Small angle approximations

In this question you must show detailed reasoning.

Given that x is small and in radians, use the small angle approximation for cosθ to show that

Given that x is small and in radians, use

• the answer to part (a)

• the small angle approximations for sin θ and tan θ

to show that

where a and b are constants to be found.

Hence, given that x is very small, deduce an approximate value for

giving a reason for your answer.

Given that θ is small and in radians, use the small angle approximations to find an approximate numerical value of

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation y = f(x) where x

Given that

• the curve has a stationary point with x coordinate α

• α is small

use the small angle approximation for cosx to estimate the value of α to 3 decimal places.

The point P(0, 3) lies on C

Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

Given that θ is small and measured in radians, use the small angle approximations to show that

where a, b and c are integers to be found.

Given that θ is small and measured in radians, use the small angle approximations to show that

where a, b and c are integers to be found.

Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.

Use Diagram 1 to show why the equation

has only one real root, giving a reason for your answer.

Given that the root of the equation is , and that is small,

use the small angle approximation for cos x to estimate the value of to 3 decimal places.

The curve C, in the standard Cartesian plane, is defined by the equation

The curve C passes through the origin O

Find the value of at the origin.

Use the small angle approximation for sin 2 y to find an equation linking x and y for points close to the origin.

Explain the relationship between the answers to (a) and (b)(i).

Given that θ is small and is measured in radians, use the small angle approximations to find an approximate value of