maths.free › Trigonometry › Graphs of sine, cosine and tangent

Graphs of sine, cosine and tangent

Amplitude, period, phase shift and the shape of periodic motion.

y = A sin(Bx + C) + D has amplitude |A|, period 2π/B, phase shift −C/B and midline D. Picture it: the graph is a wave; changing each parameter stretches, squeezes or slides it. Think it: every periodic signal (sound, light, tides) is a sum of such waves (Fourier), so these four parameters describe all of them.

Esempio svolto · y = sin(x) + cos(2x)

Algebra · Functions

Graph and analyse sin(x) + cos(2x)

y = \sin{\left(x \right)} + \cos{\left(2 x \right)}

Passo dopo passo

  1. \sin{\left(x \right)} + \cos{\left(2 x \right)}

    An expression in x. Here is what it does.

  2. x = - \frac{5 \pi}{6}, x = - \frac{\pi}{6}, x = \frac{\pi}{2}

    Real zeros (where the graph crosses the axis).

  3. \frac{d}{dx} = - 2 \sin{\left(2 x \right)} + \cos{\left(x \right)}

    Derivative (slope).

Rivela la risposta
\sin{\left(x \right)} + \cos{\left(2 x \right)}

Ora tu Scegli un problema, o digitare o disegnare il proprio. Ogni passo, un'immagine, la risposta nascosta fino a quando non si chiede.

Conserva il tuo lavoro

Un account gratuito aggiunge note su ogni lezione, un record di ciò che hai finito, i tuoi problemi risolti in un unico luogo, e un tutor che puoi chiedere su questa pagina. La matematica stessa è aperta a tutti, firmato o no.

Iscriviti Accesso

Simboli utilizzati qui

Toccare qualsiasi simbolo per la definizione completa, un'immagine, e che cosa ogni lettera in esso significa.

Come: Graphs of sine, cosine and tangent

  1. An expression in x. Here is what it does.
  2. Real zeros (where the graph crosses the axis).
  3. Derivative (slope).

Domande che la gente chiede

Why radians instead of degrees?

A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.

Why does sin x = 1/2 have infinitely many solutions?

Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.

How do I remember the exact values?

Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.

Altro in Trigonometry

The unit circleTrigonometric equationsTrigonometric identitiesDegrees and radiansRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesInverse trigonometric functions