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人気のある三角関数 >

10sin(20t-pi/6)=8

解

10 sin (20 t - \frac{π}{6}) = 8

解

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

+1

度

t = 4.15650 \dots^{\circ} + 1 8^{\circ} n, t = 7.84349 \dots^{\circ} + 1 8^{\circ} n

解答ステップ

10 sin (20 t - \frac{π}{6}) = 8

以下で両辺を割る

10

10 sin (20 t - \frac{π}{6}) = 8

以下で両辺を割る

10

\frac{10 sin ( 20 t - \frac{π}{6} )}{10} = \frac{8}{10}

簡素化

sin (20 t - \frac{π}{6}) = \frac{4}{5}

sin (20 t - \frac{π}{6}) = \frac{4}{5}

三角関数の逆数プロパティを適用する

sin (20 t - \frac{π}{6}) = \frac{4}{5}

以下の一般解

sin (20 t - \frac{π}{6}) = \frac{4}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

解く

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn : t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

\frac{π}{6}

を右側に移動します

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

両辺に

\frac{π}{6}

を足す

20 t - \frac{π}{6} + \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

簡素化

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

以下で両辺を割る

20

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

以下で両辺を割る

20

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

簡素化

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

簡素化

\frac{20 t}{20} : t

\frac{20 t}{20}

数を割る:

\frac{20}{20} = 1

= t

簡素化

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} : \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

条件のようなグループ

= \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

共通因数を約分する:

2

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 20}

数を乗じる:

6 \cdot 20 = 120

= \frac{π}{120}

= \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

解く

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn : t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

\frac{π}{6}

を右側に移動します

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

両辺に

\frac{π}{6}

を足す

20 t - \frac{π}{6} + \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

簡素化

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

以下で両辺を割る

20

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

以下で両辺を割る

20

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

簡素化

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

簡素化

\frac{20 t}{20} : t

\frac{20 t}{20}

数を割る:

\frac{20}{20} = 1

= t

簡素化

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} : \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

条件のようなグループ

= \frac{π}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

共通因数を約分する:

2

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 20}

数を乗じる:

6 \cdot 20 = 120

= \frac{π}{120}

= \frac{π}{20} + \frac{πn}{10} + \frac{π}{120} - \frac{arcsin ( \frac{4}{5} )}{20}

条件のようなグループ

= \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

10進法形式で解を証明する

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

グラフ

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