Section 7.4 Trigonometric Identities

Transcription

1 9 Section 7.4 Trigonometric Identities Definition Two functions f and g are said to be identically equal if f(x) g(x) for all x in the domain of f and g. We say that f(x) g(x) is an identity. Determine if the following are identities. Ex. a (3x + 2) 2 9x 2 +2x + 4 Ex. b tan 2 (θ) + sec 2 (θ) Ex. c 4x 5 23 Ex. d (x +3)(x 2) (x 2) x + 3 a) Yes, since if we expand (3x + 2) 2, we get 9x 2 +2x + 4. Both expressions are defined for all real numbers. b) Yes, we established that identity in the previous chapter. c) No, since 4x 5 only equals 23 when x 7. It is not true for all real numbers. d) These two functions are identical except at x 2. (undefined 5). We will see in Calculus, we can fix this problem, be defining a piecewise defined function. We have already established quite a few trigonometric identities that you should already know. Fundamental Identities cot(θ) csc(θ) sec(θ) cot(θ) sin 2 (θ) + cos 2 (θ) tan 2 (θ) + sec 2 (θ) cot 2 (θ) + csc 2 (θ) sin( θ) cos( θ) tan( θ) csc( θ) csc(θ) sec( θ) sec(θ) cot( θ) cot(θ) Keep in mind that we may be working with a variation of one of these identities. For instance, instead of using sec(θ), we might need to use sec(θ) or instead of cot2 (θ) + csc 2 (θ), we might need to use cot 2 (θ) csc 2 (θ). To prove an identity, we will need to establish a list of tools we can employ to simplify trigonometric expressions.

2 92 Objective : Use Algebra to Simplify Trigonometric Expressions. There are six major tools we use when working trigonometric expressions. Below is a list each tool and an example to see how the tool is used. Tools ) Rewriting all the trigonometric functions in terms of sine and cosine. cot(θ) sec(θ) (cot(θ) & sec(θ) ) csc(θ) 2) Multiplying by a conjugate to make a difference of squares and using a Pythagorean identity: + + cos 2 (θ) sin 2 (θ) sin 2 (θ) sin 2 (θ) (the conjugate of + is ) (expand) (sin 2 (θ) cos 2 (θ)) (divide both terms by sin 2 (θ)) (csc(θ) & cot(θ) ) csc(θ) cot(θ) 3) Finding a common denominator and adding the fractions together. csc(θ) csc(θ) cot(θ) + csc(θ) csc(θ)+cot(θ) LCD (csc(θ) cot(θ))(csc(θ) + cot(θ)) csc(θ) csc(θ) cot(θ) csc(θ)+cot(θ) csc(θ)+cot(θ) csc2 (θ)+csc(θ)cot(θ) csc 2 (θ) cot 2 (θ) 2csc 2 (θ) csc 2 (θ) cot 2 (θ) + csc(θ) csc(θ)+cot(θ) csc(θ) cot(θ) csc(θ) cot(θ) + csc2 (θ) csc(θ)cot(θ) csc 2 (θ) cot 2 (θ) ( csc 2 (θ) cot 2 (θ)) 2csc2 (θ) 2csc 2 (θ)

3 4) Expanding and simplifying. (+) 2 sec 2 (θ) (+)(+) sec2 (θ) tan2 (θ)+2 + sec 2 (θ) tan2 (θ) sec 2 (θ) ) Factoring and simplifying. cos 2 (θ) cos 2 (θ) (+)( ) ( ) (+) + (tan 2 (θ) sec 2 (θ) ) (factor the numerator and denominator) (divide by ) (sec(θ) ) + sec(θ) 6) Multiplying top and bottom of a fraction by a form of one: + + cot(θ) + cot(θ) ( and cot(θ) ) 93 Objective 2: Establishing Identities To establish an identity, we begin with one side of the identity, usually the more complicated side, and manipulate that side until we get the same expression as we have on the other side. It is important in this process is to keep examining your goal and try to use our tools to get closer to the goal.

4 Establish the identity: Ex. 2 csc(θ) cos 2 (θ) sin 2 (θ) The left side is more complicated. Since we have mostly sine and cosine functions, lets use our first tool to write everything in terms of sine and cosine. Ex. 3 Ex. 4 csc(θ) cos 2 (θ) (csc(θ) ) cos2 (θ) (simplify) cos 2 (θ) (sin 2 (θ) + cos 2 (θ) cos 2 (θ) sin 2 (θ)) sin 2 (θ) The identity has been established. ( cos 2 ( θ))( + cot 2 (θ)) The right side is more complicated, so we will start with that side. ( cos 2 ( θ))( + cot 2 (θ)) (cos( θ), so cos 2 ( θ) cos 2 (θ)) ( cos 2 (θ))( + cot 2 (θ)) (expand) + cot 2 (θ) cos 2 (θ) cos 2 (θ)cot 2 (θ) (use sine & cosine) + cos2 (θ) cos 2 (θ) cos 2 (θ) cos2 (θ) sin 2 (θ) sin 2 (θ) We seem to have reached a dead end. Let's go back to before we expanded and try a different approach: ( cos 2 (θ))( + cot 2 (θ)) But, cos 2 (θ)) sin 2 (θ) and + cot 2 (θ) csc 2 (θ) (sin 2 (θ))(csc 2 (θ)) (csc(θ) (sin 2 (θ))( ) sin 2 (θ) ) The Identity has been established. sin2 ( θ) cos( θ) We will begin with the left side. sin2 ( θ) cos( θ) (sin( θ), but sin 2 ( θ) sin 2 (θ) & cos( θ) ) 94

