Small angle approximations

Show that, for small values of x, the graph with equation

can be approximated by the straight line with equation of the form

y = ax + 1

where a is a constant to be found.

Show that, for small values of θ measured in radians

where A, B and C are constants to be found.

Use your answer to part (a) to find an approximation for

cos 0.28 + 2 sin 0.21 – tan 0.14

Give your answer to three decimal places.

Given that is a small angle, find an approximation for cos 2

Circle your answer.

Find the first two terms, in ascending powers of x, of the binomial expansion of

Hence, for small values of x, show that

where A, B and C are constants to be found.

Show that

for small values of x.

Hence, show that the area between the graph with equation

the positive x-axis and the line x 025 can be approximated by

where m and n are integers to be found.

Explain why

is not a suitable approximation for

Explain how

may be approximated by

for suitable values of a and b.

Using small angle approximations, show that for small, non-zero, values of x

where A is a constant to be determined.

Show that the first two terms of the binomial expansion of 4 2x² are

State the range of values of x for which the expansion found in part (a) is valid.

Hence, find an approximation for

giving your answer to five decimal places.

Fully justify your answer.

A student decides to use this method to find an approximation for

Explain why this may not be a suitable method.

Turn over for Section B

Section B

all questions in the spaces provided.

Show that, for small values of x, the graph of y 5 4 sin 12 tan can be approximated by a straight line.