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Pre-Cal 40S March 13, 2007

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Sum and Difference Identities and more practice with what has come before ...

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Pre-Cal 40S March 13, 2007

1. 1. Prove each of the following identities l—tana _ cota-l cos3t9—stn30=1-2stn°9 0l(l+ t-s. 'st‘9-stfa - (aroz-. <trt= < t—’ZS)rt‘t9 3775 elf’ D 4» yr“ <3?/ J 5rn°< C0ﬁ(( _ ‘*"“(‘°’%T—) 62.t~1>
2. 2. Even and Odd Identities sin(-x) = -sin(x) cos(-x) = cos(x) tan(—x) = -tan(x) The sine and tangent functions are ODD functions. Cosine is an EVEN function. Some Strategies for Proving Trigonometric Identities (1) Work with the more complicated side of the identity first. (2) Rewrite both sides of the identity exclusively in terms of sine and cosine. (3) Use a Pythagorean identity to make an appropriate substitution. (4) Simplify complex fractions or rewrite fractions sums or differences with a single denominator. (5) Use factoring (especially differences of squares). All of the above are just suggestions or "rules of thumb. " F eel free to disregard any or all of the above at any time.
3. 3. The Sam and Difference Identities _. =/*sin(a + B) = sinaeosﬁ+ cosasin/3 Sin(a _ I3) = Sin (1 cosﬁ — Cos a Sin ﬂ Notice the patterns and get ready ___y; oS(a+ = cos acos ﬂ _ sina sinﬁ I0 dance . .. UH. ’ Sit! !! Dance! cos a-13) = cosacos/3+sinasinﬂ SINE 11/ [AGES cosINE (C) JANN Bust-A-Move(+I-)
4. 4. . : ~L‘ “ tﬁlltqcns, sum
5. 5. ~«»t2tcotgt~~t; tmtgt as (2,: be or 3 (Q94) 6'05 m _ .12) ‘g Pjir ,1 co" '4 W C0 S ( IJ— as oo\$ jg; (
6. 6. cg‘-. .“'_£(_ as , x’__n. ¥:t0:¥—+ x)= -sinx ¢ vnc 0 °°‘°"" iﬁirxc-. (>— S Mk ‘Si rt-7L, 62-ED
7. 7. sin(a- = Sin S n R C 6) 90 N sin(g+6)+sin(J61_g)= coS6 05 ‘ S “Q C0311 O C-059 — sme (—t) O + Si‘-'1 9 S; at9— (QED.
8. 8. (yt, ﬂ)/ C056 51:1 6 T‘r€)* - ﬂ gar-(Z 4/gln , t‘/ ” 0) / Mgﬂﬂ In » t e/ 4' 4/5"‘? i 4”’? {,0}? {mg / % " / i/iéwda , Lé”5@ ,2 4093 / @.
9. 9. Prove each of the following identities <5-‘t. VztL. ‘(0S“iX\$ ivflt (os‘l) (ﬁt/ 1i' (M11) I 5°tvt‘{ - (oft QE. . sit = sinzt —cos: t
10. 10. Given: sina = utt. t> Find sin(a+ /3) S’ V‘ cos/3=-l5—% With cosa<0 and sin/3>0 sin (‘H-lei riaotcas (7 + cos“ _sin[’> -7.¢ :21 55"’ ('5 -7’ I3 ‘g P~
11. 11. Given: sina = 3 cos /3 = — — Find cos(a + /3) 5 13 cos (°‘ 7%)’ With cosa<0 and Sin/3>0
12. 12. cos/3=-1% With cosa<0 and sin/3>0 5 mac ctsﬁ- COSt@StFt<><
13. 13. Given: smt. z=s-3-5 cosy -2.-~— L K t 5 I3: