Open Aplicações da trigonometria

❯♥✐✈❡%&✐❞❛❞❡ ❋❡❞❡%❛❧ ❞❛ +❛%❛,❜❛

❈❡♥/%♦ ❞❡ ❈✐1♥❝✐❛& ❊①❛/❛& ❡ ❞❛ ◆❛/✉%❡③❛

❉❡♣❛%/❛♠❡♥/♦ ❞❡ ▼❛/❡♠</✐❝❛

▼❡&/%❛❞♦ +%♦✜&&✐♦♥❛❧ ❡♠ ▼❛/❡♠</✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ +❘❖❋▼❆❚

❆♣❧✐❝❛&'❡) ❞❛ ❚,✐❣♦♥♦♠❡1,✐❛

♣♦"

❊❣❞❡♠♦& ❇(✐❧❤❛♥.❡ ❞❡ ❖❧✐✈❡✐(❛

#♦❜ ♦"✐❡♥(❛*+♦ ❞♦

1(♦❢✳❉(✳ ◆❛♣♦❧❡7♥ ❈❛(♦ ❚✉❡&.❛

❚!❛❜❛❧❤♦ ❛♣!❡)❡♥+❛❞♦ ❝♦♠♦ !❡/✉✐)✐+♦ ♣❛!❛ ❛ ❝♦♥❝❧✉)2♦ ❞♦ ♠❡)+!❛❞♦ ♣!♦✜)✲ )✐♦♥❛❧ ❡♠ ♠❛+❡♠5+✐❝❛ ❡♠ !❡❞❡ ♥❛❝✐♦✲ ♥❛❧ 6❘❖❋▼❆❚✲❈❈❊◆✲❯❋6❇ ❞♦ ♣❡!A♦❞♦ ✷✵✶✸✳✶✳

❆❜!✐❧✴✷✵✶✺ ❏♦2♦ 6❡))♦❛ ✲ 6❇

†_{❖ ♣"❡$❡♥&❡ &"❛❜❛❧❤♦ ❢♦✐ "❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆4❊❙✱ ❈♦♦"❞❡♥❛89♦ ❞❡ ❆♣❡"❢❡✐8♦❛♠❡♥&♦ ❞❡}

O48a Oliveira, Egdemos Brilhante de.

Aplicações da trigonometria / Egdemos Brilhante de Oliveira.- João Pessoa, 2015.

64f. : il.

Orientador: Napoleón Caro Tuesta Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Teorema de Pitágoras. 3. Trigonometria - aplicações. 4. Circunferência trigonométrica. 5. Séries de Fouries.

❆❣"❛❞❡❝✐♠❡♥*♦,

❆ ♠❡✉ ❉❡✉% ❏❡%✉% ❈(✐%*♦✱ ♣❡❧❛ ❜❡♥23♦ 4✉❡ ♠❡ ❢♦✐ ❛*(✐❜✉6❞❛ *♦(♥❛♥❞♦✲♠❡ ❝❛♣❛③ ❞❡ ❝♦♥❝❧✉✐( ❡%*❡ *(❛❜❛❧❤♦✳ ❆♦ ♣(♦❢❡%%♦( ❉♦✉*♦( ♦(✐❡♥*❛❞♦( ◆❛♣♦❧❡>♥ ❈❛(♦ ❚✉❡%*❛ ♣❡❧❛ ❞❡❞✐❝❛23♦ ❛ ♠✐♠ ❝♦♥❝❡❞✐❞❛ ♣❛(❛ (❡❛❧✐③❛23♦ ❞♦ ♠❡%♠♦✱ ❛ ♣(♦❢❡%%♦(❛ ❚✉❝❛ ❡ ❛♠✐❣♦% ♣♦( ❛❝❡✐*❛(❡♠✱ ♣(♦♥*❛♠❡♥*❡✱ ♦ ❝♦♥✈✐*❡ ♣❛(❛ ♣❛(*✐❝✐♣❛(❡♠ ❞❡%%❡ ❚❈❈✱ ❨♦✲ ❧❛♥❞❛ ❡%♣♦%❛✱ ■(♠3♦% ❙♦♥✐❛✱ ▼✐❣✉❡❧ ✭❡♠ ♠❡♠>(✐❛✮ ❡ ●❡(❛❧❞♦ ✭❈✉♥❤❛❞♦✮ ♣♦( ♠❡ ✐♥❝❡♥*✐✈❛(❡♠ ❡♠ *✉❞♦ 4✉❡ ❢❛2♦✳

❉❡❞✐❝❛&'(✐❛

❆ ♣"♦❢✳ ❚✉❝❛ ❡ ❛♠✐❣♦. /✉❡ ❞❡ ❢♦"♠❛ ❞✐"❡1❛ ♦✉ ✐♥❞✐"❡1❛♠❡♥1❡ 1✐✈❡"❛♠ ✉♠❛ ♣❛"1✐❝✐♣❛45♦ ❢✉♥❞❛♠❡♥1❛❧ ♣❛"❛ /✉❡ ❡..❡ 1"❛❜❛❧❤♦ ♣✉❞❡..❡ .❡ 1♦"♥❛" "❡❛❧✐❞❛❞❡❀ ❡ ❡♠ ❡.♣❡❝✐❛❧ ❛ ❨♦❧❛♥❞❛ ✭❡.♣♦.❛✮✱ ❙♦♥✐❛ ▼❛"✐❛ ✭✐"♠5✮✱ ●❡"❛❧❞♦ ✭❝✉♥❤❛❞♦✮ ❡ ▼✐❣✉❡❧ ✭✐"♠5♦ ❡♠ ▼❡♠A"✐❛✮ ♣♦" .❡"❡♠ ♣❡..♦❛. ❞❡ ✉♠❛ ■♠♣♦"1C♥❝✐❛ ✐♥❞✐.♣❡♥.D✈❡❧ ❡♠ 1✉❞♦ /✉❡ ❢❛4♦ ♥❛ ✈✐❞❛✳

❘❡"✉♠♦

❆♣❧✐❝❛&'❡) ❞❛ +,✐❣♦♥♦♠❡+,✐❛ ❚,❛③ ❝♦♠♦ ❢♦❝♦✿

• ❈♦♥+❡①+♦ ❤✐)+8,✐❝♦✿ ,❡)❣❛+❛♥❞♦ ✉♠ ♣♦✉❝♦ ❛) ❝♦♥+,✐❜✉✐&'❡) ❞❡ ♣♦✈♦) ❛♥+✐❣♦)

❡ ❛❧❣✉♥) ♥♦♠❡) ❞❡ ♣❡))♦❛) <✉❡ ❝♦♠ )✉❛) ❞❡)❝♦❜❡,+❛) ❝♦♥+,✐❜✉=,❛♠ ♣❛,❛ ♦ ❞❡)❡♥✈♦❧✈✐♠❡♥+♦ +❡8,✐❝♦ ❞❛ +,✐❣♦♥♦♠❡+,✐❛❀

• ❋✉♥❞❛♠❡♥+❛&@♦ +❡8,✐❝❛✿ ✈✐)❛♥❞♦ ❡①♣❧✐❝✐+❛, ,❡❧❛&'❡) +,✐❣♦♥♦♠A+,✐❝❛) ✉+✐❧✐③❛❞❛)

♥❛) ❡)❝♦❧❛) ♣B❜❧✐❝❛) ❞❡ ♥=✈❡❧ ♠A❞✐♦✱ ❜❡♠ ❝♦♠♦✱ ❢♦,+❛❧❡❝❡, ❛ ✐♠♣♦,+D♥❝✐❛ ❞❡)+❛ ♠❛+❡♠E+✐❝❛ ❜E)✐❝❛ ♣❛,❛ ❛))✉♥+♦) ❞❡ ♥=✈❡❧ )✉♣❡,✐♦,✳

• ❆) ❛♣❧✐❝❛&'❡)✿ ✈✐)❛♥❞♦ ❛+❡♥❞❡, ❛♦ <✉❡)+✐♦♥❛♠❡♥+♦ ❢❡✐+♦ ♣❡❧❛ )♦❝✐❡❞❛❞❡ ❡♠

,❡❧❛&@♦ G) ❛♣❧✐❝❛&'❡) ❞❛ +,✐❣♦♥♦♠❡+,✐❛ ❜E)✐❝❛✳ ✳

❆❜"#$❛❝#

❚!✐❣♦♥♦♠❡(!② ❛♣♣❧✐❝❛(✐♦♥. ❜!✐♥❣. ❢♦❝✉. ♦♥✿ ❍✐.(♦!✐❝❛❧ ❝♦♥(❡①(✱ !❡.❝✉✐♥❣ .♦♠❡ ❝♦♥(!✉❜(✐♦♥. ♦❢ ❛♥❝✐❡♥( ♣❡♦♣❧❡. ❛♥❞ .♦♠❡ ♥❛♠❡. ♦❢ ♣❡♦♣❧❡ ✇✐(❤ (❤❡✐! ✜♥❞✐♥❣. ❝♦♥✲ (!✐❜✉(❡❞ (♦ (❤❡ (❤❡♦!❡(✐❝❛❧ ❞❡✈❡❧♦♣❛♠❡♥( ♦❢ (!✐❣♦♥♦♠❡(!②❀ ❚❤❡♦!❡(✐❝❛❧ ❢!❛♠✇♦!❦✿ ❛✐♠✐♥❣ ❡♣❧✐❝✐( (!✐❣♦♥♦♠❡(!✐❝ !❛(✐♦. ✉.❡❞ ✐♥ (❤❡ ❛✈❡!❛❣❡ ❧❡✈❡❧ ♦❢ ♣✉❜❧✐❝ .❝❤♦♦❧.✱ ❛. ✇❡❧❧ ❛. .(!❡♥❣(❤❡ (❤❡ ✐♠♣♦!(❛♥❝❡ ♦❢ ❜❛.✐❝ ♠❛(❤ ❢♦! (♦♣✲❧❡✈❡❧ ✐..✉❡.❀ ❆♣♣❧✐❝❛(✐♦♥.✿ (♦ ♠❡❡( (❤❡ ❝❤❛❧❧❡♥❣. !❛✐.❡❞ ❜② (❤❡ ❝♦♠♣❛♥② ✐♥ !❡❧❛(✐♦♥ (♦ ❛♣♣❧✐❝❛(✐♦♥ ♦❢ ❜❛.✐❝ (!✐❣♦♥♦♠❡(!②✳

❙✉♠#$✐♦

✶ ❈♦♥$❡①$♦ ❍✐)$*+✐❝♦ ✶

✶✳✶ ●#❡❣♦' ❡ ❊❣)♣❝✐♦' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❚#✐❣♦♥♦♠❡1#✐❛ ❡'❢4#✐❝❛ ❡ ♣❧❛♥❛# ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✷ ❋✉♥❞❛♠❡♥$❛34♦ ❚❡*+✐❝❛ ✻

