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TRIGONOMETRI. IDIKATOR: MEMBUKTIKAN KESAMAAN TRIGONOMETRI MENYEDERHANAKAN PERSAMAAN TRIGONOMETRI SERTA MENCARI PENYELESAIAN PERSAMAAN DAN PERTIDAKSAMAAN. BY : ULIYA FATIMAH (09320008). TRIGONOMETRI. MATERI: Perbandingan Trigonometri dan Teorema Pythagoras

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Trigonometri

TRIGONOMETRI

IDIKATOR:

MEMBUKTIKAN KESAMAAN TRIGONOMETRI

MENYEDERHANAKAN PERSAMAAN TRIGONOMETRI SERTA MENCARI PENYELESAIAN PERSAMAAN DAN PERTIDAKSAMAAN

BY : ULIYA FATIMAH

(09320008)


Trigonometri

TRIGONOMETRI

MATERI:

Perbandingan TrigonometridanTeorema Pythagoras

Nilai Perbandingan Trigonometri untuk Sudut Istimewa

Perbandingan Trigonometri dalam Kuadran

Identitastrigonometri


Trigonometri

TRIGONOMETRI

Perbandingan Trigonometri& Teorema Pythagoras

Ketahuilah , pada Pythagoras hanya berlaku pada segi tiga siku-siku dan sisimiring atau disebut dengan hipotenusa sama dengan jumlah padakeduasisisiku-sikusegitiga.


Trigonometri1

TRIGONOMETRI

AC2 = AB2+ BC2

C

Contoh:

Hitunglah panjang sisi x yang belum diketahui, pada segitiga siku-siku di samping ini (panjang segitiga dalam cm)

HIPOTENUSA

X

B

A

jawab


Trigonometri2

TRIGONOMETRI

Jawab:

C

AC2 = AB2 + BC2

X2 = 122 + 52

= 144 + 25

= 169

X = 13

HIPOTENUSA

X

15

B

A

5


Trigonometri3

TRIGONOMETRI

C

Sin α =

Cos α =

HIPOTENUSA

Tan α =

B

A


Trigonometri4

TRIGONOMETRI

C

cosec α =

HIPOTENUSA

Sec α =

Cotan α =

B

A


Trigonometri5

TRIGONOMETRI

contoh:

1. Di titik R (8, 15) membentuksudut α, tentukan sec α ?

Sec α =

r2= x2 + y2

= 82 + 152

= 64 + 225

r

y = 15

=

α

r = 17

x = 8

Sec α = 17 /8


Trigonometri6

TRIGONOMETRI

2. Nilai Perbandingan Trigonometri untuk Sudut Istimewa


Trigonometri7

TRIGONOMETRI

Contoh:

Buktikan sin245 + cos245 = 1

jawab:

sin245 + cos245 = 1

(½ )2 + (½)2 = 1

¼ 2 + ¼ 2 = 1

2/4 + 2/4 = 1

4/4=1

Terbukti, sin245 + cos245 = 1


Trigonometri8

TRIGONOMETRI

3

PERBANDINGAN TRIGONOMETRI DALAM KUADRAN


Trigonom etri

TRIGONOM ETRI


Trigonometri9

TRIGONOMETRI

sin α=

=

=

= +

Kuadran II

Kuadran I

r= +

r= +

y = +

y = +

x = +

α

α

x = -

y = -

y = -

r=+

r = +

Kuadran IV

Kuadran III


Trigonom etri1

TRIGONOM ETRI

Contoh:

  • Cos α = -4/5 dan tan α positif, berapa nilai sin αsin ....

sin α ==

x = -4

α

y2 = r2 - x2

y2 = 52 – (-4)2

y2 = 25 – 16

=

y = -3

y = ?

r = 5

Jadi, Sin α =


Trigonom etri2

TRIGONOM ETRI

Contoh:

  • Cos α = -4/5 dan tan α positif, berapa nilai sin αsin ....

sin α ==

x = -4

y2 = r2 - x2

y2 = 52 – (-4)2

y2 = 25 – 16

=

y = -3

α

y = ?

r = 5

Jadi, Sin α =


Trigonometri10

TRIGONOMETRI

4

IDENTITAS TRIGONOMETRI


Trigonometri11

TRIGONOMETRI

Sin α = =

cos α = =

tan α = =

cosec α = =

Sec α = =

cotann α = =


Trigonom etri3

TRIGONOM ETRI

Hubungan antar pembanding

a. Cosec α =

b. Sec α =

c. Cotan α =


Trigonometri12

TRIGONOMETRI

a. Cosec α =

Cosec α =

Cosec α =

b. Sec α =

Sec α =

Sec α =

c. Cotan α =

Cotan α =

Cotan α =


Trigonometri13

TRIGONOMETRI

2. IdentitasdariHubunganTeorema Pythagoras (x2 + y2 = r2 )

a) x2 + y2 = r2(sama-sama dibagi r2)

x2 / r2 + y2 / r2 = r2 / r2

x2 / r2 + y2 / r2 = 1

cos2 α + sin2 α = 1


Trigonometri14

TRIGONOMETRI

2. IdentitasdariHubunganTeorema Pythagoras (x2 + y2 = r2 )

b) x2 + y2 = r2(sama-sama dibagi y2)

x2 / y2 + y2 / y2 = y2 / y2

x2 / y2 + y2 / y2 = 1

cotan2α+1= cosec2 α


Trigonometri15

TRIGONOMETRI

Contoh 1 :

jika 2 sin2 x + 3 cos x = 0 dan 0° < x < 180° maka nilaix adalah.............

Jawab :

2 sin2 x + 3 cos x = 0

2(1- cos2 x) + 3 cos x = 0

2cos2 x - - 3 cos x - 2 = 0

(2 cos x + 1 ) ( cos x – 2 ) = 0

Cos x = - ½ cos x = 2 (tidak memenuhi)


Trigonometri16

TRIGONOMETRI

Contoh 2:

Dari pertidaksamaan berikut sinx . sin2 x + cos2x < ½ berapakah nilai dari x

Jawab:

sinx . sin2 x + cos2x < ½

sin x .(sin2 x + cos2x) < ½

sin x . 1 < ½

sin x < ½

x< 30°


Terimakasih

TERIMAKASIH

SEMOGA YANG KITA PELAJARI DAPAT BERMANFAAT

AMIIIIIIN..


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