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Solusi Pengerjaan Soal Pelajaran Sulitmu Gratis!

Urutkan

Buktikanlah. b.

Pembahasan Soal:

Ingat tan space x equals fraction numerator sin space x over denominator cos space x end fraction dan rumus selisih sudut pada sinus berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis A minus B right parenthesis end cell equals cell sin space A space cos space B minus cos space A space sin space B end cell end table

Berdasarkan rumus tersebut, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space A space minus space tan space B end cell equals cell fraction numerator sin space A over denominator cos space A end fraction minus fraction numerator sin space B over denominator cos space B end fraction end cell row blank equals cell fraction numerator sin space A over denominator cos space A end fraction cross times fraction numerator cos space B over denominator cos space B end fraction minus fraction numerator sin space B over denominator cos space B end fraction cross times fraction numerator cos space A over denominator cos space A end fraction end cell row blank equals cell fraction numerator sin space A space cos space B over denominator cos space A space cos space B end fraction minus fraction numerator sin space B space cos space A over denominator cos space B space cos space A end fraction end cell row blank equals cell fraction numerator sin space A space cos space B space minus space cos space A space sin space B over denominator cos space A space cos space B end fraction end cell row blank equals cell fraction numerator sin space left parenthesis A minus B right parenthesis over denominator cos space A times space cos space B end fraction space bold left parenthesis bold terbukti bold right parenthesis end cell end table

Jadi, terbukti bahwa  tan space A space minus space tan space B equals fraction numerator sin space left parenthesis A minus B right parenthesis over denominator cos space A times space cos space B end fraction.

0

Roboguru

Buktikanlah. a.

Pembahasan Soal:

Ingat rumus jumlah dan selisih sudut berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis A plus B right parenthesis end cell equals cell cos space A space cos space B minus sin space A space sin space B end cell row cell sin space left parenthesis A minus B right parenthesis end cell equals cell sin space A space cos space B minus cos space A space sin space B end cell end table

Berdasarkan rumus tersebut, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cos space left parenthesis A plus B right parenthesis over denominator sin space left parenthesis A minus B right parenthesis end fraction end cell equals cell fraction numerator cos space A space cos space B space minus space sin space A space sin space B over denominator sin space A space cos space B space minus space cos space A thin space sin space B end fraction end cell row blank equals cell fraction numerator cos space A space cos space B space minus space sin space A space sin space B over denominator sin space A space cos space B space minus space cos space A thin space sin space B end fraction cross times fraction numerator fraction numerator 1 over denominator cos space A space cos space B end fraction over denominator fraction numerator 1 over denominator cos space A space cos space B end fraction end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator cos space A space cos space B space over denominator cos space A space cos space B end fraction end style minus space begin display style fraction numerator sin space A space sin space B over denominator cos space A space cos space B end fraction end style over denominator begin display style fraction numerator sin space A space up diagonal strike cos space B end strike over denominator cos space A space up diagonal strike cos space B end strike end fraction end style space minus space begin display style fraction numerator up diagonal strike cos space A end strike thin space sin space B over denominator up diagonal strike cos space A end strike space cos space B end fraction end style end fraction end cell row blank equals cell fraction numerator 1 minus tan space A times tan space B over denominator tan space A space minus space tan space B end fraction space bold left parenthesis bold terbukti bold right parenthesis end cell end table

Jadi, terbukti bahwa  fraction numerator cos space left parenthesis A plus B right parenthesis over denominator sin space left parenthesis A minus B right parenthesis end fraction equals fraction numerator 1 minus tan space A times tan space B over denominator tan space A space minus space tan space B end fraction.

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Roboguru

Buktikan bahwa:

Pembahasan Soal:

Ingat rumus perkalian trigonometri 2 space cos alpha space cos beta equals cos open parentheses alpha plus beta close parentheses plus cos open parentheses alpha minus beta close parentheses.