✷✳✶ 7♦'1✉❧❛❞♦ ❞♦ ❚#❛'♣♦#1❡ ❞❡ ❙❡❣♠❡♥1♦' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ 7♦'1✉❧❛❞♦ ❞♦ ❚#❛♥'♣♦#1❡ ❞❡ ➶♥❣✉❧♦' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❈♦♥❣#✉?♥❝✐❛ ❞❡ ❚#✐@♥❣✉❧♦' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸✳✶ ❈❛'♦ ❞❡ ❈♦♥❣#✉?♥❝✐❛ ▲❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸✳✷ ❈❛'♦ ❞❡ ❈♦♥❣#✉?♥❝✐❛ ❆▲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹ ❙❡♠❡❧❤❛♥F❛ ❞❡ ❚#✐@♥❣✉❧♦' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✹✳✶ ❈❛'♦ ❞❡ ❙❡♠❡❧❤❛♥F❛ ✭❆❆✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✺ ❚#✐@♥❣✉❧♦ ❘❡1@♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✺✳✶ ❊❧❡♠❡♥1♦' ◆♦1M✈❡✐' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✺✳✷ ❚❡♦#❡♠❛ ❞❡ 7✐1M❣♦#❛' ✭❡♥✉♥❝✐❛❞♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✺✳✸ ❉❡♠♦♥'1#❛FQ♦ ❞♦ ❚❡♦#❡♠❛ ❞❡ 7✐1M❣♦#❛' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✻ ❉❡❝❧✐✈✐❞❛❞❡ ♦✉ ■♥❝❧✐♥❛FQ♦ ❞❡ ✉♠❛ ❘❛♠♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼ ❘❛③T❡' ❚#✐❣♦♥♦♠41#✐❝❛' ❞❡ ✉♠ @♥❣✉❧♦ ❛❣✉❞♦ ♥♦ 1#✐@♥❣✉❧♦ #❡1@♥❣✉❧♦ ✳ ✶✸ ✷✳✼✳✶ ❚❛♥❣❡♥1❡ ❞❡ ✉♠ ➶♥❣✉❧♦ ❆❣✉❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✼✳✷ ❙❡♥♦ ❡ ❈♦''❡♥♦ ❞❡ ✉♠ ➶♥❣✉❧♦ ❆❣✉❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✼✳✸ ❆#❝♦ ❞❡ ❈✐#❝✉♥❢❡#?♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✼✳✹ ❯♥✐❞❛❞❡ ❞❡ ▼❡❞✐❞❛ ●#❛✉✭◆♦1❛FQ♦✿◦_{✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺}

✷✳✼✳✺ ❯♥✐❞❛❞❡ ❞❡ ▼❡❞✐❞❛ ✶ ❘❛❞✐❛♥♦ ✭◆♦1❛FQ♦✿ ✶ ❘❛❞✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✼✳✻ ❈♦♥✈❡#'Q♦✿ ●#❛✉ ❡ ❘❛❞✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✽ ❚#✐❣♦♥♦♠❡1#✐❛ ♥❛ ❈✐#❝✉♥❢❡#?♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✽✳✶ ❈✐#❝✉♥❢❡#?♥❝✐❛ ❚#✐❣♦♥♦♠41#✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✽✳✷ ❙❡♥♦ ❡ ❈♦''❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✽✳✸ ❚❛♥❣❡♥1❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✽✳✹ ❖✉1#❛' ❘❡❧❛FT❡' ❚#✐❣♦♥♦♠41#✐❝❛' ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✽✳✺ ❘❡❧❛FT❡' ❚#✐❣♦♥♦♠41#✐❝❛' 7❛#❛ ❛ ❙♦♠❛ ❞❡ ❉♦✐' ❆#❝♦' ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✽✳✻ ❋✉♥FT❡' ❙❡♥♦✱ ❈♦''❡♥♦✱ ❚❛♥❣❡♥1❡ ♥♦ ❈❛♠♣♦ ❘❡❛❧ ❡ ❙✉❛' ■♥✈❡#'❛' ✷✺

✷✳✽✳✼ ❚❛❜❡❧❛ ❞❡ ❱❛❧♦,❡- ◆♦/0✈❡✐- ❞❡ [0,2π]. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✽✳✽ ●,0✜❝♦- ❡ ❙✉❛- 9,♦♣,✐❡❞❛❞❡- ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✽✳✾ ❋✉♥>?❡- ■♥✈❡,-❛- ❚,✐❣♦♥♦♠C/,✐❝❛- ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✽✳✶✵ ❋✉♥>F♦ ❆,❝♦ ❙❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✽✳✶✶ ❋✉♥>F♦ ❆,❝♦ ❈♦--❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✽✳✶✷ ❋✉♥>F♦ ❆,❝♦ ❚❛♥❣❡♥/❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✾ ❊①♣✭①✮ ❡ ❘❡❧❛>F♦ ❞❡ ❊✉❧❡, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✾✳✶ 9,♦♣,✐❡❞❛❞❡ ▼✉❧/✐♣❧✐❝❛/✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✾✳✷ ❘❡❧❛>F♦ ❞❡ ❊✉❧❡, ♣❛,❛ x ❘❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✷✳✾✳✸ ❙❡♥♦ ❡ ❈♦--❡♥♦ ❈♦♠♣❧❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✾✳✹ ◆♦>?❡- 9❛,❛ ❙C,✐❡ ❞❡ ❋♦✉,✐❡, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✸ ❆♣❧✐❝❛'(❡* ✸✻

✸✳✶ ❆,❝♦ -♦♠❛ ❡ ❢✉♥>F♦ /,✐❣♦♥♦♠C/,✐❝❛ ✐♥✈❡,-❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❊R✉❛>?❡- /,✐❣♦♥♦♠C/,✐❝❛- ♥♦ ❝❛♠♣♦ ❝♦♠♣❧❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸ ■❞❡♥/✐❞❛❞❡- ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✹ ■❣✉❛❧❞❛❞❡- ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✺ ❆♣❧✐❝❛>F♦ ✶ ♣❛,❛ ❛ ❙C,✐❡ ❞❡ ❋♦✉,✐❡, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✻ ❆♣❧✐❝❛>F♦ ✷ ♣❛,❛ ❛ ❙C,✐❡ ❞❡ ❋♦✉,✐❡, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✼ ▲❡✐ ❞♦ ❝♦--❡♥♦ ❡ ▲❡✐ ❞♦ -❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✽ ❆♣❧✐❝❛>F♦ ❞❛ ▲❡✐ ❞♦ ❝♦--❡♥♦ ❡ ▲❡✐ ❞♦ -❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✾ ❘❛③?❡- /,✐❣♦♥♦♠C/,✐❝❛- ❡ /❡♦,❡♠❛ ❞❡ 9✐/0❣♦,❛- ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

▲✐"#❛ ❞❡ ❋✐❣✉*❛"

✶✳✶ ❤❡①%❣♦♥♦ )❡❣✉❧❛) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ .✐)0♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ■♥❝❧✐♥❛67♦ ❞❛ ♣✐)0♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✹ ❍✐♣❛)❝♦ ❞❡ ❇✐<=♥✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✺ ❈❛❧❡♥❞%)✐♦ ❞♦ ❩♦❞=❛❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✻ ❚)✐0♥❣✉❧♦ ❊D❢F)✐❝♦ABC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶✳✼ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞)✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✽ ◆❛D✐) ❛❧✲❉✐♥ ❛❧✲❚✉D✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✶ ❙❡❣♠❡♥<♦D ❈♦♥❣)✉❡♥<❡D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ➶♥❣✉❧♦D ❈♦♥❣)✉❡♥<❡D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❚)✐0♥❣✉❧♦D ❈♦♥❣)✉❡♥<❡D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✹ ❈♦♥❣)✉P♥❝✐❛ ▲❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✺ ❈♦♥❣)✉P♥❝✐❛ ❆▲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✻ ❚)✐0♥❣✉❧♦D ❙❡♠❡❧❤❛♥<❡D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✼ ❚)✐0♥❣✉❧♦D ❘❡<0♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✽ ❆❧<✉)❛ AD ❘❡❧❛<✐✈❛ ❛ ❍✐♣♦<❡♥✉D❛ ♥♦ ❚)✐0♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✾ ❆♣❧✐❝❛67♦ ❞♦ ❚❡♦)❡♠❛ ❞❡ .✐<%❣♦)❛D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✵ ❊❧❡♠❡♥<♦D ◆♦<%✈❡✐D ♥♦ ❚)✐0♥❣✉❧♦ ❘❡<0♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✶ ❘❛♠♣❛ ❡ D✉❛D ❱❛)✐❛6W❡D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✷ ❘❛♠♣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✸ ❚)✐0♥❣✉❧♦ ❆❇❈ ❝♦♠ ➶♥❣✉❧♦ ❋✐①♦ θ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷✳✶✹ ➶♥❣✉❧♦ ❋✐①♦ θ ♥♦ ❚)✐0♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷✳✶✺ ❆)❝♦ ❆❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✻ ❈✐)❝✉♥❢❡)P♥❝✐❛ ❚)✐❣♦♥♦♠F<)✐❝❛ ❞❡ ❈❡♥<)♦ ❖ ❡ ❘❛✐♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✼ .♦♥<♦ P ❡ ❙✉❛D .)♦❥❡6W❡D ❖)<♦❣♦♥❛✐D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷✳✶✽ ❘❡<❛ t .❛)❛❧❡❧❛ ❛♦ ❊✐①♦ oy ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✶✾ ❚)✐0♥❣✉❧♦D ❙❡♠❡❧❤❛♥<❡D OP Q ❡OT A ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✷✵ ❚)✐0♥❣✉❧♦D ❙❡♠❡❧❤❛♥<❡D OP Q ❡OBT′ _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾}

✷✳✷✶ ❘❡<❛ s ❚❛♥❣❡♥<❡ ♥♦ .♦♥<♦ P ❡ ❚)✐0♥❣✉❧♦D ❙❡♠❡❧❤❛♥<❡D OP Q❡ OP S ✳ ✷✵

✷✳✷✷ ❘❡<❛ s′ _{❚❛♥❣❡♥<❡ ♥♦ .♦♥<♦ P ❡ ❚)✐0♥❣✉❧♦D ❙❡♠❡❧❤❛♥<❡D OP Q❡ OP S}′ _✷✶

✷✳✷✸ .)♦❥❡6W❡D ❖)<♦❣♦♥❛✐D ❞♦D .♦♥<♦D D ❡E ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷✳✷✹ ➶♥❣✉❧♦ ❋✐①♦ α ♥♦ ❚-✐.♥❣✉❧♦ EHD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✷✺ ➶♥❣✉❧♦ ❋✐①♦ β ♥♦ ❚-✐.♥❣✉❧♦OED ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✷✻ ❚-✐.♥❣✉❧♦ ❘❡4.♥❣✉❧♦OF E ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷✳✷✼ ❚-✐.♥❣✉❧♦ ❘❡4.♥❣✉❧♦DHE ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✷✳✷✽ ❚-✐.♥❣✉❧♦ ❘❡4.♥❣✉❧♦OF E ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✷✳✷✾ ❈✐-❝✉♥❢❡-;♥❝✐❛ ❚-✐❣♦♥♦♠>4-✐❝❛ ❞❡ ❈❡♥4-♦ O ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✸✵ ❊✐①♦ ❞❛B ❚❛♥❣❡♥4❡B t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✸✶ ❙❡♥E✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸✷ ❈♦BB❡♥E✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸✸ ❚❛♥❣❡♥4E✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸✹ F❛-✐❞❛❞❡ ❮♠♣❛- ♥♦ ❊✐①♦ ❞❛B ❚❛♥❣❡♥4❡B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸✺ F-♦❥❡JK♦ ❖-4♦❣♦♥❛❧ ♥♦ ❊✐①♦ ❞♦B ❙❡♥♦B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸✻ ❙❡♥4✐❞♦ ■♥✈❡-B♦ ❞❛ F-♦❥❡JK♦ ❞❛ ❋✐❣✉-❛ ✷✳✸✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✼ ❋✉♥JK♦ ❙❡♥♦ ❞❡

