Part V · Complex Numbers, Trigonometry, and Advanced Functions

Chapter 15 — Trigonometric, Sinusoidal, and Complex Exponential Functions

Why This Matters

Sinusoids model periodic signals. Their amplitude, frequency, phase, and rate of change all have geometric meaning on the unit circle. Complex exponentials turn trigonometric identities and phase shifts into ordinary algebra.

15.1 Sine and cosine from the unit circle

On the unit circle, the point at radian angle $\theta$ has coordinates

(\cos\theta,\sin\theta).

Therefore

\sin^2\theta+\cos^2\theta=1.

Both functions are $2\pi$ -periodic. Sine is odd, $\sin(-\theta)=-\sin\theta$ ; cosine is even, $\cos(-\theta)=\cos\theta$ .

The signal

v(t)=V_0+A\cos(\omega t+\phi)

has offset $$V_0$$ , amplitude $$|A|$$ , angular frequency $\omega$ radians per second, period $T=2\pi/\omega$ when $\omega>0$ , and phase $\phi$ . A phase shift moves a wave horizontally; it is not a change in amplitude.

Coordinates, not triangle mnemonics. A point rotating once around the unit circle has horizontal coordinate cos θ and vertical coordinate sin θ. Signs, periodicity, and the identity sin²θ+cos²θ=1 follow immediately from the circle equation, including in quadrants where a right-triangle picture is awkward.

(\cos\theta)^2+(\sin\theta)^2=1

Changing phase moves a sinusoid while preserving its unit-circle origin.

15.2 Tangent, reciprocal functions, and related rates

The remaining elementary trigonometric functions are ratios or reciprocals of sine and cosine:

\tan x=\frac{\sin x}{\cos x},\qquad \sec x=\frac1{\cos x},\qquad \csc x=\frac1{\sin x},\qquad \cot x=\frac{\cos x}{\sin x}.

Each formula carries a restriction. For example, $\tan x$ and $\sec x$ are undefined where $\cos x=0$ , namely at $x=\pi/2+k\pi$ ; $\csc x$ and $\cot x$ are undefined where $\sin x=0$ , namely at $x=k\pi$ . Tangent and cotangent have period $\pi$ , whereas secant and cosecant have period $2\pi$ . Since $|\sin x|\le1$ and $|\cos x|\le1$ , any defined value of $\sec x$ or $\csc x$ has magnitude at least $$1$$ .

The quotient rule gives

(\tan x)'=\frac{\cos^2x+\sin^2x}{\cos^2x}=\sec^2x, \qquad (\cot x)'=-\csc^2x.

The reciprocal rule similarly gives

(\sec x)'=\sec x\tan x,\qquad (\csc x)'=-\csc x\cot x.

These formulas are valid only at inputs where the relevant function is defined. Their reverse forms supply useful integrals on any interval that does not cross a singularity:

\int\tan x\,dx=-\ln|\cos x|+C,qquad \int\cot x\,dx=\ln|\sin x|+C,

\int\sec x\,dx=\ln|\sec x+\tan x|+C,qquad \int\csc x\,dx=\ln|\csc x-\cot x|+C.

The four parameters of a physical sinusoid. A general AC waveform has the mathematical form $v(t) = V_0 + A\cos(\omega t + \phi)$ .

DC Offset $$V_0$$ : vertical midline of the oscillation; average value over one complete period.
Amplitude $$A$$ : peak deviation from the midline ( $V_{\text{peak}} = V_0 + |A|$ , $V_{\text{trough}} = V_0 - |A|$ ).
Angular Frequency $\omega$ : rotation speed in $\text{rad/s}$ ; related to standard frequency $$f$$ (in $\text{Hz}$ ) by $\omega = 2\pi f$ , and period $T = \frac{2\pi}{\omega} = \frac{1}{f}$ .
Phase Angle $\phi$ : horizontal time shift; the peak occurs when $\omega t + \phi = 0 \implies t_{\text{peak}} = -\phi/\omega$ .

