Compound & Double Angle Formulae (CIE A Level Maths: Pure 3)

Express tan in terms of tan and tan

Hence show that tan.

Starting with the identity

            coscos  cos sin  sin 

and using the substitution , show that cos cos² sin² 

Hence, or otherwise, show that cos sin² .

Using an appropriate trigonometric identity, show that 

where  and  are constants.

Hence show that .

Use appropriate double angle formulae to solve the following equations in the given intervals.

cos² sin²     

Show that

Show that , where and are constants.

Use your result from part (i) to show that  .

Hence solve the equation  for  .

Using the identities

            sin sin  cos sin  cos    and

            cos  sin² 

show that sin  sin sin³ 

Hence, or otherwise, solve the equation

             sin sin³ 

Show that can be written in the form where and .

Sketch the graph of sin cos for .
Label any points where the graph intercepts the coordinate axes.

Show that cosec cosec sec .

The alternating voltage, , in an electrical circuit,  seconds after it is switched on, is modelled by the function

Show that

can be written as

where and .

After how many seconds does the voltage first equal -55 volts?

If , then

      sinsinsinsin sin sin sin 

By using a suitable counter-example with , prove that sinsin sin is not true in general.

Express cos in terms of cosines and sines of 315° and 30°.

Hence show that cos.

Show that

      sin  sin  cos 

(You may use the identity sin sin cos cos sin .)

Show that  cos sin   can be written in the form   cos,  where  and  are constants with    and    .
Give  in the form   where  is an integer, and give  correct to three significant figures.

Solve the equation

         sin sin              

Solve the equation

         cos sin²            

Show that

Show that can be written as  , where  to three significant figures.

Hence solve the equation

          sin cos

giving your answers correct to 3 significant figures.

By letting , use the identity for to derive an expression for in terms of tan .

Hence, or otherwise, solve the equation

Sketch the graph of  for .

Be sure to label any points where the graph intercepts the coordinate axes, and state the coordinates of any maximum and minimum points.

Show that

      cot tan sec² 

The alternating voltage, , in an electrical circuit  seconds after it is switched on is modelled by the function

Express

in the form

where and are constants to be found.   and is acute.

Find the voltage when the circuit is switched on.

Prove that   is not true in general.

Find values for and , with  and , for which      is true.

Use the identities and   to show that

Hence show that .

Show that

State clearly any trigonometric identities you use to show this result.

Given that  , where  and  are positive constants, is to be written in the form  ,  find expressions for:

in terms of and 

in terms of and 

Solve the equation
         

Solve the equation
         tan tan 

Show that

         tan tan sec 

Show that can be written in the form  where    , and  radians correct to three significant figures.

Use your result from part (a), and the properties of the sine and cosine functions, to solve the equation

Use an identity for cos to derive an identity for cos , in terms of .

Hence, or otherwise, solve the equation

The diagram below shows two right-angled triangles.
Angles  and  have been labelled.

Given that   , find the exact values of sin ,cos  and tan .

Explain briefly why is not a solution to the equation  .

By using an appropriate approximation, determine the value of

The alternating voltage, , in a domestic electrical circuit,  seconds after it is switched on is modelled by the function

Express

in the form

where and are constants to be found. and is  acute.

In the UK, domestic electricity runs at a frequency, , of 50 Hertz (Hz).

The constant  , is given by