Jawab:
= 1/4 (√3 + 1 - √6 + √2)
(sudah tak cek kalkulator sama)
Penjelasan dengan langkah-langkah:
= Cos 165°(Cos 135° + Tan 15°)
= Cos (180°-15) (Cos (180-45°) + Sin (15°)/Cos(15))
#Cos (180-A) = -Cos (A)
= Cos (15) Cos (45) - Sin (15)
= Cos (15) (1/2 √2) - Sin (15)
= (1/2 √2) Cos (15) - Sin (15)
Sin (A-B) = Sin A.cos B - cos A.sin B
Sin 15° = Sin (45 - 30)°
= Sin 45°.cos 30° - cos 45°.sin 30°
= 1/2√2 . 1/2√3 - 1/2√2 . 1/2
= 1/4 (√6 -√2)
Cos (A - B) = Cos A cos B + sin A sin B
Cos 15° = Cos (45 - 30)°
= Cos 45°.cos 30° + Sin 45°.sin 30°
= 1/2√2 . 1/2√3 + 1/2√2 . 1/2
= 1/4 (√6 +√2)
Jadi
= (1/2 √2) (1/4 (√6 +√2)) - 1/4 (√6 -√2)
=1/8 (√(4x3)+ 2) - 1/4 (√6 -√2)
= 1/8 (2√3 + 2 - 2√6 + 2√2)
= 1/4 (√3 + 1 - √6 + √2)
Perhatikan
tan 15°
=> (180° - 165°)
=> - tan 165°
=> - sin 165°/cos 165°
cos 135°
= cos (180° - 45°)
= - cos 45°
= - 1/2 √2
Sehingga :
cos 165° (cos 135 + tan 15°)
=> cos 165° (-1/2 √2 - sin 165°/cos 165°)
=> -1/2 √2 • cos 165° - sin 165°
Perhatikan
cos 165°
=> cos (120° + 45°)
=> cos 120° cos 45° - sin 120° sin 45°
=> cos (180° - 60°) - 1/2 √2 - sin (180° - 60°) • 1/2√2
=> 1/2√2 • (-cos 60°) -1/2√2 • sin 60°
=> 1/2√2 • (-1/2) -1/2√2 • 1/2√3
=> -1/4√2 - 1/4√2 •√3 √3
=> -1/4 √2 (1 + √3)
sin 165°
=> sin (120°+45)°
=> sin 120° cos 45° + sin 120° sin 45°
=> sin (180°-60°)-1/2√2+cos (180°-60°)•1/2√2
=> 1/2√2•sin 60° + 1/2√2•(-cos 60°)
=> 1/2√2•1/2√3+1/2√2•(-1/2)
=> 1/4√2•√3-1/4√2
=> 1/4√2(√3-1)
Sehingga
-1/2√2•cos 165°-sin 165°
=> -1/2√2•(1/4√2(1+√3))-1/4√2(√3-1)
=> 2/8 (1+√3)-(1/4√6-1/4√2)
=> 1/4(1+√3)-(1/4√6-1/4√2)
=> 1/4+1/4√3-1)4√6+1/4√2
=> 1/4 (1 + √3 - √6 + √2)
- ` Jadi, nilai dari cos 165°(cos135° + tan 15°) adalah 1/4 (1 + √3 - √6 + √2)
- `Semoga dapat membantu
