Trigonometry, Phasors, and Complex Numbers

Introduction

Trigonometry and complex numbers provide a common mathematical language for periodic motion, rotation, vibration, alternating-current (AC) circuits, signal processing, and robotic orientation. In mechatronics, a motor shaft may rotate continuously, a sensor may produce a sinusoidal voltage, and a controller may need to represent both magnitude and timing of oscillatory signals. These apparently different problems can be modeled using angles, sine and cosine functions, complex numbers, and phasors.

This chapter develops the subject from angular measurement to complex arithmetic and engineering applications. Radians are used as the fundamental angular unit because they connect angle directly with arc length and make differentiation and integration of trigonometric functions especially simple.

Learning Objectives

By the end of this chapter, you should be able to:

• convert between degrees and radians and interpret angular displacement;
• use fundamental trigonometric identities and solve basic trigonometric equations;
• construct and interpret sinusoidal models;
• represent complex numbers in rectangular, polar, and exponential forms;
• apply Euler’s formula to multiplication, division, powers, and roots;
• use phasors to analyze sinusoidal voltages, currents, impedances, and phase relationships;
• connect trigonometric and complex-number methods with oscillation, signals, and robotic motion;
• identify assumptions, units, limitations, and common sources of error.

1. Angles, Radians, and the Unit Circle

An angle measures rotation. One complete revolution is 360° or 2π radians. A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Thus, if an arc has length s and radius r,

where θ is measured in radians. The formula applies to circular arcs and requires consistent length units for s and r. Since a full circumference is 2πr, one revolution corresponds to 2π radians.

The unusual characters above are not intended; use the standard conversion:

For clarity, the conversion is θ_rad = θ_deg × π/180 and θ_deg = θ_rad × 180/π.

For a circle of radius r, the arc length and sector area are

with θ in radians. The second formula applies to a circular sector.

On the unit circle, the point corresponding to angle θ has coordinates (cos θ, sin θ). Consequently,

The tangent function is defined where cos θ ≠ 0:

The signs of the functions depend on the quadrant. This unit-circle interpretation is more reliable than memorizing isolated triangle rules because it applies to all angles, including negative angles and angles greater than one revolution.

Worked Example 1: Angular speed and arc length

A robotic joint rotates at 120 revolutions per minute. Find its angular speed in radians per second and the distance travelled by a point 0.15 m from the axis in 4 s.

The angular speed is

ω = 120 × 2π/60 = 4π ≈ 12.57 rad/s.

The angular displacement in 4 s is θ = ωt = 16π rad. The arc length is

s = rθ = 0.15 × 16π ≈ 7.54 m.

The result assumes constant angular speed and a point fixed at radius 0.15 m.

2. Trigonometric Identities and Equations

Identities are equations true for every value in their domain. The principal identities include

where sec θ = 1/cos θ, csc θ = 1/sin θ, and cot θ = cos θ/sin θ when the denominators are nonzero.

The addition and subtraction formulas are

The double-angle formulas include

These identities are useful for simplifying models, deriving control relationships, and converting products into sums. For example,

Solving trigonometric equations

To solve sin θ = k, a real solution exists only when −1 ≤ k ≤ 1. If θ₀ = sin⁻¹(k), the general solutions are

where n is any integer. Similar quadrant reasoning applies to cosine and tangent. A specified interval, such as 0 ≤ θ < 2π, must always be stated because trigonometric functions are periodic.

3. Sinusoidal Models

A sinusoidal quantity can be written as

where:

The maximum and minimum values are C + A and C − A. The period and ordinary frequency are

where T is in seconds and f is in hertz. The formula assumes ω ≠ 0. A negative amplitude can be replaced by a positive amplitude with a phase shift of π.

The velocity and acceleration of a sinusoidal displacement are obtained by differentiation:

Thus, the oscillatory part of displacement has acceleration opposite to displacement. This is the defining kinematic property of ideal simple harmonic motion.

Worked Example 2: Interpreting a sensor signal

Suppose a displacement sensor produces x(t) = 2 + 0.8cos(10πt − π/6) mm. The mean displacement is 2 mm, amplitude is 0.8 mm, angular frequency is 10π rad/s, and phase is −π/6. Its frequency is f = 10π/(2π) = 5 Hz and period is 0.2 s. The range is 1.2 to 2.8 mm. The phase term indicates that the waveform is shifted to the right by (π/6)/(10π) = 1/60 s relative to 2 + 0.8cos(10πt).

