Embedded Files
Trig Identities
The following trig identities are used frequently in solving equations and simplifying expressions.
Cofunction Identities
Addition formulas
Double Angle formulas
= =
Half Angle formulas
Product to Sum formulas
Sum to Product formulas
STEP 2008, Math II, #6: A curve has the equation , where
(i) Find the period of .
(ii) Determine all values of
in the interval for which . Find a value of in this interval at which
the curve touches the
-axis without crossing it.
(iii) Find the value or values of
in the interval for which .
STEP 2003, Math I, #4: Solve the inequality
where
and .
A solution to this problem is posted below.
Red Question: (P. Zeitz) Suppose
Prove that
for all positive integer
.
STEP 2005, Math I, #4: (a) Given that
and that
,
show that
,
and evaluate
.
(b) Prove the identity
.
Hence evaluate
, given that
and that
.
STEP 2011, Math I, #3: Prove the identity
(*)
(i) By differentiating (*), or otherwise, show that
(ii) By setting
in (*), or otherwise, obtain a similar identity for
and deduce that
Show that
STEP 2010, Math I, #3: Show that
and deduce that
Show also that
The points
, , and have coordinates , , and
respectively, where
, and and are positive.
Given that neither of the lines and is vertical, show that these lines are parallel if and only if
Step 2007, Math II, #4: Given that
, and are non-zero, show that the equation
reduces to the form
,
where
and are independent of and , if and only if .
Determine all values of
in the range , for which:
(i)
(ii)
(iii)
STEP 2009, Math II, #3: Prove that
(*)
(i) Use (*) to find the value of
Hence show that
(ii) Show that
(iii) Use (*) to show that
STEP 1998, Math I, #6: Let
with and let . Given that
find
and and show that
and
Guess general expressions for
and (for ) as products of cosines and verify that they satisfy the given equations.
STEP 2007, Math II, #5: In this question,
denotes , denotes , and so on.
(i) The function
is defined, for , by
.
Find by direct calculation
and , and determine .
(ii) Show that
where
and is any positive integer.
(iii) The function is defined, for by
.
Find an expression for for any positive integer .
STEP 2007, Math III, #1: In this question, do not consider the special cases in which the denominators of any of your expressions are zero.
Express in terms of , where , etc.
Given that , , and are the four roots of the equation
(where
) , find an expression in terms of , , and for .
The four real numbers , , and lie in the range and satisfy the equation
where
and are independent of . Show that for some integer .
[The following hint for the final result has been added to help those using this paper for practice; it leads to a stronger result.
Hint: You may use without proof the following identities:
and ]
Viète's solution of the cubic
We've already seen a general solution of the cubic due to the Italian mathematicians of the mid-16th century. Here's an
alternative solution using a trig identity that was published by Francoise Viète in 1591. (My source for this is Needham VCA.)
Show by any means that .
We've previously shown that any cubic equation can be reduced to the following form:
to get the form
Use the substitution
Now show that if then the general solutions of the original equation are
where
.
is an integer and
Why does this proof fall apart when ?
Is there any way to solve the equation using Viète's method? (He would NOT have had complex numbers and
their roots available to him but we do!)
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