Trig Equations

STEP 1999, Math II, #5: Show that if

is a solution of the equation

,

then either

or

has one other value which you should find.

Prove carefully that if , then .

STEP 2003, Math I, #3: (i) Show that

if and only if

(ii) Solve the equation

(iii) Show that

if and only if where is defined by

for , and is any integer.

STEP 2004, Math III, #5: Show that if then either or . By choosing suitable values of , , and , give an example to show that if

.

STEP 2001, Math II, #4: Let

Show that if , then the only values of for which are given by , where is an integer.

[You may assume that .]

Now let

is a positive integer and . Find an expression for the largest root of the equation , distinguishing between the cases where is even and is odd.