4. Complex Numbers

A complex number has the form

where a is the real part, b is the imaginary-part coefficient, and j = √(−1). In electrical engineering, j is used instead of i, which may denote current. The conjugate is

and the modulus or magnitude is

The argument is arg(z) = atan2(b,a), which selects the correct quadrant. Using ordinary tan⁻¹(b/a) without quadrant correction can give an incorrect angle.

Complex arithmetic follows ordinary algebra together with j² = −1. Addition and subtraction operate component by component. Multiplication is

Division is most safely performed by multiplying numerator and denominator by the denominator’s conjugate:

provided c² + d² ≠ 0.

Polar and exponential forms

If r = |z| and θ = arg(z), then

Euler’s formula states

Therefore,

The angle is dimensionless; radians are understood. Polar form is especially efficient for multiplication and division:

De Moivre’s theorem gives

for integer n. The n roots of re^jθ are

They are equally spaced around a circle in the complex plane.

Worked Example 3: Complex conversion and arithmetic

Convert z = −3 + 4j to polar form. Its magnitude is r = √(9 + 16) = 5. Since the point lies in quadrant II,

θ = atan2(4,−3) ≈ 2.214 rad, or 126.87°. Hence z = 5∠126.87°.

For w = 2∠30°, the product is

zw = 10∠156.87°. Multiplication has added the phases and multiplied the magnitudes.

5. Phasors and Sinusoidal Steady State

A phasor is a complex representation of the amplitude and phase of a sinusoid at one angular frequency. If

then its peak-amplitude phasor is

and the physical signal is recovered from

The phasor does not represent an arbitrary transient waveform. It applies to linear time-invariant systems in sinusoidal steady state, where all quantities have the same angular frequency. Peak and root-mean-square (RMS) conventions must not be mixed. The RMS phasor is V_rms = V_m/√2 for a sinusoid, and average power calculations commonly use RMS values.

For a resistor, inductor, and capacitor,

where R is resistance in ohms, L inductance in henries, C capacitance in farads, and ω angular frequency in rad/s. Impedance has units of ohms. Ohm’s law becomes

For a series RLC circuit,

The impedance angle is the phase difference between voltage and current. At series resonance,

provided the ideal lumped-element model applies. At this frequency, the inductive and capacitive reactances cancel, leaving Z = R. Real circuits include losses, parasitic effects, and frequency limits.

Average real power in a sinusoidal circuit is

where φ is the voltage-current phase difference and cos φ is the power factor. Apparent power is S = V_rms I_rms in volt-amperes, and reactive power is Q = V_rms I_rms sin φ in volt-amperes reactive.

Worked Example 4: Series RL circuit

A 10 Ω resistor and a 0.10 H inductor are connected in series to a 230 V RMS, 50 Hz source. The angular frequency is ω = 2πf = 100π rad/s. The inductive reactance is X_L = ωL = 31.42 Ω, so

Z = 10 + j31.42.

The magnitude and angle are |Z| = √(10² + 31.42²) ≈ 32.97 Ω and φ = atan2(31.42,10) ≈ 72.3°. Therefore,

I = 230/32.97 ≈ 6.98 A RMS, with current lagging voltage by approximately 72.3°. The real power is

P = I²R ≈ (6.98)²(10) ≈ 487 W.

6. Applications in Mechatronics

Oscillation and vibration

Mass–spring systems, rotating imbalance, flexible robot links, and motor torque ripple often produce approximately sinusoidal responses near an operating point. Complex frequency methods simplify the analysis of amplitude and phase, but damping and nonlinearities may make the response non-sinusoidal. Fourier methods can then represent a periodic signal as a sum of harmonics.

Signals and control

A sinusoidal test input reveals a system’s gain and phase at a selected frequency. Frequency-response measurements are naturally represented by complex numbers. Engineers must distinguish phase wrapping, measurement delay, sensor bandwidth, sampling limitations, and noise from actual system dynamics.

