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Jika cos ( A − B ) = cos A ⋅ cos B + sin A ⋅ sin B , hitunglah: cos ( sin − 1 ( 4 1 ​ ) − sin − 1 ( 3 1 ​ ) )

Jika , hitunglah:

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Y. Umi

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

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Pembahasan

Diketahui . Misalkan dan . Artinya dan . Dengan menggunakan pythagoras diperoleh Untuk , diperoleh Jadu, terbukti .

Diketahui cos open parentheses sin to the power of negative sign 1 end exponent open parentheses 1 fourth close parentheses minus sign sin to the power of negative sign 1 end exponent open parentheses 1 third close parentheses close parentheses. Misalkan sin to the power of negative 1 end exponent open parentheses 1 fourth close parentheses equals A dan sin to the power of negative 1 end exponent open parentheses 1 third close parentheses equals B. Artinya

sin space A equals 1 fourth dan sin space B equals 1 third. Dengan menggunakan pythagoras diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space italic A end cell equals cell depan over miring end cell row blank equals cell 1 fourth end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row samping equals cell square root of miring squared minus sign depan squared end root end cell row blank equals cell square root of 4 squared minus sign 1 squared end root end cell row blank equals cell square root of 16 minus sign 1 end root end cell row blank equals cell square root of 15 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space B end cell equals cell samping over miring end cell row blank equals cell fraction numerator square root of 15 over denominator 4 end fraction end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space B end cell equals cell depan over miring end cell row blank equals cell 1 third end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row samping equals cell square root of miring squared minus sign depan squared end root end cell row blank equals cell square root of 3 squared minus sign 1 squared end root end cell row blank equals cell square root of 9 minus sign 1 end root end cell row blank equals cell square root of 8 end cell row blank equals cell 2 square root of 2 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space B end cell equals cell samping over miring end cell row blank equals cell fraction numerator 2 square root of 2 over denominator 3 end fraction end cell end table

Untuk cos open parentheses sin to the power of negative sign 1 end exponent open parentheses 1 fourth close parentheses minus sign sin to the power of negative sign 1 end exponent open parentheses 1 third close parentheses close parentheses, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell cos open parentheses sin to the power of negative sign 1 end exponent open parentheses 1 fourth close parentheses minus sign sin to the power of negative sign 1 end exponent open parentheses 1 third close parentheses close parentheses end cell equals cell cos open parentheses italic A minus sign B close parentheses end cell row blank equals cell cos space italic A middle dot cos space B and sin space italic A middle dot sin space B end cell row blank equals cell fraction numerator square root of 15 over denominator 4 end fraction middle dot fraction numerator 2 square root of 2 over denominator 3 end fraction plus 1 fourth middle dot 1 third end cell row blank equals cell fraction numerator 2 square root of 30 plus 1 over denominator 12 end fraction end cell end table

Jadu, terbukti table attributes columnalign right center left columnspacing 0px end attributes row cell cos open parentheses sin to the power of negative sign 1 end exponent open parentheses 1 fourth close parentheses minus sign sin to the power of negative sign 1 end exponent open parentheses 1 third close parentheses close parentheses end cell equals cell fraction numerator 2 square root of 30 plus 1 over denominator 12 end fraction end cell end table.

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