1. Express each of the following in the forms stated, where A, B, C, R are positive:

(a) ;

(b) ;

(c)

(d)

2. (i) Express in the form A.cos(q - a) where A>0, and hence solve the equation for 0 < q < 2p.

(ii) Express in the form where R>0 and hence solve the equation for 0 < x < 2p

3. (i) Express in the form Acos(q + a) where A>0, hence state the greatest value of this expression and give the first positive value of q for which it occurs.

(ii) Express in the form A.sin(x + b) where A>0 and hence state the greatest and least value of f(x). Show that

4. Express in the form , where R>0 and 0 < e < 2p. Hence:

(i) State the maximum and minimum values of , and give the smallest positive value of q for which each of these occurs.

(ii) Solve the equation = 1 for 0 < q < 2p

5. Express sinx + cosx in the form A.sin(x + a) where A>0 and hence solve the equation sinx + cosx = 1 for 0 < q < 2p

(i) What is the greatest value of sinx + cosx, and give the value of x in the interval 0 < q < 2p for which this occurs.

(ii) By an appropriate change of origin, sketch the graph of y = sinx + cosx for 0 < q < 2p.

6. If 4cosx - 3sinx = C.cos(x + e) for all x, find the values of C and e. Hence

(i) Find the roots of the equation 4cosx - 3sinx = 3 for 0 < q < 2p as accurately as possible.

(ii) Sketch the graph of y = 4cosx - 3sinx for 0 < q < 2p.

7. (i) Without calculus, state the greatest and least values of the expression

sint - cost .

(ii) A particle moves along the x-axis so that its velocity v(t) at time t is given by

v(t) = cost + sint. Obtain an expression for the displacement x(t) in terms of t.

Show that

(iii) By expressing sint = cost in an appropriate form, solve the equation

sint - cost = for 0 < t < 2p.

8. Use the auxiliary angle method tto solve each of the following equations for 0 < q < 360.

(a) (b) (c)

9. By using the 't' method where , solve the following equations for 0 < q < 360:

(a) (b) (c)