Diperoleh perhitungan sebagai berikut 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos 3 theta space cos theta end cell equals cell 1 half open parentheses 2 space cos 3 theta space cos theta close parentheses end cell row blank equals cell 1 half open parentheses cos left parenthesis 3 theta plus theta right parenthesis plus cos left parenthesis 3 theta minus theta right parenthesis close parentheses end cell row blank equals cell 1 half left parenthesis cos 4 theta plus cos 2 theta right parenthesis end cell row blank equals cell 1 half cos 4 theta plus 1 half cos 2 theta end cell row cell cos 6 theta space cos 2 theta end cell equals cell 1 half left parenthesis 2 space cos 6 theta space cos 2 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos left parenthesis 6 theta plus 2 theta right parenthesis plus cos left parenthesis 6 theta minus 2 theta right parenthesis right parenthesis end cell row blank equals cell 1 half left parenthesis cos 8 theta plus cos 4 theta right parenthesis end cell row blank equals cell 1 half cos 8 theta plus 1 half cos 4 theta end cell row cell cos 12 theta space cos 4 theta end cell equals cell 1 half left parenthesis 2 space cos 12 theta space cos 4 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos left parenthesis 12 theta plus 4 theta right parenthesis plus cos left parenthesis 12 theta minus 4 theta right parenthesis right parenthesis end cell row blank equals cell 1 half left parenthesis cos 16 theta plus cos 8 theta right parenthesis end cell row blank equals cell 1 half cos 16 theta plus 1 half cos 8 theta end cell row cell cos 24 theta space cos 8 theta end cell equals cell 1 half left parenthesis 2 space cos 24 theta space cos 8 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos left parenthesis 24 theta plus 8 theta right parenthesis plus cos left parenthesis 24 theta minus 8 theta right parenthesis right parenthesis end cell row blank equals cell 1 half left parenthesis cos 32 theta plus cos 16 theta right parenthesis end cell row blank equals cell 1 half cos 32 theta plus 1 half cos 16 theta end cell end table

Sehingga diperoleh 

begin mathsize 12px style cos 3 theta space cos theta minus cos 6 theta space cos 2 theta plus cos 12 theta space cos 4 theta minus cos 24 theta space cos 8 theta equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 half cos 8 theta plus 1 half cos 4 theta close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 16 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 8 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 half cos 32 theta plus 1 half cos 16 theta close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 4 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 8 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 4 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 16 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 8 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 32 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 16 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 8 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 16 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 8 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 32 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 16 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 16 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 32 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell up diagonal strike 1 half cos 16 theta end strike end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 32 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank theta end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses cos 2 theta minus cos 32 theta close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table end style

ingat rumus jumlah dan selisih trigonometri
table attributes columnalign right center left columnspacing 0px end attributes row cell cos alpha space minus cos beta end cell equals cell negative 2 space sin 1 half open parentheses alpha plus beta close parentheses space sin 1 half open parentheses alpha minus beta close parentheses end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half open parentheses cos 2 theta minus cos 32 theta close parentheses end cell row blank equals cell 1 half open parentheses negative 2 sin 1 half open parentheses 2 theta plus 32 theta close parentheses space sin 1 half open parentheses 2 theta minus 32 theta close parentheses close parentheses end cell row blank equals cell 1 half open parentheses negative 2 sin 1 half open parentheses 34 theta close parentheses space sin 1 half open parentheses negative 30 theta close parentheses close parentheses end cell row blank equals cell 1 half open parentheses negative 2 sin 17 theta space sin open parentheses negative 15 theta close parentheses close parentheses end cell row blank equals cell negative sin 17 theta space sin open parentheses negative 15 theta close parentheses end cell row blank equals cell negative sin 17 theta minus space sin 15 theta end cell row blank equals cell sin 17 theta space sin 15 theta end cell end table

Dengan demikian, terbukti bahwa cos 3 theta space cos theta minus cos 6 theta space cos 2 theta plus cos 12 theta space cos 4 theta minus cos 24 theta space cos 8 theta equals sin 17 theta space sin 15 theta

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Roboguru

Diketahui matriks . Tentukan: b. Elemen-elemen kolom ke-1 dan kolom ke-2.

Pembahasan Soal:

Elemen kolom ke-1 adalah begin mathsize 14px style 1 comma space 4 comma space 0 end style.

Elemen kolom ke-2 adalah begin mathsize 14px style 3 comma space 6 comma space 7 end style.

Dengan demikian, elemen-elemen kolom ke-1 dan kolom ke-2 berturut-turut adalah begin mathsize 14px style 1 comma space 4 comma space 0 end style dan begin mathsize 14px style 3 comma space 6 comma space 7 end style.

