4. THE FUNDAMENTAL TRIGONOMETRIC IDENTITIES trigonometric ...

4. THE FUNDAMENTAL TRIGONOMETRIC IDENTITIES

A trigonometric equation is, by definition, an equation that involves at least one trigonometric function of a variable. Such an equation is called a trigonometric identity if it is true for all values of the variable for which both sides of the equation are defined. An equation that is not an identity is called a conditional equation. For instance, the trigonometric equation

csc t = 1 sin t

is an identity, since it is true for all values of t ( except, of course, for those values of t for which csct or 1 sin t is undefined). On the other hand, the trigonometric equation

sin t = cost is a conditional equation, since there are values of t (for instance, t=0) for which it isn't true.

Figure 4.1

y

unit circle

P = ( x , y ) = (cos , sin )

O -

x

Q = ( x , ?y )

= (cos(- ), sin(- ))

Now we are going to derive the trigonometric identities

sin(? ) = ? sin

and

cos(? ) = cos .

Figure 4.1 shows the angle and the corresponding angle ? both in standard position.

Evidently, the points P and Q , where the terminal sides of these angles intersect the unit

circle, are mirror images of each other across the x axis. Therefore, if P = ( x , y ) then it

follows that Q = ( x , ?y ). In section 2, we showed that

P = ( x , y ) = (cos , sin ).

Likewise,

Q = ( x , ?y ) = (cos(? ) , sin(? ) ).

Therefore, sin(? ) = ?y = ? sin

and

cos(? ) = x = cos .

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If we now combine the identities above with the quotient identity, tan = sin cos , we find that

tan (? ) = sin (-) = ? sin = ? tan .

cos (-) cos

Similar arguments apply to cot(? ), sec(? ), and csc(? ). The results are summarized in the following theorem.

Even-Odd Identities

For all values of in the domains of the functions: (i) sin(? ) = ? sin (ii) cos(? ) = cos (iii) tan(? ) = ? tan (iv) cot(? )= ? cot (ii) sec(? ) = sec (iii) csc(? ) = ? csc

Notice that only the cosine and its reciprocal the secant are even functions ? the remaining four trigonometric functions are odd. The even ? odd identities are often used to simplify expressions, as in the following example:

Example 4.1 ---------------------------- ------------------------------------------------------------

Use the even ? odd identities to simplify each expression.

(a)

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

(b) 1 + tan 2 (-t)

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

(b) 1 + tan2 (-t) = 1 + [tan(-t)]2 = 1 + (- tan t)2 = 1 + tan 2 t = sec2 t .

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Fundamental Trigonometric Identities

1. csc = 1 sin

4. tan = sin cos

7. 1+ tan2 = sec2 10. cos(? ) = cos 13. sec(? ) = sec

2. sec = 1 cos

5. cot = cos sin

8. 1+ cot2 = csc2 11. tan(? ) = ?tan 14. csc(? ) = ?csc

3. cot = 1 tan

6. cos2 + sin 2 = 1

9. sin(? ) = ?sin 12. cot(? ) = ?cot

Not only should you memorize these fourteen fundamental identities, but they should become so familiar to you that you can recognize them quickly even when they are written in equivalent forms. For instance, csc = 1 sin t can also be written as

(sin )(csc ) = 1 or sin = 1 . csc

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Incidentally, a product of values of trigonometric functions such as (sin )(csc ) is usually written simply as sin csc , unless the parentheses are necessary to prevent

confusion.

Example 4.2 ---------------------------- -----------------------------------------------------------Simplify each trigonometric expression by using the fundamental identities.

(a) (csc )(cos )

(csc )(cos ) = 1 cos = cos = cot

sin

sin

(b) tan2 t - sec2 t Because 1 + tan2 = sec2 , it follows that tan2 - sec2 = - 1 .

(c) csc4 x - 2csc2 x cot2 x + cot4 x

The given expression is the square of csc2 x - cot2 x . Because cot2 x + 1 = csc2 x , we have

csc2 x - cot2 x = 1. Therefore, csc4 x - 2 csc2 x cot2 x + cot4 x = (csc2 x - cot2 x )2 =12 = 1.

