t-formulae

Use the substitution t tan to show that

Hence show that

where A and B are constants to be determined and k is an arbitrary constant.

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

Solutions relying on calculator technology are not acceptable.

Use the substitutiont = tan to show that

where a, b and c are constants to be determined.

Hence show that

Show that the substitution t tan transforms the integral

into the integral

Hence determine

During 2029, the number of hours of daylight per day in London, H, is modelled by the equation

where x is the number of days after 1st January 2029 and the angle is in radians.

Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.

Use the substitution t tan to show that H can be written as

where a, b and c are constants to be determined.

Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.

Use the substitution t = tan to prove the identity

Use the substitution t = tan to determine the exact value of

giving your answer in simplest form.

show that f (x) can be written in the form

Hence solve, for 0 x 2π, the equation

giving your answers to 2 decimal places where appropriate.

Use the result of part (a) to show that

where K is a constant to be determined.

where k is an arbitrary constant.