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2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
We recall as well that there are four other trigonometric functions, each defined in
terms of the sine and/or cosine functions. These six trigonometric functions together offer
us a wide range of flexibility in problems involving right triangles. The tangent function is
defined by tan(θ) =
sin(θ)
cos(θ) , while the cotangent function is its reciprocal: cot(θ) =
cos(θ)
sin(θ) .
The secant function is the reciprocal of the cosine function, sec(θ) =
1
cos(θ) , and the
cosecant function is the reciprocal of the sine function, csc(θ) =
1
sin(θ) .
Because we know the derivatives of the sine and cosine function, and the other four
trigonometric functions are defined in terms of these familiar functions, we can now
develop shortcut differentiation rules for the tangent, cotangent, secant, and cosecant
functions. In this section’s preview activity, we work through the steps to find the derivative
of y = tan(x).
Preview Activity 2.4. Consider the function f (x) = tan(x), and remember that
tan(x) =
sin(x)
cos(x)
.
(a) What is the domain of f ?
(b) Use the quotient rule to show that one expression for f ′ (x) is
f
′ (x) =
cos(x) cos(x) + sin(x) sin(x)
cos 2 (x)
.
(c) What is the Fundamental Trigonometric Identity? How can this identity be used
to find a simpler form for f ′ (x)?
(d) Recall that sec(x) =
1
cos(x) . How can we express f ′ (x) in terms of the secant
function?
(e) For what values of x is f ′ (x) defined? How does this set compare to the domain
of f ?
⊲⊳
Derivatives of the cotangent, secant, and cosecant functions
In Preview Activity 2.4, we found that the derivative of the tangent function can be
expressed in several ways, but most simply in terms of the secant function. Next, we
develop the derivative of the cotangent function.
Let g(x) = cot(x). To find g ′ (x), we observe that g(x) =
cos(x)
sin(x) and apply the quotient
2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
We recall as well that there are four other trigonometric functions, each defined in
terms of the sine and/or cosine functions. These six trigonometric functions together offer
us a wide range of flexibility in problems involving right triangles. The tangent function is
defined by tan(θ) =
sin(θ)
cos(θ) , while the cotangent function is its reciprocal: cot(θ) =
cos(θ)
sin(θ) .
The secant function is the reciprocal of the cosine function, sec(θ) =
1
cos(θ) , and the
cosecant function is the reciprocal of the sine function, csc(θ) =
1
sin(θ) .
Because we know the derivatives of the sine and cosine function, and the other four
trigonometric functions are defined in terms of these familiar functions, we can now
develop shortcut differentiation rules for the tangent, cotangent, secant, and cosecant
functions. In this section’s preview activity, we work through the steps to find the derivative
of y = tan(x).
Preview Activity 2.4. Consider the function f (x) = tan(x), and remember that
tan(x) =
sin(x)
cos(x)
.
(a) What is the domain of f ?
(b) Use the quotient rule to show that one expression for f ′ (x) is
f
′ (x) =
cos(x) cos(x) + sin(x) sin(x)
cos 2 (x)
.
(c) What is the Fundamental Trigonometric Identity? How can this identity be used
to find a simpler form for f ′ (x)?
(d) Recall that sec(x) =
1
cos(x) . How can we express f ′ (x) in terms of the secant
function?
(e) For what values of x is f ′ (x) defined? How does this set compare to the domain
of f ?
⊲⊳
Derivatives of the cotangent, secant, and cosecant functions
In Preview Activity 2.4, we found that the derivative of the tangent function can be
expressed in several ways, but most simply in terms of the secant function. Next, we
develop the derivative of the cotangent function.
Let g(x) = cot(x). To find g ′ (x), we observe that g(x) =
cos(x)
sin(x) and apply the quotient