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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.

例題 · y = sin(x) + cos(2x)

Algebra · Functions

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

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

ステップバイステップ

  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).

答えを表示
\sin{\left(x \right)} + \cos{\left(2 x \right)}

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ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

どうやって: 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).

よくある質問

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.

More in Trigonometry

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