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The reciprocal and quotient identities enable us to write csc , sec , tan , and cot in terms of sin and cos . Therefore:

Any trigonometric expression can be written in terms of sines and cosines.

This fact and the Pythagorean identity cos2 + sin 2 = 1 can often be used to simplify trigonometric expressions.

Example 4.3 ---------------------------- ------------------------------------------------------------

Rewrite each trigonometric expression in terms of sines and cosines, and then simplify the result.

(a) csc t ? cos t sin t

csc t ? cot t =

1

cos t

? sin t =

1 ? cos t cos t

sec t sin t

1

sin t sin t

cos t

= 1 ? cos2 t = 1- cos2 t = sin 2 t = sin t

sin t sin t

sin t

sin t

(b) csc2 x sec2 x csc2 x + sec2 x

26

csc2 x sec2 x csc2 x + sec2 x

=

1 sin 2

x

1 cos 2

x

=

sin 2

x

cos 2

x

1 sin 2

x

1 cos 2

x

1 sin 2

x

+

1 cos 2

x

sin 2

x

cos 2

x

1 sin 2

x

+

1 cos2

x

=

1 cos2 x + sin 2 x =

1 = 1. 1

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The Pythagorean identity cos2 + sin 2 = 1 can be written as cos2 = 1 ? sin 2 or sin 2 = 1 ? cos2 .

Therefore, we have

( i ) sin = ? 1- cos2

( ii ) cos = ? 1- sin 2

In either case, the correct algebraic sign is determined by the quadrant or coordinate axis containing the terminal side of the angle in standard position. After you have rewritten a trigonometric expression in terms of sines and cosines, you can use these equations to bring the expression into a form involving only the sine or only the cosine.

Example 4.4 ---------------------------- -----------------------------------------------------------Rewrite the expression cot csc2 in terms of sin only.

cot

csc2 =

cos sin

.

1 sin 2

=

cos sin3

=

?

1 - sin 2 sin3 .

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Algebraic expressions not originally containing trigonometric functions can often be simplified by substituting trigonometric expressions for the variable. This technique, called trigonometric substitution, is routinely used in calculus to rewrite radical expressions as trigonometric expressions containing no radicals.

Example 4.5 ---------------------------- ------------------------------------------------------------

If a is a positive constant, rewrite the radical expression a2 - u 2 as a trigonometric

expression containing no radical by using the trigonometric substitution u = a sin .

Assume that ? < < , so that cos > 0.

2

2

a2 - u2 = a2 - (a sin )2 = a2 - a2 sin 2

= a2 (1 - sin 2 ) = a2 cos2 = a cos .

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Section 4 Problems---------------------- -----------------------------------------------------------In problems 1 to 6, use the even-odd identities to simplify each expression.

1. sin (? ) cos (? )

3. tan ( t ) + tan ( ? t )

5.

1 1

+ -

csc cot

(- ) (- )

2. cot2 (-u) +1 4. cos( ?x ) sec x 6. [ 1 + sin ][ 1 + sin(? )]

In problems 7 to 28, use the fundamental identities to simplify each expression.

7. sec sin 9. cot sec

8. 1 + tan

cot

10. csc2 u

1+ tan2 u

11. csc

sec

13. cot 2 - csc2

12. sin2 -1

sec

14. sec2 t -1

sec2 t

15. (csc u ? 1)( csc u + 1) 17. 1 + 1

sec2 x csc2 x

16. 1+ cot2 y

1+ tan2 y

18. (sec -1)(sec + 1)

tan

19. sin 4 t + 2 cos2 t sin 2 t + cos4 t 21. tan 4 - 2 tan 2 sec2 + sec4

20. sin 4 u + 2 cos2 u - cos4 u 22. (1+ tan2 )(1- sin 2 )

23. cos x sin3x + sin xcos3x

24. (1 - cos2 )(1 + cot2 )

25.

1 - cos t

sin t cos t sin t

26. cos + cos

1 - sin 1+ sin

27. sin t + 1 + cos t

1 + cos t sin t

28. sin + sin + cos - cos

cos + cos sin - sin

In problems 29 to 38, rewrite each trigonometric expression in terms of sines and cosines,

and then simplify the result.

29. tan x

sec x

30. (cos + tan sin ) cot

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