Trigonometric Identities

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

1

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

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.

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.

4

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

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download