Trigonometric & Polar Functions Lab

Investigate sinusoidal functions, inverse trig operations, and polar curves through guided experiments. Adjust parameters with sliders, observe real-time graph changes, collect data, and build lab reports.

Guided Experiment: Amplitude and Period Investigation

If you change the amplitude A and frequency parameter B of y = A·sin(Bx), what do you predict will happen to the graph's height and width?

Write your hypothesis in the Lab Report panel, then click Next.

Sinusoidal Function Graph

Controls

Sinusoidal Analysis

y=sin⁡(x)y = \sin(x)
Amplitude
|A| = 1.00
Period
2π/B = 6.2832
Phase Shift
C = 0.00
Vertical Shift
D = 0.00
Key Features
Midliney = 0.00
Maximumy = 1.00
Minimumy = -1.00
Range[-1.00, 1.00]

Data Table

(0 rows)
#TrialFunctionTypeAmplitude/aPeriod/bPhase/CShift/DKey Features
0 / 500
0 / 500
0 / 500

Reference Guide

Sinusoidal Parameters

The general sinusoidal function has four key parameters that control its shape.

y=Asin⁡(B(x−C))+Dy = A \sin(B(x - C)) + D
  • A (amplitude) controls the height of the wave. |A| is the distance from the midline to the peak.
  • B (frequency) controls how many cycles fit in 2π. The period is 2π/B.
  • C (phase shift) translates the graph horizontally. Positive C shifts right.
  • D (vertical shift) moves the midline up or down.

Inverse Trig Functions

Inverse trig functions reverse the standard trig functions on restricted domains.

sin⁡−1(x):[−1,1]→[−π2,π2]\sin^{-1}(x): [-1,1] \to [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]
cos⁡−1(x):[−1,1]→[0,π]\cos^{-1}(x): [-1,1] \to [0, \pi]
tan⁡−1(x):R→(−π2,π2)\tan^{-1}(x): \mathbb{R} \to (-\tfrac{\pi}{2}, \tfrac{\pi}{2})

The composition f(f⁻¹(x)) = x always holds on the domain, but f⁻¹(f(x)) = x only on the restricted domain.

Polar Curve Types

Polar curves are defined by r = f(θ). The shape depends on the equation form.

  • Cardioid r = a + a·cos(θ) passes through the origin once
  • Limaçon r = a + b·cos(θ) has an inner loop when |a| < |b|
  • Rose r = a·cos(nθ) has n petals (odd n) or 2n petals (even n)
  • Spiral r = aθ grows outward with each revolution
  • Lemniscate r² = a²·cos(2θ) is a figure-eight shape

Polar-Rectangular Conversion

Convert between polar (r, θ) and rectangular (x, y) coordinates.

x=rcos⁡θ,y=rsin⁡θx = r\cos\theta, \quad y = r\sin\theta
r=x2+y2,θ=arctan⁡yxr = \sqrt{x^2 + y^2}, \quad \theta = \arctan\frac{y}{x}

When converting θ, use atan2(y, x) to get the correct quadrant. The angle θ is measured counterclockwise from the positive x-axis.