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Soal-Soal Matematika/Trigonometri
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Sunting
<
Soal-Soal Matematika
Trigonometri
sunting
s
i
n
A
=
sisi depan
sisi miring
=
y
r
{\displaystyle sinA={\frac {\text{sisi depan}}{\text{sisi miring}}}={\frac {y}{r}}}
c
o
s
A
=
sisi samping
sisi miring
=
x
r
{\displaystyle cosA={\frac {\text{sisi samping}}{\text{sisi miring}}}={\frac {x}{r}}}
t
a
n
A
=
sisi depan
sisi samping
=
y
x
{\displaystyle tanA={\frac {\text{sisi depan}}{\text{sisi samping}}}={\frac {y}{x}}}
c
s
c
A
=
sisi miring
sisi depan
=
r
y
=
1
s
i
n
A
{\displaystyle cscA={\frac {\text{sisi miring}}{\text{sisi depan}}}={\frac {r}{y}}={\frac {1}{sinA}}}
s
e
c
A
=
sisi miring
sisi samping
=
r
x
=
1
c
o
s
A
{\displaystyle secA={\frac {\text{sisi miring}}{\text{sisi samping}}}={\frac {r}{x}}={\frac {1}{cosA}}}
c
o
t
A
=
sisi samping
sisi depan
=
x
y
=
1
t
a
n
A
{\displaystyle cotA={\frac {\text{sisi samping}}{\text{sisi depan}}}={\frac {x}{y}}={\frac {1}{tanA}}}
Sudut Istimewa
sunting
Trigonometri
Nama sudut
0°
30°
37°
45°
53°
60°
90°
Hasil interval
Sin A
0
1
2
{\displaystyle {\frac {1}{2}}}
3
5
{\displaystyle {\frac {3}{5}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
4
5
{\displaystyle {\frac {4}{5}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
1
−
1
≤
y
≤
1
{\displaystyle -1\leq y\leq 1}
Cos A
1
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
4
5
{\displaystyle {\frac {4}{5}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
3
5
{\displaystyle {\frac {3}{5}}}
1
2
{\displaystyle {\frac {1}{2}}}
0
−
1
≤
y
≤
1
{\displaystyle -1\leq y\leq 1}
Tan A
0
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
3
4
{\displaystyle {\frac {3}{4}}}
1
{\displaystyle 1}
4
3
{\displaystyle {\frac {4}{3}}}
3
{\displaystyle {\sqrt {3}}}
∞
{\displaystyle \infty }
−
∞
≤
y
≤
∞
{\displaystyle -\infty \leq y\leq \infty }
Cot A
∞
{\displaystyle \infty }
3
{\displaystyle {\sqrt {3}}}
4
3
{\displaystyle {\frac {4}{3}}}
1
{\displaystyle 1}
3
4
{\displaystyle {\frac {3}{4}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
0
−
∞
≤
y
≤
∞
{\displaystyle -\infty \leq y\leq \infty }
Sec A
1
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
5
4
{\displaystyle {\frac {5}{4}}}
2
{\displaystyle {\sqrt {2}}}
5
3
{\displaystyle {\frac {5}{3}}}
2
{\displaystyle 2}
∞
{\displaystyle \infty }
y
≤
−
1
atau
y
≥
1
{\displaystyle y\leq -1{\text{ atau }}y\geq 1}
Csc A
∞
{\displaystyle \infty }
2
{\displaystyle 2}
5
3
{\displaystyle {\frac {5}{3}}}
2
{\displaystyle {\sqrt {2}}}
5
4
