Topics
11. Differentiation
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iGCSE (2021 Edition)

11.17 Differentiating trigonometric functions

Lesson
Worksheet
Practice
Worksheet
Sine and cosine functions
1

Differentiate the following functions:

a
y = \cos 3 x
b
y = \sin 7 x
c
y = 2 - 3 \cos x
d
y = \sin x + 3 \cos x
e
y = - 3 \sin x
f
y = \sin \left(x^{6} + 5\right)
g
y = \sin \left( 4 x^{5} + 3\right)
h
y = 3 x + \sin 5 x
i
y = \sin \left(x^{4} + 3\right)
j
y = \cos \left( 3 x^{2}\right)
k
y = \sin \left(\dfrac{x}{3}\right) - 2 \cos x
l
y = \sin x^{5}
m
y = 4 \sin \left(\dfrac{t}{4}\right) + 3 \cos 4 t + t^{4}
2

Differentiate the following functions:

a
y = 4 x \cos x
b
y = \sin x \cos x
c
y = \sin ^{6}\left( 3 x\right)
d
y = \sin 4 x \left(2 + \cos x\right)
e
y = \sqrt{\cos 4 x}
f
y = \cos ^{2}\left(x + \dfrac{\pi}{2}\right)
g
y = \sin ^{2}\left( 5 x\right)
h
y = \dfrac{\sin x}{x - 8}
i
y = \left(x^{4} + 3\right) \cos 5 x
j
y = \cos 4 x \sin 6 x
k
y = x^{\frac{7}{4}} \cos 2 x
l
y = 3 \cos 4 x + 6 \sin 5 x
m
y = \dfrac{3 x}{\sin x}
n
y = x^{4} \sin \left(\dfrac{x}{2}\right)
o
y = x^{3} \cos x
p
y = 4 \sin 5 x \cos 5 x
Tangent function
3

Differentiate the following functions:

a
y = \tan \left(\dfrac{x}{4}\right)
b
y = \tan \left( 5 x + 7\right)
c
y = \tan \left( - 5 x\right)
d
y = x^{2} + \tan \left(\dfrac{x}{7}\right)
e
y= \tan(x\degree)
f
y = \tan \left(x^{5}\right)
g
y = \tan \left(x^{2} - 3 x + 7\right)
h
y = \tan \left(\dfrac{\pi x}{4}\right)
4

Differentiate the following functions:

a
y = \tan ^{2}\left( 4 x\right)
b
y = \left(\tan x - 5\right)^{5}
c
y = \tan \left(\sin x\right)
d
y = \dfrac{\tan x}{x^{2}}
e
y = \tan x \sin x
f
y = \cot x
g
y = \tan 4 x + 2 x \cos x
h
y = \sqrt{\tan 6 x}
i
y = \cos 6 x \tan 5 x
j
y = \left(3 + \tan 3 x\right)^{5}
k
y = 5 x^{6} \tan x
l
y = 4 x \tan x
Applications
5

Differentiate the function x y = \sin 5 x.

6

Consider \tan x = \dfrac{\sin x}{\cos x} and differentiate y = \tan x using the quotient rule.

7

Differentiate y = \tan(5x\degree) where x is an angle in degrees.

8

Explain why the gradient of the function y = \sin ^{2}\left( 5 x\right) + \cos ^{2}\left( 5 x\right) equal to 0 for all x.

9

If y = 4 \cos 3 x, prove that y'' + 9 y = 0.

10

Consider the graph of y = - \sin x which is the gradient function of y = \cos x. A number of points have been labelled on the graph.

Name the point on the gradient function that corresponds to the following locations on the graph of y = \cos x :

a

Where y = \cos x is increasing most rapidly.

b

Where y = \cos x is decreasing most rapidly.

c

Where y = \cos x is stationary.

\frac{1}{2}π
1π
\frac{3}{2}π
2π
x
-1
1
y
11

Consider the function y = \text{cosec } x.

a

Find the derivative of this function in terms of \sin x and \cos x.

b

Show that the derivative is equal to - \cot {x } \times \text{cosec }x.

c

State the values of x for which \dfrac{d y}{d x} is not defined, on the interval 0 \leq x \leq 2 \pi.

12

Find the derivative of y = \tan \left(\dfrac{x + 2}{x - 2}\right) using the substitution u = \dfrac{x + 2}{x - 2}.

13

There is an expansion system in mathematics that allows a function to be written in terms of powers of x. The value of \sin x and \cos x, for any value of x, can be given by the expansions below:

\sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dfrac{x^{7}}{7!} + \dfrac{x^{9}}{9!} - \ldots

\cos x = 1 - \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} - \dfrac{x^{6}}{6!} + \dfrac{x^{8}}{8!} - \ldots

Use the expansions to find:

a

\dfrac{d}{dx} \left(\sin x\right) in terms of x.

b

\dfrac{d}{dx} \left(\cos x\right) in terms of x.

c

Hence express the derivatives of \sin x and \cos x in simplest form.

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Outcomes

0606C14.3B

Use the derivatives of the standard functions sinx, cos x, tan x, together with constant multiples, sums and composite functions.

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