02 // Visual Math & Finance Solversy(θ) = R·sin(kθ), cos(θ) = sin(θ + 90°)
Visual Sine, Cosine & Tangent Wave Phase Projections
Projecting the vertical and horizontal coordinates of a rotating phasor around a circle of radius $R$ across angular domain $\theta \in [0, 2\pi k]$ generates the periodic waveforms $y_1 = R\sin(\theta)$ and $y_2 = R\cos(\theta) = R\sin(\theta + \pi/2)$.
y(θ) = R·sin(kθ), cos(θ) = sin(θ + 90°)Inputs & Parameters (3)
°
0°±1360°
u
0.5u±0.255u
Hz
1Hz±14Hz
R·cos(θ) [X-Base]
0.75
cos(60°) = 0.5
R·sin(θ) [Y-Altitude]
1.299
sin(60°) = 0.866
tan(θ) = sin / cos
1.7321
cot(θ) = 0.5774
Radians & Arc Length
1.0472 rad
0.333π rad | Arc s = 1.571
20-Row Comparison Table
Compare across:
| # | Phase Angle (θ) (°) | y = R·sin(θ) | R·cos(θ) | tan(θ) |
|---|---|---|---|---|
| 01 | 0 ° | 0 | 1.5 | 0 |
| 02 | 19 ° | 0.488 | 1.4183 | 0.3443 |
| 03 | 38 ° | 0.923 | 1.182 | 0.7813 |
| 04 | 57 ° | 1.258 | 0.817 | 1.5399 |
| 05 | 76 ° | 1.455 | 0.3629 | 4.0108 |
| 06 | 95 ° | 1.494 | -0.1307 | -11.4301 |
| 07 | 114 ° | 1.37 | -0.6101 | -2.246 |
| 08 | 133 ° | 1.097 | -1.023 | -1.0724 |
| 09 | 152 ° | 0.704 | -1.3244 | -0.5317 |
| 10 | 171 ° | 0.235 | -1.4815 | -0.1584 |
| 11 | 189 ° | -0.235 | -1.4815 | 0.1584 |
| 12 | 208 ° | -0.704 | -1.3244 | 0.5317 |
| 13 | 227 ° | -1.097 | -1.023 | 1.0724 |
| 14 | 246 ° | -1.37 | -0.6101 | 2.246 |
| 15 | 265 ° | -1.494 | -0.1307 | 11.4301 |
| 16 | 284 ° | -1.455 | 0.3629 | -4.0108 |
| 17 | 303 ° | -1.258 | 0.817 | -1.5399 |
| 18 | 322 ° | -0.923 | 1.182 | -0.7813 |
| 19 | 341 ° | -0.488 | 1.4183 | -0.3443 |
| 20 | 360 ° | -0 | 1.5 | -0 |
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Key Analytical Takeaways
- 01Cosine is identically a sine wave shifted $90^\circ$ ($\pi/2$ radians) to the left: $\cos\theta = \sin(\theta + 90^\circ)$.
- 02Increasing the amplitude $R$ stretches peak-to-trough height vertically without changing zero-crossing angles.
- 03Increasing the harmonic multiplier $k$ compresses the wavelength $\lambda = 360^\circ / k$.
Frequently Asked Questions
What is the phase difference between sine and cosine?+
Cosine leads sine by exactly 90 degrees (π/2 radians), reaching its peak of +1 at θ = 0° while sine starts at 0 and peaks at θ = 90°.
What is the period of a standard sine or cosine wave?+
One full cycle completes every 360° (2π radians), whereas tangent repeats every 180° (π radians).
Why does the tangent wave have vertical asymptotes every 180°?+
Because tan(θ) = sin(θ) / cos(θ), whenever cos(θ) crosses zero at odd multiples of 90° (±90°, ±270°), division by zero drives the tangent projection toward ±∞.