−π

2,

π

2

→[₋1,1] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✽ ❆-❝♦ ❙❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸✾ ❋✉♥JK♦ ❈♦BB❡♥♦ ❞❡[0, π]_→[₋1,1] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✹✵ ❋✉♥JK♦ ❛-❝♦✲❝♦BB❡♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹✶ ❋✉♥JK♦ ❚❛♥❣❡♥4❡ ❞❡

−π

2,

π

2

→R _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷}

✷✳✹✷ ❋✉♥JK♦ ❛-❝♦✲4❛♥❣❡♥4❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✶ ❖♥❞❛ ◗✉❛❞-❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷ ❚-✐.♥❣✉❧♦ ❆❝✉4.♥❣✉❧♦ ABC ❝♦♠ ➶♥❣✉❧♦ ❆❣✉❞♦ ❋✐①♦ ❡♠ A ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✸✳✸ ❚-✐.♥❣✉❧♦ ❖❜4✉B.♥❣✉❧♦ ABC ❝♦♠ ➶♥❣✉❧♦ ❖❜4✉B♦ ❋✐①♦ ❡♠ A ✳ ✳ ✳ ✳ ✳ ✹✹

✸✳✹ ❚-✐.♥❣✉❧♦ ■♥B❝-✐4♦ ABC ♥♦ ❈✐-❝✉❧♦ ❞❡ ❈❡♥4-♦ O ❡ ❘❛✐♦ R ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✸✳✺ ■❧❤❛BC ❡ D ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸✳✻ ❙❡❣♠❡♥4♦AB ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸✳✼ ◗✉❛❞-✐❧T4❡-♦ ABCD ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✸✳✽ ❚-✐.♥❣✉❧♦ABC ■♥4❡-♥♦ ♥❛ ❋✐❣✉-❛ ✸✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✸✳✾ ❚-✐.♥❣✉❧♦ABD ■♥4❡-♥♦ ♥❛ ❋✐❣✉-❛ ✸✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✸✳✶✵ ❚-✐.♥❣✉❧♦ACD ■♥4❡-♥♦ ♥❛ ❋✐❣✉-❛ ✸✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

▲✐"#❛ ❞❡ ❚❛❜❡❧❛"

✷✳✶ ❚❛❜❡❧❛ ❞❡ ❱❛❧♦+❡, ◆♦./✈❡✐, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

◆♦"❛$%❡'

◆♦"❛$%❡' ●❡)❛✐'

• α ✭❛❧❢❛✮❀ β ✭❜❡(❛✮❀ θ ✭(❡(❛✮ ♣❛*❛ *❡♣*❡+❡♥(❛-.♦ ❛♥❣✉❧❛*✳

• ❊①♣ ✭ex_{✮ ♣❛*❛ ❢✉♥-.♦ ❡①♣♦♥❡♥❝✐❛❧✳}

• P ✭+♦♠❛(8*✐♦✮ ♣❛*❛ +9*✐❡ ❞❡ ❢✉♥-;❡+✳

• Z

❝♦♠♦ +<♠❜♦❧♦ ♣❛*❛ ✐♥(❡❣*❛-.♦✳

• ❆❇

⌢

■♥"#♦❞✉'(♦

❆♣❧✐❝❛&'❡) ❞❛ +,✐❣♦♥♦♠❡+,✐❛ ❡♠ ✉♠ +,❛❜❛❧❤♦ ✈♦❧+❛❞♦ ♣❛,❛ ❛❧✉♥♦) ❞❡ ❡)❝♦❧❛) ♣5✲ ❜❧✐❝❛) ❞❡ ♥7✈❡❧ ♠8❞✐♦✳ ❚❡♠ ♣♦, ♦❜❥❡+✐✈♦ ,❡✈✐)❛, ❛❧❣✉♥) +<♣✐❝♦) ✐♠♣♦,+❛♥+❡)✱ ❜❡♠ ❝♦♠♦✱ ❜✉)❝❛, ✐♥❝❡♥+✐✈❛, ♦) ❛❧✉♥♦) ❛ ❞❡❞✐❝❛,❡♠ ✉♠ ♣♦✉❝♦ ♠❛✐) ❞❡ +❡♠♣♦ ❞❡ )❡✉) ❡)+✉❞♦) ♣❛,❛ ❛ +,✐❣♦♥♦♠❡+,✐❛✳

>❛,❛ ❥✉)+✐✜❝❛, ❛ ✉+✐❧✐❞❛❞❡ ❞❡❧❛ ♠❡)♠❛✱ ❡)+@♦ ✐♥❝❧✉7❞♦)✿

• ◆♦ ♣,✐♠❡✐,♦ ❝❛♣7+✉❧♦ ✉♠ ❝♦♥+❡①+♦ ❤✐)+<,✐❝♦✱ ♥♦ D✉❛❧ ❝✐✈✐❧✐③❛&'❡) ❡❣7♣❝✐❛)✱ ❣,❡✲

❣❛)✱ ♠❡)♦♣♦+F♠✐❝❛) ❡ ✐♥❞✐❛♥❛) )@♦ ❧❡♠❜,❛❞❛) ❞❡✈✐❞♦ G ❝♦♥+,✐❜✉✐&'❡) ♣❛,❛ ❛ ❤✉♠❛♥✐❞❛❞❡✳ ❆)+,H♥♦♠♦) ❝♦♠♦ ❍✐♣❛,❝♦✱ >+♦❧♦♠❡✉ ❡ ▼❡♥❡❧❛✉ )@♦ ❧❡♠❜,❛❞♦) ♣♦, ❝♦♥+,✐❜✉✐&'❡) ♣❛,❛ ❛ ❛)+,♦♥♦♠✐❛ ❡ ♣❛,❛ ❛ +,✐❣♦♥♦♠❡+,✐❛❀

• ◆♦ ❝❛♣7+✉❧♦ ❞♦✐) ❡♥❝♦♥+,❛,❡♠♦) ❞❡✜♥✐&'❡) ❡ ,❡❧❛&'❡) +,✐❣♦♥♦♠8+,✐❝❛) ❝♦♠ ✉)♦

❢,❡D✉❡♥+❡ ♥♦ ♥7✈❡❧ ♠8❞✐♦✱ ❜❡♠ ❝♦♠♦✱ ❞❡✜♥✐&'❡) ❞❡ +❡♠❛) ✉+✐❧✐③❛❞♦) ♥♦ ♥7✈❡❧ )✉♣❡,✐♦, ❝♦♠♦ ❙8,✐❡ ❞❡ ❋✉♥&'❡) ❡ ❙8,✐❡ ❞❡ ❋♦✉,✐❡,❀

• ◆♦ ❝❛♣7+✉❧♦ +,O) ❡♥❝♦♥+,❛,❡♠♦) ❛♣❧✐❝❛&'❡) ♣❛,❛ ❞✐✈❡,)❛) ,❡❧❛&'❡) ✈✐)+❛) ♥♦

❝❛♣7+✉❧♦ ❛♥+❡,✐♦,✱ ❝✉❥❛ ✜♥❛❧✐❞❛❞❡ 8 ♠♦)+,❛,✱ ♥❛ ♣,P+✐❝❛ ✱ ❛ ✐♠♣♦,+F♥❝✐❛ ❞❛ +,✐❣♦♥♦♠❡+,✐❛ ❜P)✐❝❛✱ +❛♥+♦ ♥♦ ♥7✈❡❧ ♠8❞✐♦✱ D✉❛♥+♦ ♥♦ )✉♣❡,✐♦,✳

❈❛♣#$✉❧♦ ✶

❈♦♥$❡①$♦ ❍✐.$/0✐❝♦

◆❡"#❡ ❝❛♣'#✉❧♦✱ ,❡❝♦,❞❛,❡♠♦" ❛♣❧✐❝❛01❡" ❞❛ #,✐❣♦♥♦♠❡#,✐❛ ❢❡✐#❛" ♣♦, ❝✐✈✐❧✐③❛01❡" ❛✳❈✳✳ ❉❡"#❛❝❛,❡♠♦" ❝♦♥#,✐❜✉✐01❡" ❞❡ ❛"#,;♥♦♠♦" ❝♦♠♦ ❍✐♣❛,❝♦✱ ❝♦♥"✐❞❡,❛❞♦ ♦ ♣❛✐ ❞❛ #,✐❣♦♥♦♠❡#,✐❛✱ =#♦❧♦♠❡✉✱ ♣✐♦♥❡✐,♦ ❡♠ ,❡❛❧✐③❛, ♠❡❧❤♦,❛" ♥♦" #,❛❜❛❧❤♦" ❞❡✐①❛❞♦" ♣♦, ❍✐♣❛,❝♦✱ ❡ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞,✐❛✱ ♦ ♣✐♦♥❡✐,♦ ♥❛ ❞❡✜♥✐0C♦ ♣❛,❛ ❛ #,✐❣♦♥♦♠❡#,✐❛ ❡"❢D,✐❝❛✳ =♦✈♦" ✐♥❞✐❛♥♦"✱ E,❛❜❡" ❡ ❤✐♥❞✉"✱ ❡♠❜♦,❛ ❞❡ ❢♦,♠❛ ❜,❡✈❡✱ "❡,C♦ ♠❡♥❝✐♦♥❛❞♦" ♥❡"#❡ #,❛❜❛❧❤♦ ♣♦, #❡,❡♠ ❝♦♥#,✐❜✉'❞♦ ♥♦ ❛✈❛♥0♦ ❞❛ #,✐❣♦♥♦♠❡#,✐❛ ❛♣❡,❢❡✐0♦❛♥❞♦ ❛" ❝,✐❛01❡" ❞❡✐①❛❞❛" ♣❡❧♦" ❡❣'♣❝✐♦" ❡ ❣,❡❣♦"✳

✶✳✶ ●#❡❣♦' ❡ ❊❣)♣❝✐♦'

❙❡❣✉♥❞♦ ❬✶✶❪✱ ❛ #,✐❣♦♥♦♠❡#,✐❛ D ♦ ,❛♠♦ ❞❛ ♠❛#❡♠E#✐❝❛ J✉❡ #,❛#❛ ❞♦ ❝E❧❝✉❧♦ ❞❡ K♥❣✉❧♦"✱ ♣❛,#✐❝✉❧❛,♠❡♥#❡ ❡♠ #,✐K♥❣✉❧♦" ,❡#K♥❣✉❧♦"✳ ❆#D ♦ "D❝✉❧♦ ✶✻✱ ❡❧❛ ❡,❛ ,❡❛❧♠❡♥#❡ ✉♠❛ ♣❛,#❡ ❞❛ ❣❡♦♠❡#,✐❛✱ ♠❛" ❞❡"❞❡ ❡♥#C♦ ❡❧❛ ♣❛""♦✉ ❛ "❡, ❝♦♥"✐❞❡,❛❞❛ ✉♠❛ E,❡❛ ✐♥❞❡♣❡♥❞❡♥#❡ ❞❛ ♠❛#❡♠E#✐❝❛✳