Extracting parameters from an oscilloscope trace

From

Measured trace: minimum $1\text{ V}$ , maximum $5\text{ V}$ , period $T = 2\text{ ms}$ , first maximum at $t = 0.25\text{ ms}$ .

Offset $V_0 = \frac{5 + 1}{2} = 3\text{ V}$ .
Amplitude $A = \frac{5 - 1}{2} = 2\text{ V}$ .
Frequency $\omega = \frac{2\pi}{T} = \frac{2\pi}{0.002} = 1000\pi\text{ rad/s} \approx 3142\text{ rad/s}$ ( $f = 500\text{ Hz}$ ).
Phase: $\omega(0.00025) + \phi = 0 \implies \phi = -1000\pi(0.00025) = -\frac{\pi}{4}\text{ rad} = -45^\circ$ .

Result. Model: $v(t) = 3 + 2\cos(1000\pi t - \pi/4)\text{ V}$ .

15.3 Derivatives and series

With radian measure,

\frac d{dx}\sin x=\cos x,\qquad \frac d{dx}\cos x=-\sin x.

Repeated differentiation cycles every four derivatives. Taylor’s theorem then gives

\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots,

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots.

These series converge for every real $$x$$ . In particular, $\lim_{x\to0}\sin x/x=1$ , a central small-angle fact.

The expression $\operatorname{sinc}x=\sin x/x$ is therefore assigned the continuous value $\operatorname{sinc}(0)=1$ . This normalized sinc function is common in signal processing; its value at zero comes from a limit, not division by zero.

Trigonometric functions define circular motion. On the unit circle $$x^2 + y^2 = 1$$ , a point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$ .

Pythagorean identity and angle-sum formulas

Starting point. Let $(x, y) = (\cos\theta, \sin\theta)$ lie on the unit circle $$x^2 + y^2 = 1$$ .

\[\cos^2\theta + \sin^2\theta = 1\]
Substitute coordinates directly into the circle equation $$x^2 + y^2 = 1$$ .reversible
\[1 + \tan^2\theta = \sec^2\theta\]
Divide by $\cos^2\theta$ (for $\cos\theta \neq 0$ ).reversible
\[\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\]
Double-angle formula from the angle-addition identity $\cos(\alpha + \beta)$ with $\alpha = \beta = \theta$ .reversible
\[\sin(2\theta) = 2\sin\theta\cos\theta\]
Double-angle formula for sine.reversible

Conclusion. Trigonometric identities are geometric facts about circular projection. In electronics, double-angle formulas explain frequency doubling in non-linear mixers.

Power-reduction: $\cos^2\theta = \frac{1 + \cos(2\theta)}{2}$ , $\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$ (essential for AC power integrals).
Product-to-sum: $\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$ (essential for amplitude modulation).
Derivatives: $(\sin x)' = \cos x$ , $(\cos x)' = -\sin x$ , $(\tan x)' = \sec^2 x$ .

15.4 Euler’s formula

Insert $$ix$$ into the power series for $$e^x$$ . Even powers of $$i$$ alternate between $$1$$ and $$-1$$ ; odd powers provide a factor $$i$$ . Grouping real and imaginary terms gives

e^{i\theta}=\cos\theta+i\sin\theta.

This is Euler’s formula. It does not replace real sine and cosine with “imaginary tricks”; it packages their linked behaviour into one expression.

Adding angles now follows from multiplication:

e^{i(a+b)}=e^{ia}e^{ib}.

Use Euler’s formula on both sides:

\cos(a+b)+i\sin(a+b) =(\cos a+i\sin a)(\cos b+i\sin b).

Multiplication gives a real part and an imaginary part,

(\cos a\cos b-\sin a\sin b) +i(\sin a\cos b+\cos a\sin b).

