Trigonometric equations

Express 3 cos 2 x + 4 sin 2 x in the form R cos (2 x - α), where R > 0 and 0 ° < α < 90 °.

In this question you must show detailed reasoning.

Show that the equation 2 cot²x - 9 cosec x - 3 = 0 can be expressed in the form

In this question you must show detailed reasoning.

Hence solve, for 0 < θ < π,

Give your answers correct to 3 decimal places.

The small angle approximation for sin2 θ is used to find an approximation for the smallest positive solution of the equation 2 cot²2θ - 9 cosec 2θ - 3 = 0.

Show that this approximate solution is accurate to 2 decimal places.

In this question you must show detailed reasoning.

Solve the following equations.

f ′ (x) = 0

f ′′ (x) = 0

The diagram shows the graph of the gradient function y = f ′ (x) for the domain 0 ⩽ x ⩽ 2 π.

Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points A, B, C and D.

Explain how to use the graph of the gradient function to find the values of x for which

f(x) is increasing.

Using set notation, write down the set of values of x for which f(x) is increasing in the

domain 0 ⩽ x ⩽ 2 π.

Write 2x² + 6x + 7 in the form p(x + q)² + r, where p, q and r are constants.

State the coordinates of the minimum point on the graph of y = 2x² + 6x + 7.

Hence deduce

• the minimum value of 2 tan²θ + 6 tan θ + 7,

• the smallest positive value of θ, in degrees, for which the minimum value occurs.

In this question you must show detailed reasoning.

Show that the equation m sec θ + 3 cos θ = 4 sin θ can be expressed in the form

m tan²θ - 4 tan θ + (m + 3) = 0.

It is given that there is only one value of θ, for 0 < θ < π, satisfying the equation m sec θ + 3 cos θ = 4 sin θ.

Given also that m is a negative integer, find this value of θ, correct to 3 significant figures.

Functions f and g are defined for 0 ⩽ x ⩽ 2 π by f (x) = 2 tan x and g (x) = sec x.

State the range of f.

State the range of g.

Show that fg (0.6) = 5.33, correct to 3 significant figures.

Explain why f^-1g(0.6) is not defined.

The size, P, of a population of a certain species of insect at time t months is modelled by the following formula.

P = 5000 - 1000cos (30 t)°

Write down the maximum size of the population.

Write down the difference between the largest and smallest values of P.

Without giving any numerical values, describe briefly the behaviour of the population over

time.

Find the time taken for the population to return to its initial size for the first time.

Determine the time on the second occasion when P = 4500.

A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and P converges to 5000.

Suggest a change to the model that will take account of this observation.

Express v in the form R cos(t - α), where R > 0 and 0 < α < . Give the value of α correct

to 4 significant figures.

Hence find the two smallest positive values of t for which P is moving, in either direction,

with a speed of 3 m s^-1.

In this question you must show detailed reasoning.

Solve the equation 3sin⁴φ + sin²φ = 4, for 0 ⩽ φ < 2π, where φ is measured in radians.

In this question you must show detailed reasoning.

Hence solve, for 0 < θ < 2π, 3(cot θ + cosec θ)² = 2 sec θ.