❈♦♠♦ J✉❛❧J✉❡, ♣♦❧'❣♦♥♦ ♣♦❞❡ "❡, ,❡❞✉③✐❞♦ ❛ ✉♠ ♥M♠❡,♦ ❞❡ #,✐K♥❣✉❧♦"✱ ❛ #,✐❣♦✲ ♥♦♠❡#,✐❛ ♣❡,♠✐#❡ ❛♦" ♠❛#❡♠E#✐❝♦" #,❛❜❛❧❤❛, ❝♦♠ #♦❞❛" ❛" E,❡❛" ♦✉ "✉♣❡,❢'❝✐❡" J✉❡ "❡❥❛♠ ❧✐♠✐#❛❞❛" ♣♦, ❧✐♥❤❛" ,❡#❛" ✭✈❡, ✜❣✉,❛ ✶✳✶✮✳ ❆ #,✐❣♦♥♦♠❡#,✐❛ ♣❧❛♥❛ #,❛#❛ ❞❡ E,❡❛"✱ K♥❣✉❧♦" ❡ ❞✐"#K♥❝✐❛" ❡♠ ✉♠ ♣❧❛♥♦✳ ❆ #,✐❣♦♥♦♠❡#,✐❛ ❡"❢D,✐❝❛ #,❛#❛ ❞❡ K♥❣✉❧♦" ❡ ❞✐"#K♥❝✐❛" ♥♦ ❡"♣❛0♦ #,✐❞✐♠❡♥"✐♦♥❛❧✳

❖" ❡❣'♣❝✐♦" #✐♥❤❛♠ ❛❧❣✉♠ ❝♦♥❤❡❝✐♠❡♥#♦ ❞❡ #,✐❣♦♥♦♠❡#,✐❛✱ ❝♦♠♦ ❞❡♠♦♥"#,❛ ❛ ❝♦♥"#,✉0C♦ ❞❡ "✉❛" ♣✐,K♠✐❞❡"✳ ❖ ♣❛♣✐,♦ ❞❡ ❆❤♠❡$ ✐♥❝❧✉✐ ✉♠ ♣,♦❜❧❡♠❛ J✉❡ ❞❡#❡,♠✐♥❛ ❛ $❡❦❡❞✱ ❛ ✐♥❝❧✐♥❛0C♦ ❞❛ ♣✐,K♠✐❞❡ ✭❱❡, ✜❣✉,❛ ✶✳✷✮ ❛ ♣❛,#✐, ❞❛ ❛❧#✉,❛ ❡ ❞❛ ❜❛"❡✭✈❡, ✜❣✉,❛ ✶✳✸✮✳ ❊❧❛ ❡,❛ ❡①♣,❡""❛ ❝♦♠♦ ❛ ,❡❧❛0C♦ ♦♣♦"#❛ ❞❡ ♥♦""❛ ♠❡❞✐❞❛ ❞♦ ❣,❛❞✐❡♥#❡✳ =♦,D♠✱ ♦" ❡❣'♣❝✐♦" ♥C♦ ❡,❛♠ ,✐❣♦,♦"♦" ❡♠ "❡✉ ❡"#✉❞♦ ❞❡ #,✐K♥❣✉❧♦"✳ ❈♦♠♦ ❡♠ ♦✉#,❛" E,❡❛" ❞❛ ♠❛#❡♠E#✐❝❛✱ ❡❧❡" ❡"#❛✈❛♠ ✐♥#❡,❡""❛❞♦" ❡♠ ❛♣❧✐❝❛01❡" ♣,E#✐❝❛" ❡ ♥C♦ ♥❛ #,✐❣♦♥♦♠❡#,✐❛ ♣✉,❛✳

❖" ❛♥#✐❣♦" ♠❛#❡♠E#✐❝♦" ✐♥❞✐❛♥♦" #❛♠❜D♠ "❛❜✐❛♠ ❛❧❣✉♠❛ ❝♦✐"❛ "♦❜,❡ #,✐❣♦♥♦♠❡✲ #,✐❛✳ ❖" ❙✉❧❜❛ $✉,-❛$✱ ♥♦ ❝♦♥#❡①#♦ ❞❛ ❞❡"❝,✐0C♦ ❞❡ ❛❧#❛,❡"✱ ❝♦♥#D♠ ✉♠ ❝E❧❝✉❧♦ ❞♦

✶✳✶✳ ●❘❊●❖❙ ❊ ❊●❮(❈■❖❙

❋✐❣✉$❛ ✶✳✶✿ ❤❡①,❣♦♥♦ $❡❣✉❧❛$

❋✐❣✉$❛ ✶✳✷✿ 1✐$2♠✐❞❡ ❋✐❣✉$❛ ✶✳✸✿ ■♥❝❧✐♥❛89♦ ❞❛ ♣✐$2♠✐❞❡

;❡♥♦ ❞❡ 45◦ ❝♦♠♦ _√1

2✳ ◆♦ ❡♥=❛♥=♦✱ ✜❝♦✉ ♣♦$ ❝♦♥=❛ ❞♦; ❣$❡❣♦; ♦ ❞❡;❡♥✈♦❧✈✐♠❡♥=♦ ❛♣$♦♣$✐❛❞♦ ❞❛ =$✐❣♦♥♦♠❡=$✐❛✳ ❖; ❣$❡❣♦; =♦♠❛$❛♠ ❛ ❧✐♥❤❛ $❡=❛ ❡ ♦ ❝B$❝✉❧♦ ❝♦♠♦ ❜❛;❡ ❞❡ ;✉❛ ❣❡♦♠❡=$✐❛ ❡ ❛ ♣❛$=✐$ ❞❛B ❞❡;❡♥✈♦❧✈❡$❛♠ ❛ =$✐❣♦♥♦♠❡=$✐❛✳

❆ ❝♦♥✈❡♥89♦ ❞❡ 360◦ _{❡♠ ✉♠ ❝B$❝✉❧♦ ❡ ✻✵ ♠✐♥✉=♦; ❡♠ ✉♠ ❣$❛✉ =❡✈❡ ♦$✐❣❡♠}

♥❛ ♠❛=❡♠,=✐❝❛ ❤❡❧G♥✐❝❛✱ ❛♣❛$❡♥=❡♠❡♥=❡✱ ❥, ❡;=❛✈❛ ❡♠ ✉;♦ ♥♦ =❡♠♣♦ ❞❡ ❍✐♣❛$❝♦ ❞❛ ❇✐)*♥✐❛ ✭❝✳ ✶✾✵✲✶✷✵ ❛✳❈✮ ✭✜❣✉$❛ ✶✳✹✮✳ 1$♦✈❛✈❡❧♠❡♥=❡✱ =❡✈❡ ♦$✐❣❡♠ ♥❛ ❞✐✈✐;9♦ ❛;=$♦♥O♠✐❝❛ ❜❛❜✐❧O♥✐❝❛ ❞♦ ③♦❞B❛❝♦ ✭✈❡$ ✜❣✉$❛ ✶✳✺✮ ❡♠ ✶✷ ;✐❣♥♦; ♦✉ ✸✻ ❞❡❝❛♥♦; ❡ ♦ ❝B$❝✉❧♦ ❛♥✉❛❧ ❞❡✱ ❛♣$♦①✐♠❛❞❛♠❡♥=❡✱ ✸✻✵ ❞✐❛;✳

❖ ;✐;=❡♠❛ ;✉♣❡$✐♦$ ✉;❛❞♦ ♣❡❧♦; ❜❛❜✐❧O♥✐❝♦; ♣❛$❛ $❡♣$❡;❡♥=❛$ ❢$❛8S❡; ♦ =♦$♥♦✉ ♠❛✐; T=✐❧ ❞♦ U✉❡ ♦; ;✐;=❡♠❛; ❡❣B♣❝✐♦; ♦✉ ❣$❡❣♦; ❡ ,)♦❧♦♠❡✉ ✭❝✳ ✾✵✲✶✻✽ ❞✳❈✮ ✉;♦✉ ♦ ;✐;=❡♠❛ ❞❡ ❜❛;❡ ✻✵ ❛♦ ❞✐✈✐❞✐$ ❡♠ ❣$❛✉; ❡ ♠✐♥✉=♦; ✭♣❛$)❡1 ♠✐♥✉)❛❡ ♣$✐♠❛❡✮ ❡ ❝❛❞❛ ♠✐♥✉=♦ ❡♠ ✻✵ ;❡❣✉♥❞♦; ✭♣❛$=❡; ♠✐♥✉)❛❡ 1❡❝✉♥❞❛❡✮✳

✶✳✷✳ ❚❘■●❖◆❖▼❊❚❘■❆ ❊❙❋➱❘■❈❆ ❊ 0▲❆◆❆❘

❡!❢#$✐❝♦!✳ ❖! *$✐+♥❣✉❧♦! ❡!❢#$✐❝♦! !0♦✱ ❝❧❛$❛♠❡♥*❡✱ ❡!!❡♥❝✐❛✐! ♥❛ ❛!*$♦♥♦♠✐❛✱ ♥❛ ♥4✉✲ *✐❝❛ ❡ ♥❛✈❡❣❛70♦✳

❋✐❣✉$❛ ✶✳✼✿ ▼❡♥❡❧❛✉ ❞❡ ❆❧❡①❛♥❞$✐❛

❊♥A✉❛♥*♦ ♦! +♥❣✉❧♦! ❞❡ ✉♠ *$✐+♥❣✉❧♦ ♣❧❛♥❛$ *♦*❛❧✐③❛♠ 180◦_{✱ ♦! +♥❣✉❧♦! ❞❡ ✉♠}

*$✐+♥❣✉❧♦ ❡!❢#$✐❝♦ *♦*❛❧✐③❛♠ ♠❛✐! ❞❡ 180◦_{✳ ❍4 *❛♠❜#♠ ♦✉*$❛! ❞✐❢❡$❡♥7❛! ❢✉♥❞❛♠❡♥✲}

*❛✐!✿ ❛*# ♣♦$ ✈♦❧*❛ ❞❡ ✶✷✺✵ ❡ ❞♦ *$❛❜❛❧❤♦ ❞❡ ◆❛"✐$ ❛❧✲❉✐♥ ❛❧✲❚✉"✐ ✭✶✷✵✶✲✼✹✮ ✭❋✐❣✉$❛ ✶✳✽✮✱ ❛ *$✐❣♦♥♦♠❡*$✐❛ ❡!❢#$✐❝❛ ❡!*❡✈❡ !❡♠♣$❡ ✐♥*❡❣$❛❞❛ ❝♦♠ ❛ ❛!*$♦♥♦♠✐❛✳