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Roboguru

Diketahui  dan . Tentukan nilai trigonometri berikut. c.

Pembahasan Soal:

Ingat rumus trigonometri berikut.

sin space theta over 2 equals plus-or-minus square root of fraction numerator 1 minus cos space theta over denominator 2 end fraction end root

cos space theta over 2 equals plus-or-minus square root of fraction numerator 1 plus cos space theta over denominator 2 end fraction end root

Jika diketahui tan space theta equals negative 1 half dan fraction numerator 3 straight pi over denominator 4 end fraction less or equal than theta less or equal than straight pi, maka dapat digambarkan sudut theta sebagai berikut.

Diperoleh nilai cos space theta equals negative 2 over 5 square root of 5 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space begin display style theta over 2 end style over denominator cos space begin display style theta over 2 end style end fraction end cell equals cell fraction numerator plus-or-minus square root of begin display style fraction numerator 1 minus cos space theta over denominator 2 end fraction end style end root over denominator plus-or-minus square root of begin display style fraction numerator 1 plus cos space theta over denominator 2 end fraction end style end root end fraction end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus cos space theta over denominator 1 plus cos space theta end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus open parentheses negative begin display style 2 over 5 end style square root of 5 close parentheses over denominator 1 plus open parentheses negative 2 over 5 square root of 5 close parentheses end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 plus begin display style 2 over 5 end style square root of 5 over denominator 1 minus begin display style 2 over 5 end style square root of 5 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator begin display style fraction numerator 5 plus 2 square root of 5 over denominator 5 end fraction end style over denominator begin display style fraction numerator 5 minus 2 square root of 5 over denominator 5 end fraction end style end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 5 plus 2 square root of 5 over denominator 5 minus 2 square root of 5 end fraction end root end cell end table

Dengan demikian, nilai table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space begin display style theta over 2 end style over denominator cos space begin display style theta over 2 end style end fraction end cell equals cell plus-or-minus square root of fraction numerator 5 plus 2 square root of 5 over denominator 5 minus 2 square root of 5 end fraction end root end cell end table

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Roboguru

Diketahui  dan . Tentukan nilai trigonometri berikut. b.

Pembahasan Soal:

Ingat rumus trigonometri sudut rangkap berikut.

sin space 2 theta equals 2 times sin space theta times cos space theta

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 theta end cell equals cell cos squared theta minus sin squared theta end cell row blank equals cell 1 minus 2 space sin squared theta end cell row blank equals cell 2 space cos squared theta minus 1 end cell end table

Jika diketahui tan space theta equals negative 1 half dan fraction numerator 3 straight pi over denominator 4 end fraction less or equal than theta less or equal than straight pi, maka dapat digambarkan sudut theta sebagai berikut.

Diperoleh nilai sin space theta equals 1 fifth square root of 5 dan cos space theta equals negative 2 over 5 square root of 5 sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 theta end cell equals cell 2 times sin space theta times cos space theta end cell row blank equals cell 2 times 1 fifth square root of 5 times open parentheses negative 2 over 5 square root of 5 close parentheses end cell row blank equals cell negative 4 over 5 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 theta end cell equals cell cos squared theta minus sin squared theta end cell row blank equals cell open parentheses negative 2 over 5 square root of 5 close parentheses squared minus open parentheses 1 fifth square root of 5 close parentheses squared end cell row blank equals cell 4 over 5 minus 1 fifth end cell row blank equals cell 3 over 5 end cell end table

Dapat ditentukan nilai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 theta plus cos space 2 theta end cell equals cell negative 4 over 5 plus 3 over 5 end cell row blank equals cell negative 1 fifth end cell end table

Dengan demikian, nilai table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 theta plus cos space 2 theta end cell equals cell negative 1 fifth end cell end table 

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Roboguru

Perhatikan gambar berikut. Nyatakan vektor-vektor berikut dalam . a.

Pembahasan Soal:

Gunakan penjumlahan vektor dengan metode segitiga.

begin mathsize 14px style AE with rightwards arrow on top equals AD with rightwards arrow on top plus DE with rightwards arrow on top AE with rightwards arrow on top equals straight v with rightwards arrow on top plus 1 half straight u with rightwards arrow on top end style

Dengan demikian, begin mathsize 14px style AE with rightwards arrow on top equals straight v with rightwards arrow on top plus 1 half straight u with rightwards arrow on top end style.