Robotic rotation

A planar point at radius r and angle θ has coordinates x = r cos θ and y = r sin θ. Its velocity for time-varying angle is

where ω = dθ/dt. Complex notation can represent the same point as z = re^jθ, and rotation by α is multiplication by e^jα. This is elegant for planar kinematics, although three-dimensional robot orientation requires matrices, rotation vectors, or quaternions rather than a single complex number.

Applied Case: Diagnosing a Motor-Drive Vibration

Background

An illustrative industrial case concerns a servo motor driving a belt mechanism. An accelerometer records a dominant vibration near 80 Hz. Excessive vibration can reduce positioning accuracy and accelerate bearing wear.

Problem

The measured acceleration is modeled as

a(t) = 1.6cos(160πt + 0.9) m/s². The engineering team must determine its frequency, phase, and likely relationship to a periodic drive disturbance.

Analysis

The angular frequency is 160π rad/s, so f = 160π/(2π) = 80 Hz. The acceleration amplitude is 1.6 m/s² and the phase is 0.9 rad relative to the chosen reference. The corresponding acceleration phasor is A = 1.6∠0.9 m/s². If displacement is required and the sinusoidal component is integrated twice, then the displacement amplitude is

1.6/ω² = 1.6/(160π)² ≈ 6.33 × 10⁻⁶ m, assuming zero-mean steady-state sinusoidal motion.

A spectrum or frequency-response measurement should be compared with the motor’s electrical frequency, mechanical rotation frequency, belt-pitch frequency, and structural natural frequencies. A phase comparison between an accelerometer and encoder can reveal whether the source is torque ripple, imbalance, or structural resonance.

Approach and outcome

The team applies a narrow-band filter around 80 Hz, verifies sensor calibration and sampling rate, and measures vibration at several motor speeds. If the peak tracks shaft speed, imbalance is likely; if it remains near a fixed structural frequency while amplitude changes sharply, resonance is more likely. Corrective actions may include balancing, changing servo gains, adding damping, altering operating speed, or redesigning the support.

Learning points

Phasors describe magnitude and timing but do not identify causation by themselves. Reliable diagnosis requires correct reference signals, adequate sampling, calibrated sensors, and tests over operating conditions. An apparently precise phase measurement is not meaningful when the signal-to-noise ratio is poor or the time reference is uncertain.

Limitations, Best Practices, and Contemporary Tools

Use radians in calculus and in formulas involving angular frequency. State whether amplitudes are peak or RMS. Preserve quadrant information with atan2. Check dimensions: angles are dimensionless, angular frequency is rad/s, impedance is Ω, and power is W. When adding complex quantities, use rectangular form; when multiplying or dividing, polar form is often more efficient.

Computer algebra systems, circuit simulators, oscilloscopes, and Python or MATLAB tools can calculate complex quantities and frequency responses. These tools support engineering judgment but do not replace model validation. AI-assisted tools can help generate symbolic manipulations or code, yet every result must be checked against units, limiting cases, measured data, and physical constraints. In safety-critical robotics and power systems, undocumented numerical assumptions and unverified generated code are unacceptable.

Chapter Summary

Radians connect angular displacement with arc length and provide the natural unit for calculus. The unit circle defines sine, cosine, tangent, their signs, periodicity, and identities. Sinusoidal models specify offset, amplitude, angular frequency, period, frequency, and phase; differentiation connects displacement, velocity, and acceleration.

Complex numbers combine real and imaginary components and may be written in rectangular, polar, or exponential form. Euler’s formula links exponentials with trigonometric functions, making multiplication, division, powers, and roots especially systematic. Phasors represent sinusoidal steady-state quantities by magnitude and phase. They transform differential circuit relationships into algebraic relationships involving impedance. In mechatronics, these methods support vibration diagnosis, signal analysis, AC circuit design, and planar robotic motion, provided their assumptions and limitations are respected.

Key Terms

Radian: Angular measure equal to arc length divided by radius.

Angular frequency: Rate of phase change, measured in rad/s.

Amplitude: Maximum deviation from the mean value of a sinusoid.

Phase angle: Angular displacement specifying timing relative to a reference.

Period: Time required for one complete cycle.

Frequency: Number of cycles per second, measured in hertz.

Identity: Equation true for every value in its stated domain.

Complex number: Number of the form a + jb.