Equality of complex numbers requires equality of the corresponding parts, so

\cos(a+b)=\cos a\cos b-\sin a\sin b,

\sin(a+b)=\sin a\cos b+\cos a\sin b.

Putting $b=-\pi/2$ shows $\cos(x-\pi/2)=\sin x$ : sine and cosine are the same shape with a quarter-cycle phase shift. This derivation is often more reliable than memorising several separate identities.

Engineering Connection — phasor intuition

The real part of $Ae^{i(\omega t+\phi)}$ is $A\cos(\omega t+\phi)$ . For linear steady-state sinusoidal calculations, the common time factor can be handled with complex amplitudes, then a real physical voltage or current is recovered at the end. The method has conditions; it is not a replacement for transient analysis.

One compact expression carries two coordinates. Euler's formula places cosine in the real coordinate and sine in the imaginary coordinate of a unit complex rotation. Multiplying such expressions therefore adds angles while multiplying magnitudes, which links circular motion, phasors, and trigonometric identities.

e^{i\theta}=\cos\theta+i\sin\theta

A rotating unit vector in the complex plane whose tip projects onto the real axis as cosine and the imaginary axis as sine.

15.5 Roots of unity

The $$n$$ th roots of unity solve $$z^n=1$$ . In polar form, their distinct values are

z_k=e^{2\pi ik/n},\qquad k=0,1,\ldots,n-1.

They are equally spaced around the unit circle. Complex conjugate roots occur in pairs when polynomial coefficients are real; paired factors can combine into real quadratic factors. For example, roots $e^{i\theta}$ and $e^{-i\theta}$ produce

(z-e^{i\theta})(z-e^{-i\theta}) =z^2-2(\cos\theta)z+1,

which has real coefficients. The imaginary components cancel because the roots are conjugates.

Euler's formula unites trigonometry and exponential growth. $e^{i\theta} = \cos\theta + i\sin\theta$ connects circular rotation in the complex plane to exponential arithmetic.

Derivation of Euler's formula via Taylor series

Starting point. Expand $$e^z$$ , $\cos\theta$ , and $\sin\theta$ using their Taylor series at $$0$$ .

\[e^{i\theta} = \sum_{n=0}^\infty \frac{(i\theta)^n}{n!} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdots\]
Substitute $z = i\theta$ into the exponential series.reversible
\[e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)\]
Group even powers ( $$i^2 = -1, i^4 = 1$$ ) as real part and odd powers ( $$i^1 = i, i^3 = -i$$ ) as imaginary part.reversible
\[e^{i\theta} = \cos\theta + i\sin\theta\]
Recognise the real series as $\cos\theta$ and the imaginary series as $\sin\theta$ .reversible

Conclusion. $e^{i\theta} = \cos\theta + i\sin\theta$ . Consequently, $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$ .

Euler's identity: the most beautiful equation

From

Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$ .
Set $\theta = \pi$ .

$\cos\pi = -1, \quad \sin\pi = 0$ .
$e^{i\pi} = -1 + 0i = -1$ .
$e^{i\pi} + 1 = 0$ .

Result. $e^{i\pi} + 1 = 0$ . It links the five fundamental constants of mathematics: $e, i, \pi, 1, 0$ .

15.6 Reading and constructing a sinusoid

A sinusoid is best understood as a moving unit-circle coordinate. In $A\cos(\omega t+\phi)$ , the input angle advances at $\omega$ radians each second. One complete turn requires $2\pi$ radians, which explains

T=\frac{2\pi}{|\omega|},\qquad f=\frac1T=\frac{|\omega|}{2\pi}.

The amplitude is $$|A|$$ , not necessarily $$A$$ : a negative coefficient can be absorbed as a phase shift because $-\cos\theta=\cos(\theta+\pi)$ . The horizontal shift is found by setting the internal angle to a landmark value. For example, $\cos(\omega t+\phi)$ has a maximum when $\omega t+\phi=2\pi k$ .

