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人気のある三角関数 >

15sin^2(x)-22sin(x)+8=0

解

15 sin^{2} (x) - 22 sin (x) + 8 = 0

解

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

15 sin^{2} (x) - 22 sin (x) + 8 = 0

置換で解く

15 sin^{2} (x) - 22 sin (x) + 8 = 0

仮定:

sin (x) = u

15 u^{2} - 22 u + 8 = 0

15 u^{2} - 22 u + 8 = 0 : u = \frac{4}{5}, u = \frac{2}{3}

15 u^{2} - 22 u + 8 = 0

解くとthe二次式

15 u^{2} - 22 u + 8 = 0

二次Equationの公式:

次の場合:

a = 15, b = - 22, c = 8

u_{1, 2} = \frac{- ( - 22 ) \pm ( - 22 ) ^{2} - 4 \cdot 15 \cdot 8}{2 \cdot 15}

u_{1, 2} = \frac{- ( - 22 ) \pm ( - 22 ) ^{2} - 4 \cdot 15 \cdot 8}{2 \cdot 15}

(- 22)^{2} - 4 \cdot 15 \cdot 8 = 2

(- 22)^{2} - 4 \cdot 15 \cdot 8

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 22)^{2} = 2 2^{2}

= 2 2^{2} - 4 \cdot 15 \cdot 8

数を乗じる:

4 \cdot 15 \cdot 8 = 480

= 2 2^{2} - 480

2 2^{2} = 484

= 484 - 480

数を引く:

484 - 480 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 22 ) \pm 2}{2 \cdot 15}

解を分離する

u_{1} = \frac{- ( - 22 ) + 2}{2 \cdot 15}, u_{2} = \frac{- ( - 22 ) - 2}{2 \cdot 15}

u = \frac{- ( - 22 ) + 2}{2 \cdot 15} : \frac{4}{5}

\frac{- ( - 22 ) + 2}{2 \cdot 15}

規則を適用

- (- a) = a

= \frac{22 + 2}{2 \cdot 15}

数を足す:

22 + 2 = 24

= \frac{24}{2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{24}{30}

共通因数を約分する:

6

= \frac{4}{5}

u = \frac{- ( - 22 ) - 2}{2 \cdot 15} : \frac{2}{3}

\frac{- ( - 22 ) - 2}{2 \cdot 15}

規則を適用

- (- a) = a

= \frac{22 - 2}{2 \cdot 15}

数を引く:

22 - 2 = 20

= \frac{20}{2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{20}{30}

共通因数を約分する:

10

= \frac{2}{3}

二次equationの解:

u = \frac{4}{5}, u = \frac{2}{3}

代用を戻す

u = sin (x)

sin (x) = \frac{4}{5}, sin (x) = \frac{2}{3}

sin (x) = \frac{4}{5}, sin (x) = \frac{2}{3}

sin (x) = \frac{4}{5} : x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

sin (x) = \frac{4}{5}

三角関数の逆数プロパティを適用する

sin (x) = \frac{4}{5}

以下の一般解

sin (x) = \frac{4}{5}

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

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

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

sin (x) = \frac{2}{3} : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

sin (x) = \frac{2}{3}

三角関数の逆数プロパティを適用する

sin (x) = \frac{2}{3}

以下の一般解

sin (x) = \frac{2}{3}

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

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

10進法形式で解を証明する

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

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