Imaginary unit: Quantity j satisfying j² = −1.

Conjugate: Complex number obtained by reversing the sign of the imaginary part.

Modulus: Magnitude of a complex number.

Argument: Angle of a complex number in the complex plane.

Euler’s formula: Relationship e^jθ = cos θ + j sin θ.

Phasor: Complex representation of a sinusoidal magnitude and phase.

Impedance: Complex opposition to sinusoidal current, measured in ohms.

Reactance: Imaginary component of impedance.

RMS value: Effective value of a periodic quantity for power calculations.

Power factor: Cosine of the voltage-current phase difference.

Resonance: Condition in which reactive effects cancel at a characteristic frequency.

Review Questions

Short Answer Questions

1. Why is the radian the preferred angular unit in calculus and engineering analysis?

1. Define amplitude, period, angular frequency, and phase in a sinusoidal model.

1. State Euler’s formula and explain its connection with polar form.

1. What is the difference between a complex number’s modulus and argument?

1. Under what conditions is a phasor representation valid?

Long Answer Questions

1. Derive the relationship between arc length and angular displacement, then explain its use in robotic-joint motion.

1. Explain the principal trigonometric identities and demonstrate how an addition formula can be used to simplify a signal model.

1. Convert a general complex number between rectangular, polar, and exponential forms, explaining quadrant selection.

1. Derive the impedance of a series RLC circuit and discuss the physical meaning of resonance.

1. Compare peak and RMS phasor conventions and explain why mixing them can produce incorrect power calculations.

Analytical / Application Questions

1. A rotating arm has length 0.25 m and angular position θ(t) = 3t² rad. Determine its angular velocity and tangential speed at t = 2 s.

1. A sensor signal is y(t) = 1.5 + 0.4sin(20πt − π/4). Determine its mean, amplitude, frequency, period, and value at t = 0.

1. A series circuit has R = 8 Ω, L = 40 mH, C = 100 μF, and f = 50 Hz. Compute its complex impedance in rectangular form and determine whether its current leads or lags the source voltage.

Multiple Choice Questions with Answers

Question 1

A complete revolution is equal to which angular displacement?

A. π/2 rad

B. π rad

C. 2π rad

D. 4π rad

Correct Answer: C. 2π rad

Question 2

For x(t) = 3 + 2cos(20πt), what is the frequency?

A. 10 Hz

B. 20 Hz

C. 40π Hz

D. 20π Hz

Correct Answer: A. 10 Hz

Question 3

What is the modulus of z = −6 + 8j?

A. 2

B. 10

C. 14

D. 100

Correct Answer: B. 10

Question 4

Which expression is Euler’s formula?

A. e^{jθ} = sin θ + j cos θ

B. e^{jθ} = cos θ + j sin θ

C. e^{jθ} = cos θ − j sin θ

D. e^{jθ} = θ(cos θ + j sin θ)

Correct Answer: B. e^{jθ} = cos θ + j sin θ

Question 5

The impedance of an ideal inductor is:

A. R

B. 1/(jωL)

C. jωL

D. −jωC

Correct Answer: C. jωL

Question 6

For a sinusoidal voltage with peak value 10 V, the RMS value is approximately:

A. 5.00 V

B. 7.07 V

C. 10.00 V

D. 14.14 V

Correct Answer: B. 7.07 V

Question 7

If two phasors are multiplied, their magnitudes are multiplied and their:

A. angles are added

B. angles are multiplied

C. angles are discarded

D. real parts are added only

Correct Answer: A. angles are added

Question 8

At ideal series RLC resonance, which condition holds?

A. ωL = 0

B. 1/(ωC) = 0

C. ωL = 1/(ωC)

D. R = 0 is required

Correct Answer: C. ωL = 1/(ωC)

Question 9

For z = 3e^{jπ/2}, the rectangular form is:

A. 3

B. −3j

C. 3j

D. −3

Correct Answer: C. 3j

Question 10

A phasor method is most directly applicable to:

A. arbitrary transient signals in nonlinear systems

B. sinusoidal steady state in linear time-invariant systems

C. single instantaneous measurements only

D. three-dimensional orientation using one scalar angle

Correct Answer: B. sinusoidal steady state in linear time-invariant systems