(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0


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1 TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some equation manipulating we can establish the fundamental tigonometic identities: tan θ b/a Divide both the numeato, b, and the denominato, a, by and you get tan θ (b/)/(a/) θ / θ Similaly cot θ θ / θ Recipocal Identities can be established by ug the fact that /(x/y) y/x Theefoe: / θ sec θ / θ csc θ /tan θ cot θ Fom the Pythagoean Theoem, we know that a + b If we divide both sides of that equation by, we get a a + b + b θ + θ Dividing both sides by θ, we get : + θ θ + θ θ θ tan θ sec θ Dividing both sides of θ + θ by θ, we get : cot θ + θ θ + θ θ csc θ θ
2 EvenOdd Identities ( θ)  (θ) ( θ) (θ) tan( θ)  tan(θ) csc( θ)  csc(θ) sec( θ) sec(θ) cot( θ)  cot(θ) Establishing Identities Combinations of tig functions may be equal to each othe, if we can pove it. Once poven, this type of identity can be used to educe an equation down and thus educe the eos made when calculating values ug a compute. Guidelines fo Establishing Identities ) It is almost always pefeable to stat with the side containing the moe complicated expession. ) Rewite sums o diffeences of quotients as a gle quotient. 3) Sometimes, ewiting one side in tems of only es and ines will help. 4) Always keep you goal in mind. As you manipulate one side of the expession, you must keep in mind the fom of the expession on the othe side. In class, we will do p.7 Example, Example 3, Example 8
3 3.4 SUM AND DIFFERENCE FORMULAS Theoem (α+β) α β  α (αβ) α β + α NOTE: (α+β) α + β (αβ) α β Poof of (αβ) α β + α Let s use a unit cicle so that evey point (x,y) on the cicle is the ine and e of angles in standad position (with the initial side on the positive xaxis and the teminal side with a point somewhee on the unit cicle). P ( α, α) (0,) αβ α β P ( β, ) (,0) We will use the distance fomula to stat poving ight side of the (αβ) equation. The distance fom P to P (ug the distance fomula) is d(p,p ) ( x x ) + ( y y (α β ) ) + (α )
4 αβ Now to get the left side of the equation, let s otate that pink tiangle so that it has one side on the positive xaxis. (0,) P 3 ( αβ, αβ) A (,0) Since this is the same tiangle as the pevious one (just otated), then the distance fom P to P on the pevious tiangle is the same as the distance fom P 3 to point A on this tiangle. Ug the distance fomula whee the fist point is (,0) and the second point is ( αβ, αβ) we get: ( ( α β ) ) + ( ( α ) 0) d(a,p ) β 3 Setting those two distance equations equal to each othe {d(a,p3) d(p,p )} we get: ( ( α β ) ) + ( ( α β ) 0 ) (α β ) + (α )
5 Now we can squae both sides: ( ( α β ) ) + ( ( α β ) 0 ) (α β ) + (α ) Now multiply out the squaed tems ( α β ) ( α β ) + + ( α β ) α α β + β + α α + Notice with these and tems we can apply the Pythagoean identity: θ + θ On the left side, let θ αβ and we get: ( α β ) + ( α β ) On the ight side, let θ α, and also θ β and we get: α β + α α β α Setting the ight side and left side equal to each othe we get; ( α β ) α β α Now subtacting fom both sides and then dividing both sides by  we get; (αβ) α β + α We can now pove (α+β) α β  α Use the (αβ) fomula but instead of αβ, use α(β) (α(β)) (α+β) α (β) + α (β) Now ou evenodd popeties [ (θ) θ, ( θ)  θ] will become useful: (α+β) α (β)  α β β
6 Now if we ae asked to find the exact value of a tig function of an Angle, we can pehaps combine two known angles to get an exact answe Remembe that you calculato only gives appoximate values. Which angles to we know the exact values of thei tig functions? (Look at you handy Unit Cicle Diagam) θ 0 (0 o π ad ), 30 (π/6 ad ), 45 (π/4 ad ), 60 (π/3 ad ), 90 (π/ ad ) and so on.. Example Find the exact value of is not on ou Unit Cicle Diagam, but we can use the fact that , and we know the tig functions of 30 and 45. ( ) Now you do # 5 on p. 3
7 We can use the Complementay Angle Theoem to find (αβ) and (α+β) Remembe: (π/ θ) (θ) (π/ θ) (θ) (α+β) (π/ (α+β)) ((π/ α) +β)) Shift the paentheses (π/ α) β  (π/ α) α β α Use the (αβ) fomula. Now just use the (α+β) fo (αβ) by ug this: (αβ) (α+ (β)) α (β)  α (β) And ug ou evenodd popeties again, we get: (αβ) α β α () α β + α Now let s do Examples 3, 5 Now you do #3 a, b, and c
8 Fomulas fo tan(α+β) and tan(αβ) Poof: tan(α+β) ( α + β ) ( α + β ) α β + α α b α Now divide both the numeato and denominato by α β. α β + α β α β α β α α β α α β tan ( α + β ) α + α β α α β + tan β tan β As befoe, we can use evenodd popeties to deive the othe fomula.. tan(αβ) tan(α+(β)) + tan( β ) tan( β ) Recall that tan is odd so tan(θ) tan θ, so we get: tan β tan( α β ) Now do #3d + tan β
9 SUMMARY OF SUM AND DIFFERENCE FORMULAS (α+β) α β  α (αβ) α β + α (α+β) α β α (αβ) α β + α tan ( α + β ) + tan β tan β tan( α β ) tan β + tan β Example 9 on p α β Remembe these ae angles
10 HOMEWORK p. 0 #,3, 9, 8 p.3 #5,9,5,3,7,9,59,63
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