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العربية

sin^2(x)=(1-cos^2(x))/2

Lösung

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

x = 2 πn, x = π + 2 πn

Grad

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

Schritte zur Lösung

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

Subtrahiere

\frac{1 - cos ^{2} ( x )}{2}

von beiden Seiten

sin^{2} (x) - \frac{1 - cos ^{2} ( x )}{2} = 0

Vereinfache

sin^{2} (x) - \frac{1 - cos ^{2} ( x )}{2} : \frac{2 sin ^{2} ( x ) - 1 + cos ^{2} ( x )}{2}

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

Wandle das Element in einen Bruch um:

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

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

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

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

Multipliziere aus

sin^{2} (x) \cdot 2 - (1 - cos^{2} (x)) : sin^{2} (x) \cdot 2 - 1 + cos^{2} (x)

sin^{2} (x) \cdot 2 - (1 - cos^{2} (x))

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

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

Setze Klammern

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

Wende Minus-Plus Regeln an

- (- a) = a, - (a) = - a

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

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

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

\frac{2 sin ^{2} ( x ) - 1 + cos ^{2} ( x )}{2} = 0

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

= sin^{2} (x)

sin^{2} (x) = 0

Wende Regel an

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

Allgemeine Lösung für

sin (x) = 0

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Löse

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

cot (θ) = cos (θ)

3 csc^{3} (x) - 4 csc (x) = 0

tan (θ) = \frac{6.06}{- 33.5}

csc (θ) = \frac{13}{2}, sec (θ) = \frac{13}{3}

4 cos^{2} (x) + cos (2 x) - 7 cos (x) = - 2, 0^{\circ} \leq x \leq 36 0^{\circ}