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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 xaxis, the sine of the angle gives the length of the ycomponent (rise) of the triangle, the cosine gives the length of the xcomponent (run), and the tangent function gives the slope (ycomponent divided by the xcomponent).
With this in mind, we can use the definition of a derivative,
See for graphical validation.
The same procedure can be applied to find other derivatives of trigonometric functions, the most common being
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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: Difference Quotients, Overview of Derivatives, and Financial Leverage
 function
 Appears in these related concepts: Inverse Functions, Average Value of a Function, and Functions and Their Notation
 origin
 Appears in these related concepts: Adding and Subtracting Vectors Graphically, Overview of Different Muscle Functions, and ThreeDimensional Coordinate Systems
 slope
 Appears in these related concepts: Tangent Planes and Linear Approximations, Applications of Linear Functions and Slope, and SlopeIntercept Equations
 tangent
 Appears in these related concepts: Newton's Method, Graphs of Exponential Functions, Base e, and Circular Motion
 trigonometric
 Appears in these related concepts: Hyperbolic Functions, Trigonometric Integrals, and Trigonometric Substitution
 trigonometric function
 Appears in these related concepts: Inverse Trigonometric Functions: Differentiation and Integration, Trigonometric Limits, and Further Transcendental Functions
Sources
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Cite This Source
Source: Boundless. “Derivatives of Trigonometric Functions.” Boundless Calculus. Boundless, 21 Jul. 2015. Retrieved 11 Feb. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/derivatives9/derivativesoftrigonometricfunctions462931/