Rumus Rumus Trigono metri Tanpa Tabel dan Kalkulator

1. 2 Sin phi/8 Cos phi/8 
2. 1/2 - Sin^2 15* ( * = derajat ) 
3. Cos^2 phi/12 - Sin^2 phi/12 
4. 2 Cos^2 67 1/2 derajat - 1 
5. 2 tan phi/8 per 1 - tan^2 phi/8 
6. Sin 15* cos 15* 
7. Cos^4 5/12 phi - Sin^4 5/12 phi 

 Cara Pertama

• ingat rumus sin2X = 2.sinX.cosX 

• maka 2.sin pi/8. cos pi/8 = sin 2.pi/8 = sin pi/4 = sin 45* = (1/2)√2

• ingat rumus cos 2X = 1 - 2sin²X maka
• (1/2) cos 2X = (1/2) - sin²X
• (1/2) cos 2.15* = (1/2) - sin² 15*
• jadi (1/2) - sin² 15* = (1/2). cos 30* = (1/2) (1/2) √2 = (1/4)√3
• ingat rumus cos 2X = cos² X - sin² X
• maka cos² pi/12 - sin² pi/12 = cos 2.pi/12 = cos pi/6 = cos 30* = (1/2)√3
• ingat rumus cos 2X = 2cos² X - 1
• maka 2.cos² 67 1/2* -1 = cos 2.(67 1/2)* = cos 135* = cos (180* - 45*) = -cos45* = -(1/2)√2
• ingat rumus tan2X = 2tanX / (1 - tan²X)
• maka 2tan pi/8 / (1 - tan² pi/8) = tan 2.pi/8 = tan pi/4 = tan 45* = 1
• ingat rumus sin2X = 2.sinX.cosX
• maka sinX. cosX = (1/2).sin2X
• sin15* cos15* = (1/2).sin 2.15* = (1/2) sin 30* = (1/2)(1/2) = 1/4
• cos^4 X - sin ^4X
• = (cos² X + sin² X)(cos²X - sin²X)
• = 1. (cos²X - sin²X)
• = cos²X - sin²X
• = cos 2X

jadi cos^4 5/12 pi - sin^4 5/12 pi = cos 2. 5/12 pi = cos (5/6)pi = cos 150*
= cos (180* - 30*) = -cos 30* = - (1/2) √3


Cara Kedua

•  2sinπ/8 cosπ/8 = sin (π/8 + π/8)
  = sin π/4 = ½√2 
• .½ - sin²15 = ½(sin²15 + cos²15) - sin²15
  = ½cos²15 - ½sin²15
  = ½(cos²15 - sin²15 )
  = ½cos(15+15)
  = ½cos30 = ½ ½√3 = ¼√3 
• .cos²π/12 - sin²π/12
  = cos (π/12 + π/12)
  = cos π / 6
  = ½√3 

Cara Ketiga

• 2.sin a.cos a = sin 2a

==> 2.sin phi/8 * cos phi/8
==> sin 2*phi/8
==> sin phi/4
==> sin 45 derajat
==> 1/2 akar 2

• 1/2 - sin^2 15 drjt

==> 1/2 - sin^2 (60 - 45) --> rumus utk ini adalah sin^2 (a-b) = [sin a.sin b - cos a.cos b]^2
==> 1/2 - sin^2 (60 - 45)
==> 1/2 - [sin 60.sin 45 - cos 60.cos 45]^2
==> 1/2 - [1/2 akar 3 * 1/2 akar 2 - 1/2 * 1/2 akar 2]^2
==> 1/2 - [1/4 akar 6 - 1/4 akar 2]^2
==> 1/2 - [1/4 * (akar 6 - akar 2)]^2
==> 1/2 - [ (6 - 2 - 2akar12) / 16 ]
==> 1/2 - [ (4 - 4akar3) / 16 ]
==> 1/2 - [ (1 - akar 3) / 4 ]

• rumus cos^2 a - sin^2 a = cos 2a

maka, Cos^2 phi/12 - Sin^2 phi/12 = cos 2*phi/12
==> cos phi/6
==> cos 30
==> 1/2 akar 3

• 2 sin a cos a = sin 2a

maka, sin 15 cos 15 = 1/2 * 2 sin 15 cos 15
==> 1/2 * sin 2*15
==> 1/2 * sin 30
==> 1/2 * 1/2 akar 3
==> 1/4 akar 3

• ini sama kyk pada soal yg (C)

rumus cos^2 a - sin^2 a = cos 2a
maka, Cos^4 5/12 phi - Sin^4 5/12 phi = cos 2 * 5/12 phi
==> cos 5/6 phi
==> cos 150 --> pada kuadran 2 , maka - cos 30
==> -cos 30
==> - 1/2 akar 3
NB :: ingat ya , phi = 180 derajat :)) 

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