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Mathematical Expression Editor
Introduction to Identities
From the previous section, we have found some identities. We will now summarize
what we have already found and begin to introduce new identities. These will help us
to breakdown and simplify trigonometric equations that will hopefully make our lives
easier.
Remember that an identity is an equation that is true for all possible values of for
which both sides are defined. The equation is an algebraic identity: It’s true for every
value of . The left and right sides of the equation are simply two different-looking but
entirely equivalent expressions.
Trigonometric identities are identities for which both sides of the equation are
trigonometric expressions — i.e., they’re made up of trig functions and constants
being added, subtracted, multiplied, and divided. There are many trigonometric
identities one could study; we choose to focus on those that are useful in
calculus.
The most important trigonometric identity is the fundamental trigonometric identity,
which is a trigonometric restatement of the Pythagorean Theorem. For any real angle
,
Identities are important because they enable us to view the same idea from multiple
perspectives. For example, the identity above allows us to think of as , or,
alternatively, to view as .
There are two more Pythagorean Identities, which involve the tangent, secant,
cotangent, and cosecant functions. We derived these from the fundamental
trigonometric identity by dividing both sides by either or . We will take another look
at these identities before continuing. If , then divide both sides of the fundamental
trigonometric identity by . Simplify the quotients using the Quotient and Reciprocal
Identities. If , then divide by . Simplify the quotients using the Quotient
and Reciprocal Identities. These identities prove useful in calculus when
we develop the formulas for the derivatives of the tangent and cotangent
functions.
Sums and Differences of Angles
In calculus, it is also beneficial to know a couple of other standard identities for sums
of angles or double angles.
Sum and Difference Identities for Cosine The following equations are true for all angles and .
Sum and Difference Identities for Sine The following equations are true for all angles and .
(a)
Evaluate .
(b)
Verify the identity.
Solution
(a)
In order to use the theorem to find , we need to write as a sum or difference of
angles whose cosines and sines we know. One way to do so is to write .
(b)
In a straightforward application of the theorem, we find
(a)
Evaluate .
(b)
If is a Quadrant II angle with and is a Quadrant III angle with , find .
(c)
Derive a formula for in terms of and .
Solution
(a)
As in the earlier example, we need to write the angle as a sum
or difference of common angles. The denominator of suggests a
combination of angles with denominators and . One such combination is
Applying what we know about sum and difference with sine, we get
(b)
In order to find using the theorem, we need to find and both and . To
find , we use the Pythagorean Identity . Since , we have , or . Since
is a Quadrant II angle, . We now set about finding and . We have
several ways to proceed, but the Pythagorean Identity is a quick way to
get , and, hence, . With , we get so that . Since is a Quadrant III
angle, we choose so . We now need to determine . We could use the
Pythagorean Identity , but we opt instead to use a quotient identity.
From , we have so we get . We now have all the pieces needed to find :
(c)
We can start expanding using a quotient identity and our sum formulas
Since and , it looks as though if we divide both numerator
and denominator by we will have what we want.
Naturally, this formula is limited to those cases where all of the tangents are
defined.
(a)
If for , find an expression for in terms of .
(b)
Verify the identity.
(c)
Express as a polynomial in terms of .
Solution
(a)
If your first reaction to ‘’ is ‘No it’s not, !’ then you have indeed learned
something, and we take comfort in that. However, context is everything.
Here, ‘’ is just a variable — it does not necessarily represent the -coordinate
of the point on the Unit Circle which lies on the terminal side of , assuming
is drawn in standard position. Here, represents the quantity , and what
we wish to know is how to express in terms of . We will see more of this
kind of thing in future sections, and, as usual, this is something we need
for calculus.
Since , we need to write in terms of to finish the problem. We substitute
into the Pythagorean Identity, , to get , or . Since , , and thus . Our final
answer is
(b)
We start with the right side of the identity. Note that . From this point, we use
the Reciprocal and Quotient Identities to rewrite and in terms of and :
(c)
The identity expresses as a polynomial in terms of . We are now asked to find
such an identity for . Using the sum formula for cosine, we begin with
Our goal is to express the right side in terms of only. We substitute and , which yields
Finally, we exchange for courtesy of the Pythagorean Identity, and get
and we are done.
Lists of Identities
The Pythagorean Identities
(a)
. Common Alternate Forms:
(b)
, provided .
Common Alternate Forms:
(c)
, provided .
Common Alternate Forms:
Reciprocal and Quotient Identities:
, provided ; if , is undefined.
, provided ; if , is undefined.
, provided ; if , is undefined.
, provided ; if , is undefined.
Pythagorean Conjugates
and :
and :
and :
and :
and :
and :
In this activity, we investigate how a sum of two angles identity for the sine function
helps us gain a different perspective on the average rate of change of the sine
function.
Recall that the average rate of change of a function on an interval is
a.
Let . Use the definition to write an expression for the average rate of
change of the sine function, , on the interval .
b.
Apply the sum-of-two-angles identity to .
c.
Explain why your work in (a) and (b) together with some algebra shows
that
d.
In calculus, we move from average rate of change to instantaneous rate of
change by letting approach in the expression for average rate of change.
Using a computational device in radian mode, investigate the behavior of
as gets close to . What happens? Similarly, how does behave for small
values of ? What does this tell us about as approaches ?
More Identities
In the previous sections, we saw the utility of the Pythagorean Identities. Not only
did these identities help us compute the values of the circular functions for angles,
they were also useful in simplifying expressions involving the circular functions.
We will introduce this set of identities as the “Even/Odd” identities and
we will discuss them further with trigonometric transformations in a later
section.