❋✐❣✉$❛ ✶✳✽✿ ◆❛!✐$ ❛❧✲❉✐♥ ❛❧✲❚✉!✐

✶✳✷✳ ❚❘■●❖◆❖▼❊❚❘■❆ ❊❙❋➱❘■❈❆ ❊ 0▲❆◆❆❘

❆❧✲❚✉%✐ ❢♦✐ ♦ ♣$✐♠❡✐$♦ ❛ ❧✐)*❛$ )❡✐) *✐♣♦) ❞✐)*✐♥*♦) ❞❡ *$✐-♥❣✉❧♦) $❡*-♥❣✉❧♦) ❡♠ ✉♠❛ )✉♣❡$❢0❝✐❡ ❡)❢2$✐❝❛ ❡ ♦ ♣$✐♠❡✐$♦ ❛ *$❛*❛$ ❛ *$✐❣♦♥♦♠❡*$✐❛ ❝♦♠♦ ✉♠❛ ❞✐)❝✐♣❧✐♥❛ ❞✐)❝$❡*❛✳ ❊❧❡ ❞❡)❡♥✈♦❧✈❡✉ ❛ *$✐❣♦♥♦♠❡*$✐❛ ❡)❢2$✐❝❛ ♥❛ )✉❛ ❢♦$♠❛ ❛*✉❛❧✳

❍✐♣❛*❝♦ ❢♦✐ ♦ ♣$✐♠❡✐$♦ ❛ ❝♦♠♣✐❧❛$ *❛❜❡❧❛) ❞❡ ❢✉♥78❡) *$✐❣♦♥♦♠2*$✐❝❛)✳ ❙❡✉ ✐♥*❡✲ $❡))❡ ❡$❛ ♣♦$ *$✐-♥❣✉❧♦) ✐♠❛❣✐♥;$✐♦) ✧*$❛7❛❞♦)✧)♦❜$❡ ❛ ❡)❢❡$❛ ✐♠❛❣✐♥;$✐❛ ❞♦ ❝2✉ = ♥♦✐*❡✱ $❡❧❛❝✐♦♥❛♥❞♦ ♦) ❝♦$♣♦) ❝❡❧❡)*❡) ✉♥) ❝♦♠ ♦) ♦✉*$♦) ❞❡ ♠❛♥❡✐$❛ ?✉❡ ❡❧❡ ♣♦❞✐❛ ❝❛❧❝✉❧❛$ ❡ ♣$❡✈❡$ ❛) ♣♦)✐78❡) ❞♦) ♣❧❛♥❡*❛)✳

❈❧.✉❞✐♦ 0✐1♦❧♦♠❡✉ ❊①♣❛♥❞✐✉ ♦ *$❛❜❛❧❤♦ ❞❡ ❍✐♣❛$❝♦✱ ❛♦ ❝$✐❛$ ♠❡❧❤♦$❡) *❛❜❡❧❛) *$✐❣♦♥♦♠2*$✐❝❛) ❡ ❞❡✜♥✐78❡) ❛♣$♦①✐♠❛❞❛) ♣❛$❛ ❛) ❢✉♥78❡) *$✐❣♦♥♦♠2*$✐❝❛) ✐♥✈❡$)❛) ❛$❝♦✲)❡♥♦ ❡ ❛$❝♦✲❝♦))❡♥♦✳

x

AB A′

B′ _A′_B′ _AB

AOBˆ −−→O′A′

−−→

O′_B′ −−→_O′_A′ _A′_Oˆ′_B′

AOBˆ

T1 T2 T1 ≡ T2

T1

∆ABC ≡∆A′_B′_C′ ⇔

 



AB = A′_B′ _Aˆ _{= A}ˆ′

AC = A′_C′ _Bˆ _{= B}ˆ′

BC = B′_C′ _Cˆ _{= C}ˆ′

 



AB _≡ A′_B′

ˆ

A _≡ Aˆ′

AC _≡ A′_C′

⇒∆ABC ≡∆A′_B′_C′

∆ABC ∼∆A′_B′_C′ ⇒

 



ˆ

A _≡ Aˆ′

ˆ

B _≡ Bˆ′

ˆ

C _≡ Cˆ′

a a′ =

b b′ =

c c′

ˆ

A= 90◦

AD

       

      

BC = a AC = b

AB = c .

BD = m ( c ).

CD = n ( b ).

AD = h

ACBˆ = 90◦

(5)2 = (3)2 + (4)2

25 = 9 + 16

25 = 25

ABC A

ACBˆ = 90◦

−ABCˆ = DABˆ ABCˆ = 90◦_−ACBˆ _{= DAC}ˆ

m+n =a

• ABC _∼DBA a c =

c m ⇒c

2 _=a·m(II

• ABC _∼DAC a b =

b n ⇒b

2 _=a·n(II

ii)

IIi+IIii

c2+b2 =a_·(m+n) (i) m+n=a c2+b2 =a_·a_⇒c2+b2 =a2

5%

⋄

5% 1

20

5% = 5

100 ⇒5% =

1

θ

tgθ θ

θ

tgθ= θ

θ

ABC A AB AC

⋄

θ tgθ

tgθ= 6

8 =

θ

θ

θ

6

10 = = = senθ

8

10 = = = cosθ

◦

1◦ 1

360

360◦

r 2πr

←→ r

θ _←→ 2πr

r_·θ = 2_·π_·r_·1_⇒

θ= 2·π·r

r ·1⇒ θ = 2_·π(rad).

2_·π _←→ 360◦

xoy

A

A

P

⌢

(XP, YP) P OP Px cosθ =XP

senθ =YP θ P

P

OP Px

sen2θ+ cos2θ = 1.

t A

OP t (T)

OAT

tgθ = AT

1 =AT .

tgθ = senθ

cosθ (0

t oy

OP Q OT A

OP Q OT A

P Q T A =

      

P Q = senθ

OQ = cosθ

OA = 1

T A = tgθ

senθ

tgθ =

cosθ

1 ⇒tgθ =

senθ

cosθ

θ (0◦ _{< θ <90}◦₎

OP Q OBT′

t′ _{B T}′ _t′

←→

OP P QO OBT′

OQ BT′ =

P Q BO.

 



OQ = cosθ

P Q = senθ

BO = 1

cosθ

BT′ =

senθ

1 ⇒BT

′ ₌ cosθ

senθ.

cosθ

senθ θ

cotgθ = cosθ

θ (0◦ _{< θ <90}◦₎

s P OP Q OP S

s P S s

0x OP S OQP

OS OP =

OP OQ.

OP = 1

OQ = cosθ

OS

1 =

1

cosθ ⇒OS =

1

cosθ.

1

cosθ θ

secθ = 1

cosθ

s′ _{P OP Q OP S}′

s′ _{P S}′ _s′

0y OP Q S′_OP

OS′

OP = OP P Q.

OP = 1

P Q = senθ

OS′

1 =

1

senθ ⇒OS

′ ₌ 1

senθ.

1

senθ θ

cossecθ = 1

senθ

D E

HE GF HE =GF HDEˆ

α FOEˆ

sen(α+β) DG DG =

DH+HG

• DH

HDE DH = cosαsenβ

DE = senβ.

• HG

OED OE = cosβ

OF E OE = cosβ EF =

α EHD

β OED

OF E

sen (α+β) =DE DE =DH+HG

sen(α+β) = senαcosβ+ senβcosα.

sen(α₋β) sen(2α) senα

2

cos(α+β)

cos(α+β) OG OG=OF ₋GF

• GF =HE GF EH

HDE HE = senαsenβ

HE =GF GF = senαsenβ(4)

DHE

• OF

OF E OF = cosαcosβ(5)

OF E

OG= cosαcosβ₋senαsenβ cos(α+β) = cosαcosβ₋senαsenβ.

cos(α₋β) cos 2α cosα

2

✷✳✽✳ ❚❘■●❖◆❖▼❊❚❘■❆ ◆❆ ❈■❘❈❯◆❋❊❘✃◆❈■❆

✐✐✐✮ tg(α+β)

❆ #❛♥❣❡♥#❡ ♣)♦❝✉)❛❞❛ ❝♦))❡.♣♦♥❞❡ ❛ )❛③0♦ DG

OG =

sen(α+β)

cos (α+β) ♥❛ ❋✐❣✉)❛

✷✳✷✸✳ 5♦)#❛♥#♦✿

tg(α+β) = senαcosβ+ cosαsenβ

cosαcosβ₋senαsenβ

❈♦♠♦cosαcosβ ₆= 0 (0◦ _{< α, β, (α+β)<90}◦_{)✱ ❛tg(α+β): ❡;✉✐✈❛❧❡♥#❡}

❛✿

tg(α+β) =

senαcosβ

cosαcosβ +

cosαsenβ

cosαcosβ

cosαcosβ

cosαcosβ −

senαsenβ

cosαcosβ

,

♦✉ .❡❥❛✱

tg(α+β) =

senα

cosα +

senβ

cosβ

1₋ senα

cosα

senβ

cosβ

.

5♦)#❛♥#♦✱ #❡♠♦.✿

tg(α+β) = tgα+ tgβ

1₋tgαtgβ

◆♦#❡ ;✉❡ ❛ ♣❛)#✐) ❞❡..❛ ❢A)♠✉❧❛✱ ♣♦❞❡♠♦. ❡♥❝♦♥#)❛) ♦✉#)❛. ❝♦♠♦

tg(α₋β)✱ tg 2α✱tgα

2 ❡ ❡#❝✳

✷✳✽✳✻ ❋✉♥'(❡* ❙❡♥♦✱ ❈♦**❡♥♦✱ ❚❛♥❣❡♥2❡ ♥♦ ❈❛♠♣♦ ❘❡❛❧ ❡

❙✉❛* ■♥✈❡9*❛*

❉❡✜♥✐%&♦ ✷✳✶✸ ❉❛❞♦ ✉♠ ♥'♠❡)♦ )❡❛❧ x✱ ,❡❥❛ ♦ ♣♦♥/♦ P(cosx,senx) ,✉❛ ✐♠❛❣❡♠ ♥❛ ❝✐)❝✉♥❢❡)4♥❝✐❛ /)✐❣♦♥♦♠5/)✐❝❛ ✭✈❡) ❋✐❣✉)❛✷✳✷✾✮✳

❉❡♥♦♠✐♥❛)❡♠♦. ❢✉♥C0♦ .❡♥♦✱ ❛ ❢✉♥C0♦ f : R _→ R _{#❛❧ ;✉❡} f(x) = senx. 5❛)❛ ❛

❢✉♥C0♦ f :R_→R _{#❛❧ ;✉❡} f(x) = cosx ❞❡♥♦♠✐♥❛)❡♠♦. ❢✉♥C0♦ ❝♦..❡♥♦✳

❈♦♥.✐❞❡)❡♠♦. ❛❣♦)❛ ♦ ❡✐①♦ ❞❛. #❛♥❣❡♥#❡. ✭✈❡) ❋✐❣✉)❛✷✳✸✵✮✳

OT : ♦ ♣)♦❧♦♥❣❛♠❡♥#♦ ❞♦ .❡❣♠❡♥#♦ OP ❛#: #♦❝❛) ♦ ❡✐①♦ ❞❛. #❛♥❣❡♥#❡. ♥♦

♣♦♥#♦ T✳ ✉♠❛ ✈❡③ ;✉❡ AT = tgx ♣❛)❛ x ₆= π

2 + kπ ✭k ✐♥#❡✐)♦✮✱ ❞❡✜♥✐)❡♠♦. ❛ ❢✉♥C0♦ #❛♥❣❡♥#❡ ❝♦♠♦ ❛ ❢✉♥C0♦ f : D _→ R _{#❛❧ ;✉❡} f(x) = tgx ♦♥❞❡ D =

n

x_∈R_|x₆= π

2 +kπ, k ∈Z

o

.

