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人気のある三角関数 >

7tan(x)=8sin(x)

解

7 tan (x) = 8 sin (x)

解

x = 2 πn, x = π + 2 πn, x = 0.50536 \dots + 2 πn, x = 2 π - 0.50536 \dots + 2 πn

+1

度

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 28.95502 \dots^{\circ} + 36 0^{\circ} n, x = 331.04497 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

7 tan (x) = 8 sin (x)

両辺から

8 sin (x)

を引く

7 tan (x) - 8 sin (x) = 0

サイン, コサインで表わす

7 tan (x) - 8 sin (x)

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 7 \cdot \frac{sin ( x )}{cos ( x )} - 8 sin (x)

簡素化

7 \cdot \frac{sin ( x )}{cos ( x )} - 8 sin (x) : \frac{7 sin ( x ) - 8 sin ( x ) cos ( x )}{cos ( x )}

7 \cdot \frac{sin ( x )}{cos ( x )} - 8 sin (x)

乗じる

7 \cdot \frac{sin ( x )}{cos ( x )} : \frac{7 sin ( x )}{cos ( x )}

7 \cdot \frac{sin ( x )}{cos ( x )}

分数を乗じる:

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

= \frac{sin ( x ) \cdot 7}{cos ( x )}

= \frac{7 sin ( x )}{cos ( x )} - 8 sin (x)

元を分数に変換する:

8 sin (x) = \frac{8 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x ) \cdot 7}{cos ( x )} - \frac{8 sin ( x ) cos ( x )}{cos ( x )}

分母が等しいので, 分数を組み合わせる:

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

= \frac{sin ( x ) \cdot 7 - 8 sin ( x ) cos ( x )}{cos ( x )}

= \frac{7 sin ( x ) - 8 sin ( x ) cos ( x )}{cos ( x )}

\frac{7 sin ( x ) - 8 cos ( x ) sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

7 sin (x) - 8 cos (x) sin (x) = 0

因数

7 sin (x) - 8 cos (x) sin (x) : - sin (x) (8 cos (x) - 7)

7 sin (x) - 8 cos (x) sin (x)

共通項をくくり出す

- sin (x)

= - sin (x) (- 7 + 8 cos (x))

- sin (x) (8 cos (x) - 7) = 0

各部分を別個に解く

sin (x) = 0 or 8 cos (x) - 7 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

以下の一般解

sin (x) = 0

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解く

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

8 cos (x) - 7 = 0 : x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

8 cos (x) - 7 = 0

7

を右側に移動します

8 cos (x) - 7 = 0

両辺に

7

を足す

8 cos (x) - 7 + 7 = 0 + 7

簡素化

8 cos (x) = 7

8 cos (x) = 7

以下で両辺を割る

8

8 cos (x) = 7

以下で両辺を割る

8

\frac{8 cos ( x )}{8} = \frac{7}{8}

簡素化

cos (x) = \frac{7}{8}

cos (x) = \frac{7}{8}

三角関数の逆数プロパティを適用する

cos (x) = \frac{7}{8}

以下の一般解

cos (x) = \frac{7}{8}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

すべての解を組み合わせる

x = 2 πn, x = π + 2 πn, x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

10進法形式で解を証明する

x = 2 πn, x = π + 2 πn, x = 0.50536 \dots + 2 πn, x = 2 π - 0.50536 \dots + 2 πn

グラフ

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