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Roboguru

Dengan menjabarkan bentuk aljabar dari integral, temukan hasil integrasi di bawah ini.

Pembahasan Soal:

Rumus integral adalah sebagai berikut:

integral a x to the power of n space d x equals fraction numerator a over denominator n plus 1 end fraction x to the power of n plus 1 end exponent plus c 

Penyelesaiannya adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell integral open parentheses 1 minus x cubed close parentheses squared space d x end cell equals cell integral open parentheses 1 minus x cubed close parentheses open parentheses 1 minus x cubed close parentheses space d x end cell row blank equals cell integral open parentheses 1 minus 2 x cubed plus x to the power of 6 close parentheses space d x end cell row blank equals cell x minus fraction numerator 2 over denominator 3 plus 1 end fraction x to the power of 3 plus 1 end exponent plus fraction numerator 1 over denominator 6 plus 1 end fraction x to the power of 6 plus 1 end exponent plus c end cell row blank equals cell x minus 2 over 4 x to the power of 4 plus 1 over 7 x to the power of 7 plus c end cell row blank equals cell x minus 1 half x to the power of 4 plus 1 over 7 x to the power of 7 plus c end cell end table 

Jadi, hasil integrasi begin mathsize 14px style integral open parentheses 1 minus x cubed close parentheses squared space d x end style adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell x minus 1 half x to the power of 4 plus 1 over 7 x to the power of 7 plus c end cell end table.

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Roboguru

Buktikan bahwa:

Pembahasan Soal:

Ingat rumus perkalian trigonometri 
2 space sin alpha space sin beta equals cos open parentheses alpha minus beta close parentheses minus cos open parentheses alpha plus beta close parentheses.

Diperoleh perhitungan sebagai  berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell sin 3 theta space sin theta space end cell equals cell 1 half open parentheses 2 space sin 3 theta space sin theta close parentheses end cell row blank equals cell 1 half open parentheses cos open parentheses 3 theta minus theta close parentheses minus cos open parentheses 3 theta plus theta close parentheses close parentheses end cell row blank equals cell 1 half open parentheses cos 2 theta minus cos 4 theta close parentheses end cell row blank equals cell 1 half cos 2 theta minus 1 half cos 4 theta end cell row cell sin 6 theta space sin 2 theta space end cell equals cell 1 half left parenthesis 2 space sin 6 theta space sin 2 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos open parentheses 6 theta minus 2 theta close parentheses minus cos open parentheses 6 theta plus 2 theta close parentheses right parenthesis end cell row blank equals cell 1 half left parenthesis cos 4 theta minus cos 8 theta right parenthesis end cell row blank equals cell 1 half cos 4 theta minus 1 half cos 8 theta end cell row cell sin 12 theta space sin 4 theta space end cell equals cell 1 half left parenthesis 2 space sin 12 theta space sin 4 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos open parentheses 12 theta minus 4 theta close parentheses minus cos open parentheses 12 theta plus 4 theta close parentheses right parenthesis end cell row blank equals cell 1 half left parenthesis cos 8 theta minus cos 16 theta right parenthesis end cell row blank equals cell 1 half cos 8 theta minus 1 half cos 16 theta end cell row cell sin 24 theta space sin 8 theta space end cell equals cell 1 half left parenthesis 2 space sin 24 theta space sin 8 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos open parentheses 24 theta minus 8 theta close parentheses minus cos open parentheses 24 theta plus 8 theta close parentheses right parenthesis end cell row blank equals cell 1 half left parenthesis cos 16 theta minus cos 32 theta right parenthesis end cell row blank equals cell 1 half cos 16 theta minus 1 half cos 32 theta end cell row cell sin 48 theta space sin 16 theta space end cell equals cell 1 half left parenthesis 2 space sin 48 theta space sin 16 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos open parentheses 48 theta minus 16 theta close parentheses minus cos open parentheses 48 theta plus 16 theta close parentheses right parenthesis end cell row blank equals cell 1 half left parenthesis cos 32 theta minus cos 64 theta right parenthesis end cell row blank equals cell 1 half cos 32 theta minus 1 half cos 64 theta end cell row cell sin 96 theta space sin 32 theta space end cell equals cell 1 half left parenthesis 2 space sin 96 theta space sin 32 theta right parenthesis end cell row blank equals cell 1 half left parenthesis cos open parentheses 96 theta minus 32 theta close parentheses minus cos open parentheses 96 theta plus 32 theta close parentheses right parenthesis end cell row blank equals cell 1 half left parenthesis cos 64 theta minus cos 128 theta right parenthesis end cell row blank equals cell 1 half cos 64 theta minus 1 half cos 128 theta end cell end table

