slide identitas trigonometri dasar
TRANSCRIPT
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Identitas Trigonometri
Dasar
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Rumus Kebalikan
cosec α°=
sec α°=
cot α°=
0tan
1
0sin
1
0cos
1
0tan
1
0sin
1
0cos
1
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Rumus Perbandingan
tan α°=
cot α°=
0
0
cos
sin
0
0
sin
cos
0
0
cos
sin
0
0
sin
cos
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Identitas Trigonometri
Dasar
Merupakan hubungan kebalikan
Merupakan Hubungan Perbandingan (kuosien)
Diperoleh Dari Hubungan Pythagoras
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Hubungan Kebalikan
• cosec α°= atau sin α°=
• sec α°= atau cos α°=
• cot α°= atau tan α°=
0sin
1
0tan
1
0cos
1
0 cosec
1
0sec
1
0cot
1
Back
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tan α°= 0
0
cos
sin
cot α°= 0
0
sin
cos
Rumus Perbandingan
Back
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Yang diperoleh dari Hubungan Pythagoras
a. sin² α° + cos² α° = 1b. 1 + tan² α° = sec² α°c. 1 +cot² α° = cosec² α°
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sin² α° + cos² α° = 1
Y
P (x,y)
1
O
α°
x P’ X
Bukti:P (x,y) terletak pada lingkaran satuan dengan <POX=α°Δ OPP’ merupakan segitiga siku-siku di P’.OP=1 PP’=yOP’=xcos α°= x/1=xsin α°= y/1=y
y
Y
P (x,y)
1
Oα°
x P’X
y
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Y
P (x,y)
1
O
α°
x P’ X
y
Sehingga berlaku hubungan pythagoras :(OP’)² + (PP’) ² =(OP)²x ² + y ²= 1Karena : cos α°=x dan sin α°=y,maka diperoleh :
cos²α + sin ²α =1
Back
Y
P (x,y)
1
O
α°x P’
Xy
Y
P (x,y)
1
Oα°
x P’X
y
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1 + tan² α° = sec² α°
1
Oα°
x P’ X
yY
P (x,y)
1
O
α°x P’
X
Bukti:tan α°= y/x, tan² α°= y²/x²sec α°= 1/xsec² α°=1/x²
x ² + y ²= 1
Maka diperoleh:
y
: x²
22
2
2
2 1
xx
y
x
x
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22
2
2
2 1
xx
y
x
x
221
1
xx
y
Substitusi : tan² α°= y²/x² dan sec² α°=1/x²Ke persamaan di atas, maka diperoleh :
1 + tan² α° = sec² α°
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Contoh Soal
Diketahui cosec β=2 dan β sudut di kuadran kedua. Hitunglah :a.cot βb.sin βc.cos β
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Diket : cosec β=2β sudut di kuadran ke dua
Dit: a. cot βb. sin βc. cos βjawab:a. 1 + cot² β = cosec ² β
cot² β =cosec ² β-1cot ² β = (2) ² - 1 cot ² β = 3cot β = √3 atau cot β = - √3Karena β sudut di kuadran II, diambil cot β = - √3Jadi, cot β = - √3
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TERIMAKASIH