Pertanyaan

Jika sin 2 θ = 2 sin θ ⋅ cos θ dan cos 2 θ = 1 − 2 sin 2 θ , hitunglah: b. sin ( 2 sin − 1 3 2 )

Jika $sin 2 θ = 2 sin θ \cdot cos θ$ dan $cos 2 θ = 1 - 2 sin^{2} θ$ , hitunglah:

b. $sin (2 sin^{- 1} \frac{2}{3})$

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

.

$sin open parentheses 2 sin to the power of negative sign 1 end exponent 2 over 3 close parentheses equals fraction numerator 4 square root of 5 over denominator 9 end fraction$ .

Pembahasan

Misalkan , maka .Jika , maka dapat dibentuk menjadi Untuk , dengan menggunakan pythagoras diperoleh Dengan demikian Jadi, .

Misalkan sin to the power of negative sign 1 end exponent 2 over 3 equals italic theta , maka sin space italic theta equals 2 over 3 . Jika sin space 2 theta equals 2 sin space theta times cos space theta , maka sin open parentheses 2 sin to the power of negative sign 1 end exponent 2 over 3 close parentheses dapat dibentuk menjadi

sin space 2 italic theta equals sin space italic theta middle dot cos space italic theta

Untuk sin space italic theta equals 2 over 3 , dengan menggunakan pythagoras diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space italic theta end cell equals cell depan over miring end cell row blank equals cell 2 over 3 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 2 squared end root end cell row blank equals cell square root of 9 minus sign 4 end root end cell row blank equals cell square root of 5 end cell end table

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space italic theta end cell equals cell samping over miring end cell row blank equals cell fraction numerator square root of 5 over denominator 3 end fraction end cell end table$

Dengan demikian

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 italic theta end cell equals cell 2 sin space italic theta middle dot cos space italic theta end cell row blank equals cell 2 middle dot 2 over 3 middle dot fraction numerator square root of 5 over denominator 3 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 over denominator 9 end fraction end cell end table$

Jadi, $sin open parentheses 2 sin to the power of negative sign 1 end exponent 2 over 3 close parentheses equals fraction numerator 4 square root of 5 over denominator 9 end fraction$ .