Trigonometric identities

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

Solutions relying entirely on calculator technology are not acceptable.

Prove that

Hence solve, for 0 < x < 90, the equation

Give each answer, in degrees, to one decimal place.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Show that the equation

may be written as

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

Hence, solve for 360° ≤ x 540°

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that the first three terms of a geometric series are 12 cos θ 5 + 2 sin θ and 6 tan θ

show that

Given that θ is an obtuse angle measured in radians,

solve the equation in part (a) to find the exact value of θ

show that the sum to infinity of the series can be expressed in the form

where k is a constant to be found.

Solve, for –180° θ 180°, the equation

giving your answers, where necessary, to one decimal place.

[ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Deduce the smallest positive solution to the equation

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

show that

where A is a rational constant to be found.

Prove that

Hence solve, for < x < the equation

Give any non-exact answer to 3 decimal places where appropriate.

Given that

use standard trigonometric identities, to find in terms of p,

cot(– 45)°

Write each answer in its simplest form.

Show that 2 tan x – cot x = 5 cosec x may be written in the form

stating the values of the constants a, b and c.

Hence solve, for 0 ≤ x 2, the equation 2 tan x – cot x = 5 cosec x giving your answers to 3 significant figures.

Show that

stating the value of the constant .

Given that

show that

Given that y = (x² + x³)ln 2x find the exact value of at x = 2, giving your answer in its simplest form.

Given that

show that

where g(x) is an expression to be found.

Show that cosec 2x + cot 2x = cot x, x  90n°, n

Hence, or otherwise, solve, for 0 θ 180°,

You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Differentiate

Show that (sec? 3x) can be written in the form

where yu is a constant. (3)

Use an appropriate double angle formula to show that

and state the value of the constant /. (3)

You must show all your working. Give your answers in terms of z. (6)

Express 4cosec² 2 − cosec² in terms of sin and cos .

Hence show that

Hence or otherwise solve, for 0 < <,

giving your answers in terms of .

Solve

Use the identity cos²θ + sin²θ = 1 to prove that tan²θ = sec² θ – 1.

Solve, for 0 ≤ θ < 360°, the equation

In this question solutions based entirely on graphical or numerical methods are not acceptable.

Solve for 0 x 360°, 4 cos (x + 70°) = 3 giving your answers in degrees to one decimal place.

Find, for 0 θ 2π, all the solutions of

giving your answers in radians to 3 significant figures.

Show that the equation

can be written in the form

Hence solve, for 0 ≤ x 360

giving your answers to 2 decimal places.

Solve, for 0 ≤ θ < 360°, the equation

giving your answers to 1 decimal place. You must show each step of your working.

Solve, for – ≤ x , the equation 2tan x – 3sin x = 0 giving your answers to 2 decimal places where appropriate. [ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Solve, for –180° x < 180°, tan(x – 40 °) = 1.5 giving your answers to 1 decimal place.

Show that the equation

can be written in the form

Hence solve, for 0 < 360°,

showing each stage of your working.

Solve, for 0 180

giving your answers to 1 decimal place.

Find all the values of x, in the interval 0 x 360, for which

giving your answers to 1 decimal place.

You must show clearly how you obtained your answers.

Show that the equation tan 2 x = 5 sin 2 x can be written in the form (1 – 5 cos 2 x) sin 2 x = 0

Hence solve, for 0 x 180°, tan 2 x = 5 sin 2 x giving your answers to 1 decimal place where appropriate. You must show clearly how you obtained your answers.

Solve for 0 x 360°, giving your answers in degrees to 1 decimal place,

Find, for 0 x 2, all the solutions of

giving your answers in radians. You must show clearly how you obtained your answers.

Solve, for –180° 180°,

Solve, for 0 x < 360°, 4sin x = 3tan x.