{\displaystyle {\frac {5}{4}}}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
1
y
≤
−
1
atau
y
≥
1
{\displaystyle y\leq -1{\text{ atau }}y\geq 1}
Rumus lainnya
sunting
pythagoras trigonometri
sin
2
A + cos
2
A = 1
tan
2
A + 1 = sec
2
A
1 + cot
2
A = csc
2
A
jumlah dan selisih sudut
sin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B - cos A sin B
cos (A+B) = cos A cos B - sin A sin B
cos (A-B) = cos A cos B + sin A sin B
tan (A+B) =
t
a
n
A
+
t
a
n
B
1
−
t
a
n
A
⋅
t
a
n
B
{\displaystyle {\frac {tanA+tanB}{1-tanA\cdot tanB}}}
tan (A-B) =
t
a
n
A
−
t
a
n
B
1
+
t
a
n
A
⋅
t
a
n
B
{\displaystyle {\frac {tanA-tanB}{1+tanA\cdot tanB}}}
perkalian trigonometri
2 sin A cos B = sin (A+B) + sin (A-B)
2 cos A sin B = sin (A+B) - sin (A-B)
2 cos A cos B = cos (A+B) + cos (A-B)
-2 sin A sin B = cos (A+B) - cos (A-B)
jumlah dan selisih trigonometri
sin A + sin B =
2
s
i
n
(
A
+
B
2
)
c
o
s
(
A
−
B
2
)
{\displaystyle 2sin({\frac {A+B}{2}})cos({\frac {A-B}{2}})}
sin A - sin B =
2
c
o
s
(
A
+
B
2
)
s
i
n
(
A
−
B
2
)
{\displaystyle 2cos({\frac {A+B}{2}})sin({\frac {A-B}{2}})}
cos A + cos B =
2
c
o
s
(
A
+
B
2
)
c
o
s
(
A
−
B
2
)
{\displaystyle 2cos({\frac {A+B}{2}})cos({\frac {A-B}{2}})}
cos A - cos B =
−
2
s
i
n
(
A
+
B
2
)
s
i
n
(
A
−
B
2
)
{\displaystyle -2sin({\frac {A+B}{2}})sin({\frac {A-B}{2}})}
rangkap dua sudut
sin 2A = 2 sin A cos A
cos 2A = cos
2
A-sin
2
A = 2 cos
2
A-1 = 1-2 sin
2
A
tan 2A =
2
t
a
n
A
1
−
t
a
n
2
A
{\displaystyle {\frac {2tanA}{1-tan^{2}A}}}
rangkap tiga sudut
sin 3A = 3 sin A - 4 sin
3
A
cos 3A = 4 cos
3
A - 3 cos A
tan 3A =
3
t
a
n
A
−
t
a
n
3
A
1
−
3
t
a
n
2
A
{\displaystyle {\frac {3tanA-tan^{3}A}{1-3tan^{2}A}}}
setengah sudut
sin 1/2A =
1
−
c
o
s
A
2
{\displaystyle {\sqrt {\frac {1-cosA}{2}}}}
cos 1/2A =
1
+
c
o
s
A
2
{\displaystyle {\sqrt {\frac {1+cosA}{2}}}}
tan 1/2A =
1
−
c
o
s
A
1
+
c
o
s
A
{\displaystyle {\sqrt {\frac {1-cosA}{1+cosA}}}}
=
s
i
n
A
1
+
c
o
s
A
{\displaystyle {\frac {sinA}{1+cosA}}}
=
1
−
c
o
s
A
s
i
n
A
{\displaystyle {\frac {1-cosA}{sinA}}}
aturan sinus
A
s
i
n
A
=
B
s
i
n
B
=
C
s
i
n
C
=
2
R
{\displaystyle {\frac {A}{sinA}}={\frac {B}{sinB}}={\frac {C}{sinC}}=2R}
L
=
a
⋅
b
2
s
i
n
C
=
a
⋅
c
2
s
i
n
B
=
b
⋅
c
2
s
i
n
A
{\displaystyle L={\frac {a\cdot b}{2}}sinC={\frac {a\cdot c}{2}}sinB={\frac {b\cdot c}{2}}sinA}
aturan kosinus
a
2
=
b
2
+
c
2
−
2
⋅
b
⋅
c
⋅
c
o
s
A
{\displaystyle a^{2}=b^{2}+c^{2}-2\cdot b\cdot c\cdot cosA}
b
2
=
a
2
+
c
2
−
2
⋅
a
⋅
c
⋅
c
o
s
B
{\displaystyle b^{2}=a^{2}+c^{2}-2\cdot a\cdot c\cdot cosB}
c
2
=
a
2
+
b
2
−
2
⋅
a
⋅
b
⋅
c
o
s
C
{\displaystyle c^{2}=a^{2}+b^{2}-2\cdot a\cdot b\cdot cosC}
