Further Trigonometric Equations (CIE A Level Maths: Pure 3)

Solve the equation cosec  for .

Given that

tan

find the values of such that  .

Solve the equation

Use the identity

sincos sin sin cos 

to show that

sin  cos

can be written as

sin

Hence, or otherwise, solve the equation sin cos for  .
Give your answers to three significant figures.

Write down the maximum value of sin cos  and state the first positive value of for which it occurs.  Give your value of   to three significant figures.

Show that the equation  tan² sec   can be written as

sec² sec

Hence, or otherwise, solve the equation

tan²  sec 

Give your answers to three significant figures.

Solve the equation

cos cos

State your answers as multiples of .

Sketch the graph of for  .

By adding three lines to your graph demonstrate how the equation

cot²

where  is a constant has either 0, 4 or 8 real solutions.

Solve the equation

cot² sec² 

Solve the equation  sec² tan   for  .

Given that

sin

and that

  and 

find the values of and .

Solve the equation

Show that cos sin can be written in the form  cos, where  and  is an acute angle measured in radians.

Hence, or otherwise, solve the equation cos sin ,for 
Give your answers to three significant figures.

Write down the minimum value of cos sin and the smallest positive value of for which it occurs.  Give your value of   to three significant figures.

Solve the equation

cot² cosec

giving your answers to three significant figures where appropriate.

Solve the equation

cos⁴  cos 

State your answers as multiples of .

Determine the values of the constant for which the equation

                  cosec 

has      (i) no real solutions,
            (ii) 1 real solution,
            (iii) 2 real solutions,
            (iv) 4 real solutions

Solve the equation

cot² cosec 

Give your answers to one decimal place where appropriate.

Solve the equation sin³ sin cos²   for  .

Given that

cos and  tan

and that

find the possible values of and .

Solve the equation sin cos ,for  .
Give your answers to three significant figures.

Write down the maximum value of sin cos and the second positive value of   for which it occurs.  Give your value of  to three significant figures.

Solve the equation

sec⁴ tan² 

giving your answers to three significant figures where appropriate.

Solve the equation

cosec² 

Give your answers to three significant figures.

Solve the equation

sin⁴  cos 

State your answers as multiples of .

The number of real solutions to the equation

is determined by the value of the constant .

Find the number of real solutions for all values of , given that .

Given that  is a root of , solve the equation by factorisation.

Solve the equation

Give your answers to one decimal place where appropriate.