# Derivatives of Trigonometric Functions

## Derivatives of trigonometric functions can be found using the standard derivative formula.

#### 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.

#### Terms

• a straight line that intersects a curve at two or more points

#### Figures

1. ##### Interactive Graph: Sine and Cosine Functions

The derivative of the sine function is the cosine function , and the derivative of the cosine function is negative the sine function .

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 x-axis, the sine of the angle gives the length of the y-component (rise) of the triangle, the cosine gives the length of the x-component (run), and the tangent function gives the slope (y-component divided by the x-component).

With this in mind, we can use the definition of a derivative,

$f'(x) = \lim_{h\rightarrow 0} \frac {f(x+h) - f(x)}{h}$ to calculate the derivatives of different trigonometric functions. For example, if

$f(x) = sin \: x$, then

$f'(x) = \lim_{h\rightarrow 0} \frac {sin \: (x+h) - sin \: (x)}{h}$

$= \lim_{h\rightarrow 0} \frac{cos(x)sin(h) + cos(h)sin(x) - sin(x)}{h}$

$= \lim_{h\rightarrow 0} \frac{cos(x)sin(h) + (cos(h) - 1)sin(x)}{h}$

$= \lim_{h\rightarrow 0} \frac{cos(x)sin(h)}{h} + \lim_{h\rightarrow 0} \frac {(cos(h) - 1)sin(x)}{h}$

$= cos (x) (1) + sin (x) (0)$

$= cos (x)$.

See Figure 1 for graphical validation.

The same procedure can be applied to find other derivatives of trigonometric functions, the most common being

$\frac{\mathrm{d} }{\mathrm{d} x}cos \: x = -sin \: x$

$\frac{\mathrm{d} }{\mathrm{d} x}tan \: x = sec^2 \: x$.

#### Key Term Glossary

axis
a fixed, one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis)
##### Appears in these related concepts:
derivative
a measure of how a function changes as its input changes
##### Appears in these related concepts:
function
a relation in which each element of the domain is associated with exactly one element of the co-domain
##### Appears in these related concepts:
length
distance between the two ends of a line segment
##### Appears in these related concepts:
origin
the point at which the axes of a coordinate system intersect
##### Appears in these related concepts:
slope
also called gradient; slope or gradient of a line describes its steepness
##### Appears in these related concepts:
tangent
a straight line touching a curve at a single point without crossing it there
##### Appears in these related concepts:
trigonometric
relating to the functions used in trigonometry: sin, cos, tan, csc, cot, sec
##### Appears in these related concepts:
trigonometric function
any function of an angle expressed as the ratio of two of the sides of a right triangle that has that angle, or various other functions that subtract 1 from this value or subtract this value from 1 (such as the versed sine)