Calculus & Trigonometry Grade 10+

Sine & Cosine Function Calculator

Analyze general trigonometric functions f(x) = a·sin(b(x-c))+d, determine amplitude, period, frequency, roots, extrema, use the interactive unit circle, and solve periodic modeling problems.

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Curriculum & Modeling Presets (Grade 10 to Calculus):

Parameters of the General Trigonometric Function

Amplitude |a| (Streckung y): 1

Frequenzfaktor b (Periode p = 2π): 1

Periode:

Phasenverschiebung c: 0

Presets:

Ruhelage / Mittellinie d (y-Richtung): 0

Gleichungslöser f(x) = y₀:

2D-Funktionsplotter

2D-Funktionsgraph & Schwingungsverlauf

x-Achse:

Funktion:

f(x) = sin(x)

Hochpunkte (3)

Tiefpunkte (2)

Nullstellen (6)

Wendepunkte (6)

Amplitude |a|

Auslenkung um Ruhelage

Periode p

p = 2\pi

≈ 6.2832 rad • Wellenlänge

Ruhelage d

y = 0

Symmetrieachse / Mittellinie

Wertebereich 𝕎

[-1; 1]

Min / Max Werte

Funktionsgleichung

f(x) = sin(x)

Analysis (Ableitungen & Stammfunktion)

1. Ableitung f'(x) (Steigung / Geschwindigkeit):

f'(x) = \cos(x)

2. Ableitung f''(x) (Krümmung / Beschleunigung):

f''(x) = -\sin(x)

Stammfunktion F(x):

F(x) = -\cos(x) + C

Kurvendiskussion (Punkte im Intervall)

Nullstellen (6):

N_{1}(-2\pi \mid 0)

N_{2}(-\pi \mid 0)

N_{3}(0 \mid 0)

N_{4}(\pi \mid 0)

N_{5}(2\pi \mid 0)

N_{6}(3\pi \mid 0)

Hochpunkte (3):

H_{1}(-\frac{3\pi}{2} \mid 1)

H_{2}(\frac{\pi}{2} \mid 1)

H_{3}(\frac{5\pi}{2} \mid 1)

Tiefpunkte (2):

T_{1}(-\frac{\pi}{2} \mid -1)

T_{2}(\frac{3\pi}{2} \mid -1)

Wendepunkte (6):

W_{1}(-2\pi \mid 0)

W_{2}(-\pi \mid 0)

W_{3}(0 \mid 0)

W_{4}(\pi \mid 0)

W_{5}(2\pi \mid 0)

W_{6}(3\pi \mid 0)

Historical Origin & The Challenge Solved

Why was this mathematics developed, where does it come from, and what real-world problem did it solve?

📜

Epoche & ZeitalterAncient Astronomy to Fourier Analysis (c. 150 BCE – 1822 CE)

Pioniere: Hipparchus of Nicaea, Aryabhata, Leonhard Euler, Joseph Fourier

1. Periodic Natural Phenomena & Oscillations

Ocean tides, planetary orbits, pendulums, and musical sounds repeat periodically. Classical linear mathematics could not continuously model smooth harmonic circular and wave motions.

2. Unit Circle Functions & Fourier Decomposition

Euler defined sine and cosine as continuous functions on the unit circle. In 1822, Joseph Fourier proved that ANY periodic oscillation (sound, heat, light) can be decomposed into a sum of pure sine and cosine waves.

3. MP3 Compression, Wi-Fi, 5G & MRI Scanners

Wireless telecommunications, Wi-Fi networks, MP3 audio compression, and MRI medical imaging function by converting complex waveforms into sine and cosine components.

„Fourier series are one of the most magnificent discoveries of the human mind.“ — Lord Kelvin

Real-World Applications of Periodic Phenomena

Sine and cosine functions are the foundational mathematical language for modeling waves, oscillations, and cyclical natural cycles.

Recreation & Kinematics

Ferris Wheel Modeling (Grade 10 Standard)

The height of a cabin above the ground follows a cosine oscillation h(t) = -r·cos(ωt) + h_center with radius r and center axle height h_center.

Praxis-Beispiel:

Ferris wheel with radius 15 m, axle height 18 m, and 30 s rotation period: h(t) = -15·cos((π/15)t) + 18.

Oceanography & Geophysics

Ocean Tides (Ebb and Flood)

Periodic ocean water levels caused by the Moon's gravitational pull are modeled with a sine curve having a period of p ≈ 12.4 hours.

Praxis-Beispiel:

Average depth 3.5 m with tidal range 3.6 m (amplitude 1.8 m): W(t) = 1.8·sin((2π/12.4)t) + 3.5.

Acoustics & Electronics

Sound Waves & AC Voltage

Musical notes like concert pitch A (440 Hz) and household electrical grid voltage (120V / 230V, 50/60 Hz) are pure sine oscillations y(t) = A·sin(2π·f·t).

Praxis-Beispiel:

Grid AC voltage with peak value 325 V: u(t) = 325·sin(100π·t) V.

Mathematical Guide to General Sine and Cosine Functions

The general sinusoidal model f(x) = a·sin(b(x-c))+d extends triangle trigonometry to continuous real-valued periodic oscillations.

Amplitude |a| & Midline d

The amplitude |a| specifies maximum upward and downward displacement. Parameter d shifts the wave vertically (midline y = d). Range: 𝕎 = [d-|a|; d+|a|].

Frequency Factor b & Period p

Factor b compresses (b > 1) or stretches (b < 1) the wave horizontally. The period is p = 2π / |b| (in degrees: p = 360° / |b|). Angular frequency is ω = |b|.

Phase Shift c

Parameter c shifts the graph horizontally: (x - c) shifts right, (x + c) shifts left. Note that cos(x) = sin(x + π/2).

Derivative & Integral Cycle

In radians, the derivative cycle is: sin(x) → cos(x) → -sin(x) → -cos(x) → sin(x). With chain rule: (a·sin(b(x-c)))' = a·b·cos(b(x-c)).

Frequently Asked Questions (FAQ) on Sine and Cosine Functions

What is the key difference between sine and cosine?▼

Cosine is simply a sine wave shifted left by 90° (or π/2 radians): cos(x) = sin(x + π/2). Both have identical amplitude, frequency, and wave shape.

When should I use radians (rad) vs degrees (°)?▼

In calculus, differentiation, and physics, radians are mandatory because only in radians does (sin x)' = cos x hold without conversion factors. Degrees are often used in introductory geometry.

How do you solve an equation like sin(x) = 0.5 in [0; 2π]?▼

1. Principal value via arcsine: x₁ = arcsin(0.5) = π/6 (30°). 2. Symmetric value in Quadrant II: x₂ = π - x₁ = 5π/6 (150°). Add multiples of the period + k·2π for all solutions.

How do you build a sinusoidal function for a Ferris wheel?▼

Amplitude a = radius r, midline d = axle height = bottom height + r, period p = rotation time T, b = 2π/T. If the cabin starts at the bottom, use -r·cos((2π/T)t) + d.

Why is the derivative of sin(x) equal to cos(x)?▼

At x=0, the sine curve reaches its steepest slope 1 (cos(0) = 1), and at peaks x=π/2 the slope is 0 (cos(π/2) = 0). Plotting these tangent slopes yields the cosine wave.