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Trigonometric Identities
Dr. Philippe B. Laval
Kennesaw STate University
Abstract
This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new ones.
1
Basic Trigonometric Identities
1.1
Quick Review
You will recall that an identity is a statement which is always true. In contrast,
an equation is a statement which is only true for
√ certain values of the variable(s)
2
involved. For example, 5x + 1 = 10, 2 sin x + 3 = 0 are equations. (x + y) =
2
2
x + 2xy + y is an identity. We already know some identities. Some are
definitions. Others have been proven. We begin by listing all the identities we
should know.
1.2
Known Identities
1. Pythagorean Identities
sin2 α + cos2 α = 1
1 + tan2 α = sec2 α
1 + cot2 α = csc2 α
1
2. Reciprocal Identities
sin α =
cos α =
tan α =
cot α =
sec α =
csc α =
1
csc α
1
sec α
sin α
cos α
cos α
sin α
1
cos α
1
sin α
3. Even-Odd Identities
sin (−α) = − sin α
cos (−α) = cos α
tan (−α) = − tan α
4. Cofunction Identities
−α
= cos α
2
π
−α
= sin α
cos
2
sin
π
We have already proven all these identities, except the cofunction identities.
We have already mentioned them when we studied transformations of the graphs
of sine and cosine. There is a nice way to prove them using a triangle. Consider
the triangle below:
In this triangle, we have:
sin α =
cos β
=
a
c
a
c
Hence,
sin α = cos β
2
But since
α+β+
It follows that
β=
π
=π
2
π
−α
2
Therefore, we have
sin α = cos
π
−α
2
The proof is similar for the other cofunction identity. Try it.
These identities will be used as our starting point for proving more identities.
Before we do this, you may have already asked yourself: what are identities used
for? One answer is that learning how to prove identities is a good exercise for
the brain. But identities are useful for other reasons. Very often, identities
allow you to simplify expressions. The simpler an expression is, the easier it is
to work with. Identities are also used in solving trigonometric equations.
1.3
Guidelines for Proving Identities
The primary strategy used is to transform one side of the equation into the
other side. This ”transformation” is made by using the rules of algebra as
well as identities you already know. It may require several steps. During this
transformation, keep the following in mind:
1. Memorize the basic identities. Known identities are often used to prove
new ones.
2. It is usually easier to start with the more complicated side.
3. It is sometimes useful to rewrite everything in terms of sines and cosines.
4. Use algebra and the identities you know. In particular, factor, bring fractional expressions to a common fraction, rationalize the denominator, ...
We illustrate this with a few examples.
1 + tan2 x
= tan2 x.
Example 1 Show that
csc2 x
We start with the more complicated side, and transform it into the other side.
1 + tan2 x
csc2 x
sec2 x
csc2 x
1
2x
cos
=
1
sin2 x
sin2 x
=
cos2 x
= tan2 x
=
3
Example 2 Show that cos x (sec x − cos x) = sin2 x
We start with the more complicated side, and transform it into the other side.
1
− cos x
cos x (sec x − cos x) = cos x
cos x
1 − cos2 x
= cos x
cos x
= 1 − cos2 x
= sin2 x
Example 3 Express
1−
1−
1
csc x
2
1
csc x
2
+ cos2 x in terms of sin x
+ cos2 x = (1 − sin x)2 + cos2 x
= 1 − 2 sin x + sin2 x + cos2 x
= 2 − 2 sin x
2
Other Identities
2.1
2.1.1
Sum and Difference Identities
The Identities
Proposition 4 Let α and β be two real numbers (or two angles). Then we
have:
1. sin (α + β) = sin α cos β + cos α sin β
2. sin (α − β) = sin α cos β − cos α sin β
3. cos (α + β) = cos α cos β − sin α sin β
4. cos (α − β) = cos α cos β + sin α sin β
5. tan (α + β) =
tan α + tan β
1 − tan α tan β
6. tan (α − β) =
tan α − tan β
1 + tan α tan β
2.1.2
Proof of cos (α − β) = cos α cos β + sin α sin β
We prove the fourth identity with the help of a graphical method. Given α and
β, the angle α − β can be represented as shown on the picture below.
4
We now concentrate on α − β, and represent it for various values of α and β, in
such a way that α − β remains constant. Two possible such representations are
shown in the picture below.
Because a − β remained constant, the distance between A and B, denoted
d (A, B) is the same as the distance between A and B , denoted d (A , B ). The
reader will recall
that if the coordinates of A are (x, y) and those of B are (x , y ),
then d (A, B) =
(x − x)2 + (y − y)2 . Therefore, we can write:
d (A, B) = d (A B )
2
(d (A, B))
2
= (d (A B ))
5
Using the coordinates on the picture above, we can compute these distances.
