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人気のある三角関数 >

2sin^2(45-a)=1-sin^2(a)

解

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

解

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

+1

ラジアン

a = 0 + 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

解答ステップ

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

三角関数の公式を使用して書き換える

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

三角関数の公式を使用して書き換える

sin (4 5^{\circ} - a)

角の差の公式を使用する:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

簡素化

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a) : \frac{2 cos ( a ) - 2 sin ( a )}{2}

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

sin (4 5^{\circ}) cos (a) = \frac{2 cos ( a )}{2}

sin (4 5^{\circ}) cos (a)

簡素化

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

次の自明恒等式を使用する:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

36 0^{\circ} n

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}

= \frac{2}{2}

= \frac{2}{2} cos (a)

分数を乗じる:

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

= \frac{2 cos ( a )}{2}

cos (4 5^{\circ}) sin (a) = \frac{2 sin ( a )}{2}

cos (4 5^{\circ}) sin (a)

簡素化

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

次の自明恒等式を使用する:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

36 0^{\circ} n

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} sin (a)

分数を乗じる:

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

= \frac{2 sin ( a )}{2}

= \frac{2 cos ( a )}{2} - \frac{2 sin ( a )}{2}

規則を適用

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

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

2 (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} = 1 - sin^{2} (a)

簡素化

2 (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} : (cos (a) - sin (a))^{2}

2 (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

(\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} = \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

(\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2} = \frac{cos ( a ) - sin ( a )}{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2}

共通項をくくり出す

2

= \frac{2 ( cos ( a ) - sin ( a ) )}{2}

キャンセル

\frac{2 ( cos ( a ) - sin ( a ) )}{2} : \frac{cos ( a ) - sin ( a )}{2}

\frac{2 ( cos ( a ) - sin ( a ) )}{2}

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( cos ( a ) - sin ( a ) )}{2}

指数の規則を適用する:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{cos ( a ) - sin ( a )}{2 ^{1 - \frac{1}{2}}}

数を引く:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{cos ( a ) - sin ( a )}{2 ^{\frac{1}{2}}}

累乗根の規則を適用する:

a^{\frac{1}{n}} = n a

2^{\frac{1}{2}} = 2

= \frac{cos ( a ) - sin ( a )}{2}

= \frac{cos ( a ) - sin ( a )}{2}

= (\frac{cos ( a ) - sin ( a )}{2})^{2}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( cos ( a ) - sin ( a ) ) ^{2}}{( 2 ) ^{2}}

(2)^{2} : 2

累乗根の規則を適用する:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分数を乗じる:

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

= \frac{1 \cdot 2}{2}

共通因数を約分する:

2

= 1

= 2

= \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

= 2 \cdot \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

分数を乗じる:

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

= \frac{( cos ( a ) - sin ( a ) ) ^{2} \cdot 2}{2}

共通因数を約分する:

2

= (cos (a) - sin (a))^{2}

(cos (a) - sin (a))^{2} = 1 - sin^{2} (a)

(cos (a) - sin (a))^{2} = 1 - sin^{2} (a)

両辺から

1 - sin^{2} (a)

を引く

cos^{2} (a) - 2 cos (a) sin (a) + 2 sin^{2} (a) - 1 = 0

三角関数の公式を使用して書き換える

- 1 + cos^{2} (a) + 2 sin^{2} (a) - 2 cos (a) sin (a)

ピタゴラスの公式を使用する:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= 2 sin^{2} (a) - 2 cos (a) sin (a) - sin^{2} (a)

簡素化

= sin^{2} (a) - 2 cos (a) sin (a)

sin^{2} (a) - 2 cos (a) sin (a) = 0

因数

sin^{2} (a) - 2 cos (a) sin (a) : sin (a) (sin (a) - 2 cos (a))

sin^{2} (a) - 2 cos (a) sin (a)

指数の規則を適用する:

a^{b + c} = a^{b} a^{c}

sin^{2} (a) = sin (a) sin (a)

= sin (a) sin (a) - 2 sin (a) cos (a)

共通項をくくり出す

sin (a)

= sin (a) (sin (a) - 2 cos (a))

sin (a) (sin (a) - 2 cos (a)) = 0

各部分を別個に解く

sin (a) = 0 or sin (a) - 2 cos (a) = 0

sin (a) = 0 : a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

sin (a) = 0

以下の一般解

sin (a) = 0

sin (x)

36 0^{\circ} n

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}

a = 0 + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

a = 0 + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

解く

a = 0 + 36 0^{\circ} n : a = 36 0^{\circ} n

a = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

a = 36 0^{\circ} n

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

sin (a) - 2 cos (a) = 0 : a = arctan (2) + 18 0^{\circ} n

sin (a) - 2 cos (a) = 0

三角関数の公式を使用して書き換える

sin (a) - 2 cos (a) = 0

cos (a), cos (a) \neq = 0

で両辺を割る

\frac{sin ( a ) - 2 cos ( a )}{cos ( a )} = \frac{0}{cos ( a )}

簡素化

\frac{sin ( a )}{cos ( a )} - 2 = 0

基本的な三角関数の公式を使用する:

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

tan (a) - 2 = 0

tan (a) - 2 = 0

2

を右側に移動します

tan (a) - 2 = 0

両辺に

2

を足す

tan (a) - 2 + 2 = 0 + 2

簡素化

tan (a) = 2

tan (a) = 2

三角関数の逆数プロパティを適用する

tan (a) = 2

以下の一般解

tan (a) = 2

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

すべての解を組み合わせる

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = arctan (2) + 18 0^{\circ} n

10進法形式で解を証明する

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

グラフ

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