Sehingga diperoleh

begin mathsize 12px style sin 3 theta space sin theta plus sin 6 theta space sin 2 theta plus sin 12 theta space sin 4 theta plus sin 24 theta space sin 8 theta plus sin 48 theta space sin 16 theta plus sin 96 theta space sin 32 theta equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 2 theta minus 1 half cos 4 theta end cell end table plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 4 theta minus 1 half cos 8 theta end cell end table plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 8 theta minus 1 half cos 16 theta end cell end table plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 16 theta minus 1 half cos 32 theta end cell end table plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 32 theta minus 1 half cos 64 theta end cell end table plus table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 64 theta minus 1 half cos 128 theta end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 2 theta minus 1 half cos 128 theta end cell end table equals 1 half open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos 2 theta minus cos 128 theta end cell end table close parentheses end style

Ingat rumus jumlah dan selisih trigonometri 
cos alpha minus cos beta equals 2 space sin 1 half open parentheses alpha plus beta close parentheses space sin 1 half open parentheses alpha minus beta close parentheses.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half open parentheses size 12px cos size 12px 2 size 12px theta size 12px minus cos 128 theta close parentheses end cell row blank equals cell 1 half open parentheses negative 2 space sin 1 half open parentheses size 12px 2 size 12px theta plus 1 size 12px 28 size 12px theta close parentheses space sin 1 half open parentheses size 12px 2 size 12px theta minus 1 size 12px 28 size 12px theta close parentheses close parentheses end cell row blank equals cell 1 half open parentheses negative 2 space sin 1 half open parentheses size 12px 130 size 12px theta close parentheses space sin 1 half open parentheses negative 1 size 12px 26 size 12px theta close parentheses close parentheses end cell row blank equals cell 1 half open parentheses negative 2 space sin 65 theta space sin begin mathsize 12px style left parenthesis negative 63 theta right parenthesis end style close parentheses end cell row blank equals cell negative sin 65 theta space sin begin mathsize 12px style left parenthesis negative 63 theta right parenthesis end style end cell row blank equals cell negative sin 65 theta space minus sin begin mathsize 12px style 63 theta end style end cell row blank equals cell sin 65 theta space sin begin mathsize 12px style 63 theta end style end cell row blank blank blank end table

Dengan demikian, terbukti bahwa sin 3 theta space sin theta plus sin 6 theta space sin 2 theta plus sin 12 theta space sin 4 theta plus sin 24 theta space sin 8 theta plus sin 48 theta space sin 16 theta plus sin 96 theta space sin 32 theta equals sin 65 theta space sin 63 theta

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Diketahui  dan . Tentukan nilai trigonometri berikut. a.

Pembahasan Soal:

Ingat rumus trigonometri sudut rangkap berikut.

tan space 2 theta equals fraction numerator 2 space tan space theta over denominator 1 minus tan squared theta end fraction

Jika diketahui tan space theta equals negative 1 half dan fraction numerator 3 straight pi over denominator 4 end fraction less or equal than theta less or equal than straight pi, maka dapat ditentukan nilai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space 2 theta end cell equals cell fraction numerator 2 space tan space theta over denominator 1 minus tan squared theta end fraction end cell row blank equals cell fraction numerator 2 open parentheses negative begin display style 1 half end style close parentheses over denominator 1 minus open parentheses negative begin display style 1 half end style close parentheses squared end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 1 minus begin display style 1 fourth end style end fraction end cell row blank equals cell fraction numerator negative 1 over denominator begin display style 3 over 4 end style end fraction end cell row blank equals cell negative 4 over 3 end cell end table

Dengan demikian, nilai dari table attributes columnalign right center left columnspacing 0px end attributes row cell tan space 2 theta end cell equals cell negative 4 over 3 end cell end table

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