Trigonometric equations

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

Show that

Hence solve, for 0 θ 180

giving your answers to one decimal place, where appropriate.

Given that for all values of k, where 0 < k < 1, the equation

has exactly 6 solutions in the interval 0 ≤ x < 2π

deduce the value of n

deduce the number of solutions of the equation

in the interval 0 ≤ x < 5π, justifying your answer.

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 should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Given that 1 + cos 2θ + sin 2θ ≠ 0 prove that

Hence solve, for 0 < x < 180°

giving your answers to one decimal place where appropriate.

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

Solutions relying entirely on calculator technology are not acceptable.

Given that 1 + cos 2θ + sin 2θ ≠ 0 prove that

Hence solve, for 0 < x < 180°

giving your answers to one decimal place where appropriate.

Express 2cosθ – sinθ in the form Rcos (θ + α), where R > 0 and 0 α Give the exact value of R and the value of α in radians to 3 decimal places.

Figure 6 shows the cross-section of a water wheel.

The wheel is free to rotate about a fixed axis through the point C.

The point P is at the end of one of the paddles of the wheel, as shown in Figure 6.

The water level is assumed to be horizontal and of constant height.

The vertical height, H metres, of P above the water level is modelled by the equation

H = 3 + 4 cos (0.5 t) – 2 sin (0.5 t)

where t is the time in seconds after the wheel starts rotating.

Using the model, find

the maximum height of P above the water level,

the value of t when this maximum height first occurs, giving your answer to one decimal place.

In a single revolution of the wheel, P is below the water level for a total of T seconds.

According to the model,

find the value of T giving your answer to 3 significant figures.

(Solutions based entirely on calculator technology are not acceptable.)

In reality, the water level may not be of constant height.

Explain how the equation of the model should be refined to take this into account.

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

Show that cosecθ − sinθ ≡ cosθcotθ θ ≠ (180n)° n

Hence, or otherwise, solve for 0 < x < 180° cosec x − sin x = cos x cot (3 x − 50°)

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

Show that

Hence solve, for −90° x ≤ 180°, the equation

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

f(x) = 10e^–0.25x sin x, x 0

Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4

Figure 3 shows a sketch of part of the curve with equation y = f(x).

Sketch the graph of H against t where

showing the long-term behaviour of this curve.

The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find

the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

Prove

Hence solve, for 90° θ 180°, the equation

giving any solutions to one decimal place.

The depth of water, D metres, in a harbour on a particular day is modelled by the formula

where t is the number of hours after midnight.

A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.

Find the depth of the water in the harbour when the boat enters the harbour.

Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Solve, for 0≤ θ <360°, the equation

giving your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

Prove that

Hence solve, for < x < the equation

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

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 – ≤ ,

giving your answers in terms of .

Solve, for 0 x 360,

giving your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)

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.

Solve, for 0 x 180°,

giving your answers to 1 decimal place. You should show each step in your working.

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.

Solve, for 0 ≤ x < 360°,