Trigonometric Identities

[Pages:12]Trigonometric Identities

Dr. Philippe B. Laval Kennesaw STate University

April 20, 2005

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 forcertain values of the variable(s) involved. For example, 5x + 1 = 10, 2 sin x + 3 = 0 are equations; they are only true for certain values of x. (x + y)2 = x2 + 2xy + y2 is an identity; it is true no matter what x and y are. 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 + cos2 = 1 1 + tan2 = sec2 1 + cot2 = csc2

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2. Reciprocal Identities 3. Even-Odd Identities

sin

=

1 csc

cos

=

1 sec

tan

=

sin cos

cot

=

cos sin

sec

=

1 cos

csc

=

1 sin

sin (-) = - sin cos (-) = cos tan (-) = - tan

4. Cofunction Identities

sin

2

-

cos

2

-

= cos = sin

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 = a

c

cos

=

a c

2

Hence,

sin = cos

But since

+

+

2

=

It follows that

=

2

-

Therefore, we have

sin = cos

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.

Example

1

Show

that

1 + tan2 x csc2 x

= tan2 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

=

cos2 x 1

sin2 x

=

sin2 x cos2 x

= tan2 x

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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.

cos x (sec x - cos x)

=

cos x

1 cos

x

-

cos

x

=

cos x

1 - cos2 x cos x

= 1 - cos2 x

= sin2 x

Example 3 Express

1

-

1 csc x

2

+ cos2 x in terms of sin x

1

-

1 csc

x

2

+ 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 Sum and Difference Identities

2.1.1 The Identities

Proposition 4 Let and be two real numbers (or two angles). Then we have:

1. sin ( + ) = sin cos + cos sin

2. sin ( - ) = sin cos - cos sin

3. cos ( + ) = cos cos - sin sin

4. cos ( - ) = cos cos + sin sin

5.

tan ( + ) =

tan + tan 1 - tan tan

6.

tan ( - ) =

tan - tan 1 + tan tan

2.1.2 Proof of cos ( - ) = cos cos + sin sin

We prove the fourth identity with the help of a graphical method. Given and , the angle - can be represented as shown on the picture below.

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We now concentrate on - , and represent it for various values of and , in such a way that - remains constant. Two possible such representations are shown in the picture below.

Because a - 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 ) (d (A, B))2 = (d (A B ))2

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Using the coordinates on the picture above, we can compute these distances.

(d (A, B))2 = (cos - cos )2 + (sin - sin )2 = cos2 - 2 cos cos + cos2 + sin2 - 2 sin sin + sin2 = cos2 + sin2 + cos2 + sin2 - 2 (sin sin + cos cos ) = 1 + 1 - 2 (sin sin + cos cos ) = 2 - 2 (sin sin + cos cos )

and

(d (A B ))2 = (cos ( - ) - 1)2 + (sin ( - ) - 0)2 = cos2 ( - ) - 2 cos ( - ) + 1 + sin2 ( - ) = cos2 ( - ) + sin2 ( - ) - 2 cos ( - ) + 1 = 1 - 2 cos ( - ) + 1 = 2 - 2 cos ( - )

Since the two distances are equal, we have

2 - 2 cos ( - ) = 2 - 2 (sin sin + cos cos ) cos ( - ) = sin sin + cos cos

2.1.3 Proof of cos ( + ) = cos cos - sin sin We write + = - (-) and use the identity for cos ( - ).

cos ( + ) = cos ( - (-)) = cos cos (-) + sin sin (-) = cos cos - sin sin

since cos (-) = cos and sin (-) = - sin .

2.1.4 Proof of sin ( + ) = sin cos + cos sin

We use the cofunction identities.

sin ( + )

=

cos

2

- ( + )

= cos

2

-

-

We now use the difference identity for cosine.

sin ( + )

=

cos

2

-

cos + sin

2

-

sin

= sin cos + cos sin

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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:

t

0

6

4 3 2

sin t

0

1 2

2 2

3 2

1

cos t 1

3 2

21 22

0

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)

= sin30 cos45 + cos 30 sin 45

=

1 2

2 2

+

3 2

2 2

=

2 4

+

6 4

=

2+ 6 4

Example

6

Find

the

exact

value

of

cos

12

First, we express 12 in trigonometric functions.

terms of angles

Since

12

=

3

-

for which we

4

,

we

have

know

the

value

of

the

cos

12

=

cos

3

-

4

=

cos 3

cos4

+ sin

3

sin

4

=

1 2

2 2

+

3 2

2 2

=

2+ 6 4

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2.1.6 Application: Simplifying Expressions of the Form A sin + B cos

If we could find an angle such that cos = A and sin = B, the we would have

A sin + B cos = cos sin + sin cos = sin ( + )

If this is going to work, then we must have A2 +B2 = 1 since cos2 +sin2 = 1. What about if A and B are such that A2+B2 = 1? Here is the trick to remember:

A sin + B cos =

A2 + B2

A A2 +

B2

sin

+

B A2 +

B2

cos

First, we note that A

2

+

B

2

= 1. We can find an angle

A2 + B2

A2 + B2

such that cos = A

and sin = B . To see this, simply

A2 + B2

A2 + B2

draw a triangle in which one of the angles is , the length of the side opposite

is A, the length of the side adjacent is B. So, we have:

Proposition 7 If A and B are real numbers, then

A sin + B cos = A2 + B2 sin ( + )

where satisfies

cos = A

and sin = B

A2 + B2

A2 + B2

Example 8 Express 3 sin + 4 cos in the above form. From what we saw above,

3 sin + 4 cos = 32 + 42 sin ( + ) = 5 sin ( + )

where sin = be in the first

4 5

and

cos

quadrant.

=

3 5.

Since

both

sin

and

Using a calculator, we find

cos are positive, should that 53.1. Thus,

3 sin + 4 cos = 5 sin ( + 53.1)

2.2 Double and Half-Angle Identities

In this

tan

2

.

section, we derive The first three are

identities known as

for the

sin 2, cos 2, double-angle

tan 2, sin identities.

2,

cos

2

The last

, and three

are the half-angle identities.

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