O

t

[0

,

2

π

]

.

2

π

90

π

2

180

π

270

3

π

2

P = 2π;

f :R_→R f(x) = cosx.

f :nx_∈R_|x₆= π

2 +kπ, k∈Z

o

→R f(x) = tgx

P =π;

f : R _→ R f(x) = senx

x _∈ R senx = 3. f :

h −π₂,π

2

i

→[₋1,1] f(x) = senx

−π

2,

π

2

→[₋1,1]

f f−1

f

f−1 _[−1,1] h₋π

2,

π

2

i

x_∈[₋1,1]

y_∈h₋π

2,

π

2

i

y= arcsenx_⇔seny =x y_∈h₋π

2,

π

2

i

f :R_→R f(x) = cosx ∄x_∈R x =

4 f : [0, π]_→[₋1,1] f(x) = cosx

[0, π]_→[₋1,1]

f f−1

[₋1,1] f−1 _{[0, π]}

x_∈[₋1,1] y_∈[0,₋π] y

f : _{x _∈ R_|x ₆= π

2 +kπ, k ∈ Z} → R f(x) = tgx

0◦ _{6= π tg 0}◦ _{= tgπ f :} i₋π

2,

π

2

h → R

f(x) = tgx

−π

2,

π

2

f f−1

R f−1_, i₋π

2,

π

2

h

x _∈ R

y_∈i₋π

2,

π

2

h

y x

y= arctgx_⇐⇒tgy=x y_∈i₋π

2,

π

2

h

ex _(x∈R)

(x) =ex = 1 +x+x

2

2! +

x3

3! +· · ·+

xn

n! +. . .

✷✳✾✳ ❊❳%✭❳✮ ❊ ❘❊▲❆➬➹❖ ❉❊ ❊❯▲❊❘

✷✳✾✳✷ ❘❡❧❛'(♦ ❞❡ ❊✉❧❡- ♣❛-❛

x

❘❡❛❧

ei·x = cosx+i_·senx

❡ ♣❛#❛

e−i·x = cos(₋x) +i_·sen(₋x).

❈♦♠♦ ❛ ❢✉♥*+♦ ❝♦--❡♥♦ . ♣❛# ❡ ❛ ❢✉♥*+♦ -❡♥♦ . /♠♣❛#✱ ✈❡♠✿

e−i·x= cos(x)₋i_·sen(x).

✷✳✾✳✸ ❙❡♥♦ ❡ ❈♦33❡♥♦ ❈♦♠♣❧❡①♦

(1) ei·x = cosx+i_·senx (x_∈R)

❝♦♠♦✿

(2) e−i·x= cos(x)₋i_·sen(x) (x_∈R).

❋❛③❡♥❞♦✿

(1) + (2) ❡(1)₋(2)✱ -❡❣✉❡✲-❡ 8✉❡✿

❉❡ (1) + (2)✱ ✈❡♠✿ cosx= e

i·x_+e−i·x

2

❉❡ (1)₋(2)✱ ✈❡♠✿ senx= e

i·x

−e−i·x

2_·i .

❛"❛ ❝❛❞❛ z =x+y_·i ❝♦♠ x ❡ y ❘❡❛✐*✱ ❞❡✜♥❛♠♦*✿

        

cos(z) = e

i·z_+e−i·z

2 ❡

sen(z) = e

i·z_−e−i·z

2_·i

✱❝♦♠ ♦ *❡♥♦ ❡ ❝♦**❡♥♦* ❝♦♠♣❧❡①♦*.

✷✳✾✳✹ ◆♦&'❡) *❛,❛ ❙.,✐❡ ❞❡ ❋♦✉,✐❡,

❉❡✜♥✐%&♦ ✷✳✶✺ ✭❙"#✐❡ ❞❡ ❋✉♥*+❡,✮ ❯♠❛ ❙"#✐❡ P∞

m=0

fm✱ ✭♠ ♥❛2✉#❛❧✮✱ ♥❛ 4✉❛❧ ❝❛❞❛

fm " ✉♠❛ ❢✉♥*7♦ ❞❡♥♦♠✐♥❛#❡♠♦, ❙"#✐❡ ❞❡ ❢✉♥*+❡,✳

❊①❡♠♣❧♦✿

❛✮ P∞

n=0

xn _{✭n ♥❛3✉"❛❧✮✱ x ✈❛"✐7✈❡❧ "❡❛❧✱ ❝♦♠ |x|<1}

❜✮ P∞

k=1

1

x2_+k2 ✭k ♥❛3✉"❛❧✮✱ x✈❛"✐7✈❡❧ "❡❛❧✳

✷✳✾✳ ❊❳%✭❳✮ ❊ ❘❊▲❆➬➹❖ ❉❊ ❊❯▲❊❘ ❝✮ a0 2 + ∞ P n=1

[ancos(nx) +bnsen(nx)]✱an❡bn♥#♠❡%♦' %❡❛✐' ❞❛❞♦' ❡n♥❛+✉%❛❧ (n ≥1)

✭❙0%✐❡ +%✐❣♦♥♦♠0+%✐❝❛✮

⋄

❖❜$❡&✈❛)*♦ ✷✳✻ ❉✐③❡♠♦& '✉❡ ❛ ❙+,✐❡ ❞❡ ❢✉♥01❡& P∞

m=0

fm ❝♦♥✈❡%❣❡ ✉♥✐❢♦%♠❡♠❡♥+❡✱

❡♠ ✉♠ ❞♦♠.♥✐♦ D✱ / ❢✉♥01♦ t:D_→R 2❡✱ ♣❛%❛ +♦❞♦ ǫ >0 ❞❛❞♦✱ ❡①✐2+✐% ✉♠ ♥❛+✉%❛❧

n0 +❛❧ 7✉❡✱ ♣❛%❛ +♦❞♦ x∈ D✱ n > n0 ⇒

∞ P m=0

fm−t(x)

< ǫ. ◆1♦ ❞❡✈❡♠♦2 ❡27✉❡❝❡%

❞❛ ❡①✐2+9♥❝✐❛ ❞❡ ❝%✐+:%✐♦2 ❝♦♠♦ ❞❡ ❈❛✉❝❤② ❡ ♦ ❞❡ ❲❡✐❡%2+%❛22✱ ❜❛2+❛♥+❡ ✉+✐❧✐③❛❞♦2 ♣❛%❛ ❞❡❝✐❞✐% 2❡ ✉♠❛ ❝♦♥✈❡%❣9♥❝✐❛ : ✉♥✐❢♦%♠❡ ♦✉ ♥1♦✳

❉❡✜♥✐%&♦ ✷✳✶✻ ✭❈♦❡✜❝✐❡♥+❡2 ❞❡ ❋♦✉%✐❡% ❡♠ ✉♠❛ ❋✉♥01♦ g✮ ❙❡♥❞♦ g : [₋π, π]_→R

❡ g ✐♥+❡❣%G✈❡❧ ♦2 ♥H♠❡%♦2 a0 = 1

π

Z π

−π

g(x)dx✱ an=

1

π

Z π

−π

g(x) cos(nx)dx♣❛%❛ n _≥1

❡ bn =

1

π

Z π

−π

g(x) sen(nx)dx ♣❛%❛ n _≥1 ✭n ♥❛+✉%❛❧✮✱ 21♦ ❞❡♥♦♠✐♥❛❞♦2 ❝♦❡✜❝✐❡♥+❡2

❞❡ ❋♦✉%✐❡% ❞❡ g✳

❉❡✜♥✐%&♦ ✷✳✶✼ ✭❙:%✐❡ ❞❡ ❋♦✉%✐❡% ❞❡ ✉♠❛ ❢✉♥01♦ g✮ ❆ ❙:%✐❡ a0

2 +

∞ P

n=0

[ancos(nx) +

bnsen(nx)]♦♥❞❡ an ❡ bn 21♦ ❝♦❡✜❝✐❡♥+❡2 ❞❡ ❋♦✉%✐❡% ❞❡ g✱ : ❞✐+❛ ❙:%✐❡ ❞❡ ❋♦✉%✐❡% ❞❡

g✳

❖❜/❡0✈❛%&♦ ✷✳✼ ▲❡♠❜%❛%❡♠♦2 ♥❡2+❡ ♠♦♠❡♥+♦ ❞❡ ❝❡%+❛2 ✐♥+❡❣%❛✐2 ❝♦♠ ✉2♦ ❢%❡7✉❡♥+❡ ♥❛2 ❙:%✐❡2 ❞❡ ❋♦✉%✐❡%✱ ❜❡♠ ❝♦♠♦✱ ❛ %❡❣%❛ ❞❡ ✐♥+❡❣%❛01♦ ♣♦% ♣❛%+❡2✳

• Z

cos(nx)dx = sen(nx)

n +w ✭w❝♦♥$%❛♥%❡ (❡❛❧✱ n≥1 ♥❛%✉(❛❧✮

• Z

sen(nx)dx =₋cos(nx)

n +w

• Z

f(x)_·g‘(x) dx=f(x)_·g(x)₋

Z

f‘(x)_·g(x)dx

❆❞♠✐%❛♠♦$ ❛$ ❢✉♥23❡$ f ❡ g ❞❡(✐✈5✈❡✐$✱ ❝♦♥%6♥✉❛$ ❡ ✐♥%❡❣(5✈❡✐$ ❡♠ ✉♠ ✐♥%❡(✈❛❧♦ ■✳

❈❛♣#$✉❧♦ ✸

❆♣❧✐❝❛,-❡/

❈♦♥✈✐✈❡&❡♠♦( ♥❡()❡ ❝❛♣-)✉❧♦ ❝♦♠ (✐)✉❛01❡( ❝✉❥♦ ❞❡(❡♥✈♦❧✈✐♠❡♥)♦ ❞❡♣❡♥❞❡&4 ❞♦ ✉(♦ ❞❡ ✉♠❛ ♦✉ ♠❛✐( ❞❛( ❢✉♥❞❛♠❡♥)❛01❡( )❡6&✐❝❛( ✈✐()❛ ♥♦ ❝❛♣-)✉❧♦ ❛♥)❡&✐♦&✳

✸✳✶ ❆$❝♦ '♦♠❛ ❡ ❢✉♥./♦ 0$✐❣♦♥♦♠30$✐❝❛ ✐♥✈❡$'❛

✵✶✮ ❯)✐❧✐③❛♥❞♦ ❛ ❢6&♠✉❧❛ tg(x +y) = tgx+ tgy

1₋tgx_·tgy ❡ ❛ ❡:✉✐✈❛❧;♥❝✐❛ arctgy = x _⇔ tgx = y ✭x ❡ y &❡❛✐( ❡ ❞✐❢❡&❡♥)❡( ❞❡ π

2 +kπ, k ✐♥)❡✐&♦✮✳ ❱❡&✐✜:✉❡ :✉❡

π

4 = 3 arctg

1

5+ arctg

9

46✳

❙♦❧✉'(♦✿ ✸✳✶ ❙❡❥❛ π

4 = arctg

1

5 + arctga.

