How do we derive this identity? It is tricky, so follow closely.
Suppose we have the unit circle shown below. The two points
and
are on the circle.
We can get the coordinates of
and
in terms of the angles
and
.
For the triangle
, we have that
Thus the coordinates of
are
. In the same way as above, we can see that the coordinates of
are
.
The identity for
is now determined by calculating
in two ways. Using the distance formula (i.e.
or
), we can find
:
The second way we can determine
is by using the cosine rule for
:
Equating our two values for
, we have
Now let
. Then
But
. Thus
Derivation of
We can use
to show that
We know that
and
Therefore,
Derivation of
We can use
to show that
We know that
Therefore,
Derivation of
We found this identity in our derivation of the
identity. We can also use the fact that
to derive that
As
we have that
Derivation of
We know that
When
, we have that
Derivation of
We know that
When
, we have that
However, we can also write
and
by using
The
Identity
Use
to show that:
Problem-solving strategy for identities
The most important thing to remember when asked to prove identities is:
Trigonometric Identities
When proving trigonometric identities, never assume that the left hand side is equal to the right hand side. You need to
show that both sides are equal.
A suggestion for proving identities: It is usually much easier simplifying the more complex side of an identity to get the simpler side than the other way round.
Prove that
without using a calculator.
We only know the exact values of the trig functions for a few special angles (
,
,
, etc.). We can see that
. Thus we can use our double-angle identity for
to express
in terms of known trig function values.
Questions & Answers
I'm interested in biological psychology and cognitive psychology
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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