5 95 Ex. 5 sin2 (θ) sin2 (θ) sin2 (θ) sin2 (θ) cos2 (θ) ( ) (get a common denominator and subtract) (regroup the numerator) ( sin 2 (θ) cos 2 (θ)) (factor out a in the numerator) The identity has been established. +sin(x) + +sin(x) +sin(x) + +sin(x) +sin(x) + +sin(x) 2sec(x) (get a common denominator and add) +sin(x) +sin(x) cos 2 (x) (+sin(x)) + +2sin(x)+sin2 (x) (+sin(x)) If we examine the right side, we see the sec(x) (expand the numerator) so we do not want to expand the denominator, but try to get rid of + sin(x). cos2 (x)++2sin(x)+sin 2 (x) (+sin(x)) cos2 (x)+sin 2 (x)++2sin(x) (+sin(x)) ++2sin(x) (+sin(x)) 2+2sin(x) (+sin(x)) 2(+sin(x)) (+sin(x)) 2 (regroup the numerator) (sin 2 (x) + cos 2 (x) ) (factor 2 out of the numerator) (sec(x) ) 2sec(x) The identity has been established.

6 96 Ex. 6 sin(w)cos(w) tan(w) cos 2 (w) sin 2 (w) tan 2 (w) First, rewrite the right side in terms of sine and cosine: tan(w) tan 2 (w) sin( w ) cos(w) sin2 ( w ) sin( w ) cos(w) sin2 ( w ) sin(w ) cos(w) cos2 ( w ) sin2 ( w ) cos 2 cos2 ( w) ( w ) sin(w ) cos(w ) sin2 ( w ) sin(w)cos(w) cos 2 (w) sin 2 (w) (multiply top and bottom by cos 2 (w)) The identity has been established. Ex. 7 sin 3 (t)+cos 3 (t) 2cos 2 (t) sec(t) sin(t) tan(t) We need to reduce the powers of the sine and cosine function on the left side. If we examine this carefully, it is a sum of cubes. sin 3 (t)+cos 3 (t) 2cos 2 (t) (sin(t)+cos(t))(sin2 (t) sin(t)cos(t)+cos 2 (t)) 2cos 2 (t) (sin(t)+cos(t))(sin2 (t)+cos 2 (t) sin(t)cos(t)) 2cos 2 (t) (sin(t)+cos(t))( sin(t)cos(t)) 2cos 2 (t) (sin(t)+cos(t))( sin(t)cos(t)) cos 2 (t) cos 2 (t) (sin(t)+cos(t))( sin(t)cos(t)) sin 2 (t) cos 2 (t) (F 3 + L 3 (F + L)(F 2 FL + L 2 )) (regroup the numerator) (sin 2 (t) + cos 2 (t) ) (now, let's work with the denominator) (sin 2 (t) cos 2 (t)) (factor the denominator)

7 97 Ex. 8 (sin(t)+cos(t))( sin(t)cos(t)) (sin(t)+cos(t))(sin(t) cos(t)) ( sin(t)cos(t)) (sin(t) cos(t)) (looking at the right side, we need tan(t) in the denominator. Thus, we need to multiply top and bottom by cos( t) cos( t) ( sin(t)cos(t)) (sin(t) cos(t)) cos(t) sin(t)cos(t) cos( t) sin( t) cos(t) cos(t) cos( t) sec(t) sin(t) tan(t) ++ + (tan(t) sin(t) cos(t) and sec(t) cos(t) ) The identity has been established. sec(θ) + cos(t) ) Looking at the left side, we need to try to get rid of the denominator [ +]+ [ +] [ +]+ [ +]+ [ +] [ +]+ (group + ) [ +]2 +2[ +]+sin 2 (θ) [ +] 2 sin 2 (θ) (multiply top & bottom by [ + ] + ) (expand) (expand) +2+cos2 (θ)+2+2+sin 2 (θ) +2+cos 2 (θ) sin 2 (θ) sin2 (θ)+cos 2 (θ) sin 2 (θ)+2+cos 2 (θ) (sin 2 (θ) + and sin 2 (θ) cos 2 (θ)) cos 2 (θ)+2+cos 2 (θ) cos 2 (θ) 2[+++] 2[+cos 2 (θ)] +++ +cos 2 (θ) (regroup) (combine like terms) (factor out 2 from the top & bottom) (factor the top by grouping)

8 98 [+]+[+] +cos 2 (θ) [+][+] +cos 2 (θ) [+][+] [+] [+] + sec(θ) + (sec(θ) (factor out in the denominator) (divide), )