❋❛③❡♥❞♦✿

x = arctg1

5 ❡

y = arctga

⇐⇒

tgx = 1

5 ❡

tgy = a

(_∗)

♦"#❛♥#♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡✿ π

4 = arctg

1

5+ arctga✱ ✈❡♠✿

π

4 =x+y⇒tg

π

4 = tg(x+y)✳ ❈♦♠♦ tg(x+y) =

tgx+ tgy

1₋tgx_·tgy ❡ tg

π

4 = 1✱

2❡❣✉❡✲2❡ 4✉❡✿

1 = tgx+ tgy

1₋tgx_·tgy ⇒tgx+tgy= 1−tgx·tgy✳ ▼❛2✱ ♣♦"(∗) tgx=

1

5 ❡ tgy=

a✳ ❙✉❜2#✐#✉✐♥❞♦✱ ❱❡♠✿

1

5+a= 1−

1

5a⇒1 + 5a= 5−a⇒a=

2 3✳

✸✳✶✳ ❆❘❈❖ ❙❖▼❆ ❊ ❋❯◆➬➹❖ ❚❘■●❖◆❖▼➱❚❘■❈❆ ■◆❱❊❘❙❆

▲♦❣♦✱ ❝♦♠♦✿ π

4 = arctg

1

5 + arctga⇒

π

4 = arctg

1

5+ arctg

2

3 ✭❆✜*♠❛,-♦ ✶✮

❙❡❥❛✿

arctg2

3 = arctg

1

5 + arctgb.

❋❛③❡♥❞♦✿

x = arctg1

5 ❡

t = arctgb

⇐⇒

tgx = 1

5 ❡

tgt = b

,t_∈R ❡ t ₆= π

2 +kπ, k ∈Z

(_∗∗).

7♦*8❛♥8♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡✿ arctg2

3 = arctg

1

5 + arctgb✱ ✈❡♠✿

arctg2

3 =x+t⇐⇒tg(x+t) =

2 3. ❉❛>✿

tgx+ tgt

1₋tgx_·tgt =

2

3 ⇒ 3 tgx+ 3 tgt = 2−2 tgx·tgt. ▼❛@✱ ♣♦* (∗∗) tgx =

1

5 ❡ tgt =b. ❙✉❜@8✐8✉✐♥❞♦✱ ✈❡♠✿ 3

5+ 3·b= 2−

2

5 ·b⇒3 + 15·b= 10−2·b ⇒b =

7

17.

▲♦❣♦✱ ❝♦♠♦✿ π

4 = arctg

1

5 + arctg

2

3 ✭❆✜*♠❛,-♦ ✶✮✳

⇒ π₄ = arctg1

5 + arctg

1

5+ arctg

7

17 ⇒

π

4 = 2 arctg

1

5 + arctg

7

17 ✭❆✜*♠❛,-♦

✷✮✳ ❙❡❥❛✿

arctg 7

17 = arctg

1

5 + arctgc.

❋❛③❡♥❞♦✿

x = arctg1

5 ❡

w = arctgc

⇐⇒

tgx = 1

5 ❡

tgw = c

,w_∈R _❡ w₆= π

2 +kπ, k ∈Z

(_{∗ ∗ ∗}).

7♦*8❛♥8♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡✿ arctg 7

17 = arctg

1

5 + arctgc, ✈❡♠✿

arctg 7

17 =x+w⇐⇒tg(x+w) =

7

17.

❉❛>✿

✸✳✷✳ ❊◗❯❆➬(❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙ ◆❖ ❈❆▼3❖ ❈❖▼3▲❊❳❖

tgx+ tgw

1₋tgx_·tgw =

7

17 ⇒ 17 tgx + 17 tgw = 7−7 tgx·tgw. ▼❛"✱ ♣♦& (∗ ∗

∗) tgx= 1

5 ❡ tgw=c. ❙✉❜"+✐+✉✐♥❞♦✱ ✈❡♠✿ 17

5 + 17c= 7−

7

5c⇒17 + 85c= 35−7c⇒c=

9

46.

▲♦❣♦✱ ❝♦♠♦ π

4 = 2 arctg

1

5 + arctg

7

17 ✭❆✜&♠❛89♦ ✷✮

⇒ π₄ = 2 arctg1

5 + arctg

1

5+ arctg

9

46 ⇒

π

4 = 3 arctg

1

5+ arctg

9 46

❖❜"❡$✈❛'(♦ ✸✳✶ <&♦❝❡❞❡♥❞♦ ❞❡"+❛ ❢♦&♠❛✱ ❏♦❤♥ ▼❛❝❤✐♠ ✭✶✻✽✵ ✲ ✶✼✺✶✮ ❝❛❧❝✉✲ ❧♦✉ ♣❛&❛ π ✉♠❛ ❛♣&♦①✐♠❛89♦ ❝♦♠ ✶✵✵ ❝❛"❛" ❞❡❝✐♠❛✐"✳ ❖ ♠❡"♠♦ ♣&♦❝❡❞✐♠❡♥+♦

❢♦✐ ✉+✐❧✐③❛❞♦ ♣♦& ❲✐❧❧✐❛♠ ❙❤❛♥❦" ✭✶✽✶✷ ✲ ✶✽✽✷✮ ❈❛❧❝✉❧❛♥❞♦ ♣❛&❛ π ✉♠❛ ❛♣&♦✲

①✐♠❛89♦ ❝♦♠ ✼✵✼ ❝❛"❛" ❞❡❝✐♠❛✐"✳

✸✳✷ ❊$✉❛'(❡* +,✐❣♦♥♦♠2+,✐❝❛* ♥♦ ❝❛♠♣♦ ❝♦♠✲

♣❧❡①♦

✵✷✮ ❱❡"✐✜%✉❡ ♥♦ ❝❛♠♣♦ ❝♦♠♣❧❡①♦ ♦ ♥/♠❡"♦ ❞❡ 1♦❧✉23❡1 ♣❛"❛ ❛ ❡%✉❛24♦ senz = 3✱

♦♥❞❡ z =x+yi (x ❡ y "❡❛✐1).

❙♦❧✉'(♦✿ ✸✳✷ senz = 3

♣♦& ❞❡✜♥✐89♦ "❡♥z = e

iz

−e−iz

2i

✱ ❞❛O✿

eiz_−e−iz

2i = 3⇔e

iz_−e−iz _{= 6i⇔e}iz₋ 1

eiz −6i= 0 ✭♠✉❧+✐♣❧✐❝❛♥❞♦ ❛♠❜♦" ♦"

♠❡♠❜&♦" ♣♦& eiz_{✮✱ ✈❡♠✿}

(eiz₎2 _−6i(eiz_{)−1 = 0 ✭❢❛③❡♥❞♦ e}iz _{=w✮✱ +❡♠♦"✿ w}2_{−6iw−1 = 0 ✭❡P✉❛89♦}

❞♦ "❡❣✉♥❞♦ ❣&❛✉ ❡♠ ✇✮✳ ▲♦❣♦✿

∆ = (₋6i)2_{−4(1)(−1)}

∆ = 36i2_{+ 4 ✭❝♦♠♦ 4 = −4i}2_✮

∆ = 36i2_−4i2

∆ = 32i2

√

∆ =√2_·16_·i2

√

∆ = 4i√2✳ ❈♦♠♦✿ w= 6i±4i

√

2

2 ⇔w= (3±2

√

2)i.

<♦&R♠✱ ❝♦♠♦ eiz _{=w✱ ✈❡♠✿}

✸✳✸✳ ■❉❊◆❚■❉❆❉❊❙

e(x+yi)i = (3_±2√2)i(z =x+yi) _⇔ exi+yi2 = (3_±2√2)i _⇔ e−y

·exi = (3_± 2√2)i ✭ ♣"♦♣"✐❡❞❛❞❡ ♠✉❧+✐♣❧✐❝❛+✐✈❛✮ _⇔e−y_{(cosx+isenx) = (3}

±2√2)i ✭"❡❧❛/0♦ ❞❡ ❊✉❧❡"✮

❆♥❛❧✐4❛♥❞♦ ❝❛❞❛ ♣♦44✐❜✐❧✐❞❛❞❡

■✮ e−y(cosx+isenx) = (3 + 2√2)i. ◆♦+❡ 8✉❡ ♦ ♥9♠❡"♦ ♥♦ 4❡❣✉♥❞♦ ♠❡♠❜"♦

❞❛ ✐❣✉❛❧❞❛❞❡ ; ■♠❛❣✐♥<"✐♦ ♣✉"♦✱ ❧♦❣♦✿

cosx= 0✱ ✐4+♦ ;✱ x = π

2 + 2kπ ❝♦♠ k∈ Z ❡ e

−y_senπ

2 + 2kπ

= 3 + 2√2_⇔

−y = ln(3 + 2√2)_⇔y =₋ln(3 + 2√2). ▲♦❣♦✿ ♥9♠❡"♦4 ❝♦♠♣❧❡①♦4 ❞❛ ❢♦"♠❛

z =π

2 + 2kπ

−iln(3 + 2√2) ❝♦♠ k _∈Z 4❛+✐4❢❛③❡♠ ❛ ❡8✉❛/0♦ senz = 3.

■■✮ e−y_{(cosx+isenx) = (3− 2}√_{2)i. ◆♦+❡ 8✉❡ x =} π

2 + 2kπ, k ∈ Z ✭♣♦" ♠♦+✐✈♦4 ❛♥<❧♦❣♦ ❛ 4♦❧✉/0♦ ❛♥+❡"✐♦"✮ ❡ e−y_senπ

2 + 2kπ

= 3₋2√2✱ ❞❛C✿

−y = ln(3₋2√2)_⇔y =₋ln(3₋2√2). ▲♦❣♦✿ ♥9♠❡"♦4 ❝♦♠♣❧❡①♦4 ❞❛ ❢♦"♠❛

z = π

2 + 2kπ

− iln(3 ₋2√2) ❝♦♠ k _∈ Z _{4❛+✐4❢❛③❡♠ ❛ ❡8✉❛/0♦} senz = 3. D♦"+❛♥+♦✱ ♥♦ ❝❛♠♣♦ ❞♦4 ♥9♠❡"♦4 ❝♦♠♣❧❡①♦4✱ ❛ ❡8✉❛/0♦ senz = 3 ♣♦44✉✐ ✐♥✜♥✐+❛4 4♦❧✉/F❡4✳

✸✳✸ ■❞❡♥&✐❞❛❞❡)

✵✸✮ ❯!❛♥❞♦ ❛&❣✉♠❡♥+♦! +&✐❣♦♥♦♠-+&✐❝♦!✱ ♣&♦✈❡ ❝❛❞❛ ✐❞❡♥+✐❞❛❞❡ ❛❜❛✐①♦ ♣❛&❛ m ❡ n

✐♥+❡✐&♦!✿

❛✮ sen(mx) sen(nx) = 1

2[cos(m−n)x−cos(m+n)x]

❜✮ cos(mx) cos(nx) = 1

2[cos(m−n)x+ cos(m+n)x]

❝✮ sen(mx) cos(nx) = 1

2[sen(m+n)x+ sen(m−n)x]

❙♦❧✉*+♦✿ ✸✳✸ ❉❡ ❢❛+♦✿

❉❡ cos(m+n)x= cos(mx) cos(nx)₋sen(mx) sen(nx) (_∗).

