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Beliebt Algebra >

vereinfachen (sin(a)+cos(b))^2+(cos(a)+sin(b))^2

Lösung

vereinfachen

(sin (a) + cos (b))^{2} + (cos (a) + sin (b))^{2}

Lösung

2 sin (a + b) + 2

Schritte zur Lösung

(sin (a) + cos (b))^{2} + (cos (a) + sin (b))^{2}

Schreibe

(sin (a) + cos (b))^{2}

um:

sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b)

= sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b) + (cos (a) + sin (b))^{2}

Schreibe

(cos (a) + sin (b))^{2}

um:

cos^{2} (a) + 2 cos (a) sin (b) + sin^{2} (b)

= sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b) + cos^{2} (a) + 2 cos (a) sin (b) + sin^{2} (b)

Verwende die Pythagoreische Identität:

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

= 2 cos (a) sin (b) + 2 cos (b) sin (a) + 1 + 1

Addiere die Zahlen:

1 + 1 = 2

= 2 cos (a) sin (b) + 2 cos (b) sin (a) + 2

Benutze die Identität der Winkelsumme:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= 2 (1 + sin (a + b))

Faktorisiere

2 cos (a) sin (b) + 2 cos (b) sin (a) + 2 : 2 (cos (a) sin (b) + cos (b) sin (a) + 1)

= 2 (sin (a + b) + 1)

Schreibe

2 (sin (a + b) + 1)

um:

2 sin (a + b) + 2

= 2 sin (a + b) + 2

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e x p an d t^{2} μ (t - 1) + t cos (5 t)

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