(d (A, B))2
=
=
=
=
=
(cos α − cos β)2 + (sin α − sin β)2
cos2 α − 2 cos α cos β + cos2 β + sin2 α − 2 sin α sin β + sin2 β
cos2 α + sin2 α + cos2 β + sin2 β − 2 (sin α sin β + cos α cos β)
1 + 1 − 2 (sin α sin β + cos α cos β)
2 − 2 (sin α sin β + cos α cos β)
and
2
(d (A B ))
=
=
=
=
=
(cos (α − β) − 1)2 + (sin (α − β) − 0)2
cos2 (α − β) − 2 cos (α − β) + 1 + sin2 (α − β)
cos2 (α − β) + sin2 (α − β) − 2 cos (α − β) + 1
1 − 2 cos (α − β) + 1
2 − 2 cos (α − β)
Since the two distances are equal, we have
2 − 2 cos (α − β) = 2 − 2 (sin α sin β + cos α cos β)
cos (α − β) = sin α sin β + cos α cos β
2.1.3
Proof of cos (α + β) = cos α cos β − sin α sin β
We write α + β = α − (−β) and use the identity for cos (α − β).
cos (α + β) = cos (α − (−β))
= cos α cos (−β) + sin α sin (−β)
= cos α cos β − sin α sin β
since cos (−α) = cos α and sin (−α) = − sin α.
2.1.4
Proof of sin (α + β) = sin α cos β + cos α sin β
We use the cofunction identities.
π
− (α + β)
2 π
−α −β
= cos
2
sin (α + β) = cos
We now use the difference identity for cosine.
π
π
− α cos β + sin
− α sin β
sin (α + β) = cos
2
2
= sin α cos β + cos α sin β
6
2.1.5
Application: Finding the Exact Value of the Trigonometric
Functions
The sum and difference identities are often used to prove other identities, as we
will see later. You will also use them in Calculus I, so you must know them.
They can also be used to find the exact value of the trigonometric functions at
certain angles. We know the exact value of the trigonometric functions at the
following angles:
π
π
π
π
t
0
6
4
3
2
√
√
1
2
3
sin t 0
1
2
2
√
√2
3
2 1
cos t 1
0
2
2
2
For the other angles, we rely on our calculator. The sum and difference formulas
allow us to find the exact value of the trigonometric functions for additional
angles.
Example 5 Find the exact value of sin 75
sin 75 = sin (30 + 45)
= sin 30 cos 45 + cos 30 sin 45
√
√ √
1 2
3 2
=
+
2 2 √2 2
√
2
6
+
=
4
4
√
√
2+ 6
=
4
π
Example 6 Find the exact value of cos
12
π
First, we express
in terms of angles for which we know the value of the
12
π
π π
trigonometric functions. Since
= − , we have
12
3
4
π π π
cos
= cos
−
12
3
4
π
π
π
π
= cos cos + sin sin
3
4
3
4
√
√ √
3 2
1 2
+
=
2
2
2
2
√
√
2+ 6
=
4
7
2.1.6
Application: Simplifying Expressions of the Form A sin α +
B cos α
If we could find an angle β such that cos β = A and sin β = B, the we would
have
A sin α + B cos α = cos β sin α + sin β cos α
= sin (α + β)
If this is going to work, then we must have A2 +B 2 = 1 since cos2 β +sin2 β = 1.
What about if A and B are such that A2 +B 2 = 1? Here is the trick to remember:
A
B
2
2
√
sin α + √
cos α
A sin α + B cos α = A + B
A2 + B 2
A2 + B 2
2 2
B
A
+ √
= 1. We can find an angle
First, we note that √
A2 + B 2
A2 + B 2
B
A
and sin β = √
. To see this, simply
β such that cos β = √
2
2
2
A +B
A + B2
draw a triangle in which one of the angles is β, the length of the side opposite
β is A, the length of the side adjacent β is B. So, we have:
Proposition 7 If A and B are real numbers, then
A sin α + B cos α = A2 + B 2 sin (α + β)
where β satisfies
B
A
and sin β = √
cos β = √
2
2
2
A +B
A + B2
Example 8 Express 3 sin α + 4 cos α in the above form.
