further Transcendental Functions
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Cite This Source
Source: Boundless. “Derivatives of Trigonometric Functions.” Boundless Calculus. Boundless, 22 Mar. 2016. Retrieved 30 Apr. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/derivatives9/derivativesoftrigonometricfunctions462931/