❚"♦❝❛♥❞♦ n ♣♦" ₋n ❡ 4❡♥❞♦ ✐♠♣❛" ❛ ❢✉♥/0♦ 4❡♥♦ ❡ ♣❛" ❛ ❢✉♥/0♦ ❝♦44❡♥♦✱

❡4❝"❡✈❡♠♦4✿

cos(m₋n)x= cos(mx) cos(nx) + sen(mx) sen(nx) (_∗∗).

❋❛③❡♥❞♦✿ (_∗∗)₋(_∗) ❡ (_∗∗) + (_∗), ✈❡♠✿

✸✳✹✳ ■●❯❆▲❉❆❉❊❙

cos(m₋n)x₋cos(m+n)x= 2 sen(mx) sen(nx) ❡ cos(m₋n)x+cos(m+n)x=

2 cos(mx) cos(nx). ❉❛#✿

[cos(m₋n)x₋cos(m+n)x]_·1

2 = sen(mx) sen(nx) ❡ [cos(m−n)x+ cos(m+

n)x]_· 1

2 = cos(mx) cos(nx).

❉❡ sen(m+n)x= sen(mx) cos(nx) + sen(nx) cos(mx) (1).

❉❡ ♠♦❞♦ ❛♥)❧♦❣♦ ❛♦ ❡①❡-❝#❝✐♦ ❛♥0❡-✐♦-✱ ❡2❝-❡✈❡♠♦2✿

sen(m₋n)x= sen(mx) cos(nx)₋sen(nx) cos(nx) (2).

❋❛③❡♥❞♦✿ (1) + (2)✱ ✈❡♠✿

sen(m+n)x+ sen(m₋n)x= 2 sen(mx) cos(nx). ❉❛#✿ [sen(m+n)x+ sen(m₋

n)x]_· 1

2 = sen(mx) cos(nx).

✸✳✹ ■❣✉❛❧❞❛❞❡*

✵✹✮ ▼♦"#$❡ &✉❡✿

❛✮

Z π

−π

sen(mx) sen(nx) dx= 0 =

Z π

−π

cos(mx) cos(nx) dx; )❛$❛ m₆=n ✐♥#❡✐$♦"

❜✮ Z π

−π

sen(mx) cos(nx) dx= 0; ♣❛$❛ #♦❞♦ m ❡ n ✭✐♥#❡✐$♦"✮

❝

Z π

−π

sen(mx) sen(nx) dx=π=

Z π

−π

cos(mx) cos(nx) dx ♣❛$❛ m=n ✭✐♥#❡✐$♦✮✳

❙♦❧✉*+♦✿ ✸✳✹ ❛✮ m₆=n✱ ❞❛#✿

Z π

−π

sen(mx) sen(nx) dx= 1

2

sen(m₋n)x

(m₋n) −

sen(m+n)x

(m+n)

π −π Z π −π

sen(mx) sen(nx) dx= 0, ❡✿

Z π

−π

cos(mx) cos(nx) dx= 1

2

Z π

−π

[cos(m₋n)x+ cos(m+n)x] dx ❞❡ ♠♦❞♦ ❛♥)✲

❧♦❣♦✱ 0❡♠♦2✿

Z π

−π

cos(mx) cos(nx)dx= 0.

❜✮ 9-✐♠❡✐-♦ ❝❛2♦ m ₆=n✱ ✈❡♠✿

✸✳✺✳ ❆$▲■❈❆➬➹❖ ✶ $❆❘❆ ❆ ❙➱❘■❊ ❉❊ ❋❖❯❘■❊❘

Z π

−π

sen(mx) cos(nx) dx= 1

2

Z π

−π

[sen(m+n)x+ sen(m₋n)x] dx

Z π

−π

sen(mx) cos(nx) dx= 1

2

−cos(m+n)x

(m+n) +

−cos(m₋n)x

(m₋n)

π −π Z π −π

sen(mx) cos(nx) dx= 0

❙❡❣✉♥❞♦ ❝❛)♦ m=n✱ ✈❡♠✿

Z π

−π

sen(mx) cos(nx) dx=

Z π

−π

sen(2nx)

2 dx

Z π

−π

sen(mx) cos(nx) dx= 1

2

Z π

−π

[sen(2nx) dx

Z π

−π

sen(mx) cos(nx) dx= 1

2

−cos(2_2nnx)

π −π Z π −π

sen(mx) cos(nx) dx= 0.

❝✮ /❛0❛ m=n✱ ✈❡♠✿

Z π

−π

sen(mx) sen(nx) dx=

Z π

−π

sen2(nx) dx ❡

Z π

−π

cos(mx) cos(nx) dx=

Z π

−π

cos2(nx) dx.

❉❛2✿

Z π

−π

sen(mx) sen(nx) dx =

Z π

−π

[1₋cos2(nx)] dx ❡

Z π

−π

cos(mx) cos(nx) dx =

Z π −π 1 2 + 1

2cos(2nx) dx

.

❙❡❣✉❡✲)❡ 4✉❡✿

Z π

−π

sen(mx) sen(nx) dx= 2π₋

π+ 1

2·0

❡ Z π

−π

cos(mx) cos(nx) dx= 2π_·

1 2

/♦05❛♥5♦✿

Z π

−π

sen(mx) sen(nx) dx=π ❡

Z π

−π

cos(mx) cos(nx) dx=π.

✸✳✺ ❆♣❧✐❝❛)*♦ ✶ ♣❛-❛ ❛ ❙/-✐❡ ❞❡

❋♦✉-✐❡-✵✺✮ ❙❡❥❛ f : [₋π, π] _→R _{✉♠❛ ❢✉♥()♦ ✐♥,❡❣./✈❡❧ ,❛❧ 2✉❡} f(x) = ax (a _∈ R∗_{). ❆❝❤❡}

❛ ❙6.✐❡ ❞❡ ❋♦✉.✐❡. ❞❡ f(x).

✸✳✻✳ ❆$▲■❈❆➬➹❖ ✷ $❆❘❆ ❆ ❙➱❘■❊ ❉❊ ❋❖❯❘■❊❘

❙♦❧✉$%♦✿ ✸✳✺ ■✮ ❝♦❡✜❝✐❡♥(❡) ❞❡ ❋♦✉-✐❡-✳

a0 = 1

π

Z π

−π

ax dx ❡ an=

1

π

Z π

−π

ax_·cos(nx) dx

(n_≥1). ❝♦♠♦ ax ❡ ax_·cos(nx) )0♦ ❢✉♥23❡) ✐♠♣❛-❡)✱ (❡♠♦) a0 = 0 ❡ an = 0.

bn=

1

π

Z π

−π

ax_·sen(nx) dx (n _≥1). ❝♦♠♦ ax_·sen(nx) 7 ❢✉♥20♦ ♣❛-✱ (❡♠♦)✿

bn=

2

π

Z π

0

ax_·sen(nx) dx.

■♥(❡❣-❛♥❞♦ ♣♦- ♣❛-(❡)✱ )❡❣✉❡✲)❡ ;✉❡✿

bn=

2

π

"

−ax_n cos(nx)

π 0 + a n Z π 0

cos(nx) dx

#

.

bn=

2

π

"

(₋1)aπ

n

(₋1)n₊ a

n

sen(nx)

n +w

π 0 # .

bn=

2

π

h

(₋1)n+1aπ

n

i

= (₋1)n+12a

n .

▲♦❣♦✱ ❛ ❙7-✐❡ ❞❡ ❋♦✉-✐❡- ♣-♦❝✉-❛❞❛ 7✿

∞ X

n=1

(₋1)n+12a

n ·sen(nx), n≥1.

✸✳✻ ❆♣❧✐❝❛)*♦ ✷ ♣❛-❛ ❛ ❙/-✐❡ ❞❡

❋♦✉-✐❡-✵✻✮ ❯♠❛ ♦♥❞❛ &✉❛❞(❛❞❛ ) ✉♠❛ ❢✉♥+,♦ ♣❡(✐0❞✐❝❛ 2✐♠♣❧❡2 ❞❡ ❢♦(♠❛ ❛♥❛❧45✐❝❛ f(x) =

−1, ₋π _≤x_≤0

1, 0_≤x_≤π ✱f(x+ 2π) =f(x). ❊25❛ ❢♦(♠❛ ❞❡ ♦♥❞❛ ❇92✐❝❛ ) ❡♥❝♦♥✲

5(❛❞❛ ❢(❡&✉❡♥5❡♠❡♥5❡ ♥❛2 9(❡❛2 ❞❛ ❡❧❡5(;♥✐❝❛ ❡ ❞♦ ♣(♦❝❡22❛♠❡♥5♦ ❞❡ 2✐♥❛✐2✱ ❡❧❛ ❛❧5❡(♥❛ (❡❣✉❧❛(♠❡♥5❡ ❡ ✐♥25❛♥5❛♥❡❛♠❡♥5❡ ❡♥5(❡ ❞♦✐2 ♥4✈❡✐2✳ ◆❛ ✜❣✉(❛ ✸✳✶✱ 5❡♠✲2❡ ✉♠❛ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ♦♥❞❛ &✉❛❞(❛❞❛✳

❉❡5❡(♠✐♥❡ ❛ ❙)(✐❡ ❞❡ ❋♦✉(✐❡( ❞❛ ♦♥❞❛ ❡①♣♦25❛ ♥❛ ✜❣✉(❛ ✸✳✶✳

❙♦❧✉$%♦✿ ✸✳✻ ❖) ❝♦❡✜❝✐❡♥(❡) ❞❡ ❋♦✉-✐❡-✿

a0 =

a0 = 1

π

Z 0

−π−

1dx+

Z π

0

1 dx

a0 = 0

an =

1

π

Z π

−π

f(x) cos(nx)dx

an =

1

π

Z 0

−π−

cos(nx)dx+

Z π

0

cos(nx) dx

an = 0

bn=

1

π

Z π

−π

f(x) sen(nx) dx

bn=

1

π

Z 0

−π

−sen(nx) dx+

Z π

0

sen(nx) dx

bn=

1

π



−

−cos(nx)

n 0 −π +

−cos(_nnx)

π 0  

bn=

1

π

−

−_n1 + cos(nπ)

n

+

−cos(_nnπ) + 1

n

bn=

1

π

2

n −

2 cos(nπ)

n

bn=

2

nπ(1−cos(nπ)).

∞ P

n=1

2