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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.

Ишлатилган мисол · y = sin(x) + cos(2x)

Algebra · Functions

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

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

Қадамма-қадам

  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).

Жавобни кўрсатиш
\sin{\left(x \right)} + \cos{\left(2 x \right)}

Энди сизлар Мавзуни танланг, ёзинг ёки ўзингиз чизинг. Ҳар бир қадам, расм, жавоб сиз сўрагунча яширин.

Ўзингиз ишлашда давом этинг

Бепул ҳисоб ҳар бир дарсга ёзув қўшиб беради, сиз тугатган нарсалар ва ҳал қилинган масалалар бир жойда қайд этилади ва сиз бу саҳифа ҳақида устоздан сўрашингиз мумкин. Математика ўзида очиқ бўлиб, унга кириш ёки киришни истамаган ҳар кимга очиқ.

Ройхатдан ўтиш Кириш

Бу ерда қўлланилган белгилар

Барча белгилар учун тўлиқ таъриф, сурат ва ҳар бир ҳарфнинг маъносини кўриш учун белгини босинг.

Қандай: 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).

Одамлар сўрайдиган саволлар

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.

Кўпроқ Trigonometry

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