From what we saw above,
32 + 42 sin (α + β)
3 sin α + 4 cos α =
= 5 sin (α + β)
4
3
where sin β = and cos β = . Since both sin β and cos β are positive, β should
5
5
be in the first quadrant. Using a calculator, we find that β ≈ 53.1◦ . Thus,
3 sin α + 4 cos α = 5 sin (α + 53.1)
2.2
Double and Half-Angle Identities
α
α
In this section, we derive identities for sin 2α, cos 2α, tan 2α, sin , cos , and
2
2
α
tan . The first three are known as the double-angle identities. The last three
2
are the half-angle identities.
8
Proposition 9 (double-angle identities) Let α be any angle. Then
1. sin 2α = 2 sin α cos α
2. cos 2α = cos2 α − sin2 α = 1 − 2 sin2 α = 2 cos2 α − 1
3. tan 2α =
2 tan α
1 − tan2 α
Proposition 10 (half-angle identities) Let α be any angle. Then
α
1 − cos α
1. sin = ±
2
2
1 + cos α
α
2. cos = ±
2
2
1 − cos α
α
3. tan = ±
2
1 + cos α
4. tan
sin α
α
=
2
1 + cos α
5. tan
1 − cos α
α
=
2
sin α
We now show how these identities can be derived.
2.2.1
Proof of sin 2α = 2 sin α cos α
The trick to remember here is to write 2α = α + α and use the sum identity for
sine.
sin 2α = sin (α + α)
= sin α cos α + cos α sin α
= 2 sin α cos α
The proof is similar for cosine and tangent, we leave it as an exercise.
2.2.2
Note on the identity for cos 2α
You will notice that this identity is in fact three identities. The first one is proven
the same way the identity for sin 2α was proven. To go from cos2 α − sin2 α to
1 − 2 sin2 α or 2 cos2 α − 1, we simply use the Pythagorean identity. For example
cos2 α − sin2 α = 1 − sin2 α − sin2 α
= 1 − 2 sin2 α
9
The identities
cos 2α = 1 − 2 sin2 α
cos 2α = 2 cos2 α − 1
are important. We can rewrite them as
sin2 α =
cos2 α =
1 − cos 2a
2
1 + cos 2α
2
These two new identities are often used in Calculus II. They allow us to decrease
the power of either sine or cosine.
2.2.3
α
α
1 − cos α
Proof of sin = ±
and cos
2
2
2
1 − cos 2β
, it follows that
2
1 − cos 2β
sin β = ±
2
α
This is true for every β, so it is true for β = . So, we obtain
2
α
1 − cos α
sin = ±
2
2
α
The identity for cos is proven the same way. You will note that for these two
2
identities, we will need to know the quadrant of the angle to determine the sign
of the expression.
Since sin2 β =
α
2
The first identity is proven by using the half-angle identities for sine and cosine,
and the definition of tangent. The proof is left as an exercise. The proof of the
remaining two identities can be found in your book.
2.2.4
Proof of the half-angle identities for tan
2.2.5
Examples
Example 11 Find the exact value of sin 22.5.
45
, we have
Since 22.5 =
2
45
sin 22.5 = sin
2
1 − cos 45
= ±
2
10
Since 22.5 is in the first quadrant, sin 22.5 > 0. Therefore,
1 − cos 45
sin 22.5 =
2
√
2
1 −
2
=
2
√
2− 2
=
4
√
1
=
2− 2
2
Example 12 Express sin2 x cos2 x in terms of the first power of cosine.
sin2 x cos2 x =
=
=
=
2.3
1 − cos 2x 1 + cos 2x
2
2
1 − cos2 2x
4
1 + cos 4x
1−
2
4
1 − cos 4x
8
Practice Problems
1. Prove that sin (α − β) = sin α cos β − cos α sin β
2. Prove that tan (α + β) =
of sine and cosine)
3. Prove that tan (α − β) =
tan α + tan β
(Hint: write tan (α + β) in terms
1 − tan α tan β
tan α − tan β
1 + tan α tan β
4. Prove that cos 2α = cos2 α − sin2 α = 1 − 2 sin2 α = 2 cos2 α − 1
2 tan α
1 − tan2 α
1 + cos α
α
6. Prove that cos = ±
2
2
1 − cos α
α
7. Prove that tan = ±
2
1 + cos α
5. Prove that tan 2α =
11
8. Prove that tan
α
sin α
=
2
1 + cos α
9. Prove that tan
1 − cos α
α
=
2
sin α
10. Do # 1, 8, 11, 17, 23, 45, 47, 49, 51, 57, 69 on pages 436, 437.
11. Do # 3, 5, 7, 9, 15, 17, 23 on pages 445, 446.
12. Do # 1, 3, 9, 17, 20, 21 on page 452.
12