[Figure: Plot

v(t)=V_0+A\cos(\omega t+\phi)

over two periods. Mark the midline

\(V_0\)

, peak values

V_0\pm|A|

, period

\(T\)

, and the phase-shifted first maximum.]

Example — decode a voltage waveform

For

v(t)=2+3\sin(100\pi t-\pi/6)\ \mathrm V,

the midline is $$2$$ V and the amplitude is $$3$$ V. The angular frequency is $100\pi$ rad/s, hence

f=\frac{100\pi}{2\pi}=50\ \mathrm{Hz},\qquad T=0.020\ \mathrm s.

The phase $-\pi/6$ means the sine waveform is delayed relative to $3\sin(100\pi t)$ by

\frac{\pi/6}{100\pi}=\frac1{600}\ \mathrm s.

It is incorrect to call $-\pi/6$ “a delay of $\pi/6$ seconds”: phase is an angle and becomes a time shift only after division by angular frequency.

From first principles. A sinusoid records one coordinate of uniform circular motion. Amplitude sets the vertical radius, angular frequency sets how quickly the point turns, and phase states where the rotation begins.

v(t)=V_0+A\cos(\omega t+\phi)

A unit-circle coordinate linked to a sinusoid over two periods.

15.7 Identities as geometry and algebra

The Pythagorean identity comes from the unit-circle equation $$x^2+y^2=1$$ . Dividing it by $\cos^2x$ , where $\cos x\ne0$ , gives

1+\tan^2x=\sec^2x.

Dividing by $\sin^2x$ , where $\sin x\ne0$ , gives

1+\cot^2x=\csc^2x.

The restrictions matter because division by a zero coordinate is not allowed. The addition formulas in Section 15.4 generate useful double-angle formulas:

\sin2x=2\sin x\cos x,\qquad \cos2x=\cos^2x-\sin^2x.

Using $\sin^2x+\cos^2x=1$ , cosine also has the alternatives $\cos2x=1-2\sin^2x=2\cos^2x-1$ . Choose the version that contains the quantity already present in a problem.

Example — simplify without changing the domain

For inputs where $\cos x\ne0$ ,

\frac{1-\sin^2x}{\cos x} =\frac{\cos^2x}{\cos x} =\cos x.

The original expression is undefined at $\cos x=0$ , whereas $\cos x$ itself is defined there. Thus this is a simplification on the original domain, not an unqualified equality of functions on all real numbers.

Radians make calculus formulas natural. In calculus, all trigonometric derivatives require angles to be measured in radians. The identity $\lim_{\theta\to0} \frac{\sin\theta}{\theta} = 1$ is true only in radians.

Why radians are the natural angle unit

Starting point. By definition of radian measure on a circle of radius $$r$$ : arc length $s = r\theta$ .

\[\text{For } r = 1: \quad s = \theta\]
On the unit circle, angle in radians equals arc length along the circumference.reversible
\[\lim_{\theta\to0} \frac{\sin\theta}{\theta} = 1\]
The chord length $2\sin(\theta/2)$ approaches the arc length $\theta$ as $\theta \to 0$ .reversible
\[\frac{d}{d\theta}\sin\theta = \lim_{h\to0} \frac{\sin(\theta+h) - \sin\theta}{h} = \cos\theta\]
The standard derivative formula $\frac{d}{d\theta}\sin\theta = \cos\theta$ holds without extra conversion constants only in radians.reversible

Conclusion. If degrees were used, $\frac{d}{d\theta}\sin\theta^\circ = \frac{\pi}{180} \cos\theta^\circ$ . Radians eliminate the awkward $\frac{\pi}{180}$ scale factor throughout physics and engineering.

Conversion: $\pi\text{ radians} = 180^\circ \implies 1\text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$ .
Degrees to radians: multiply by $\frac{\pi}{180^\circ}$ .
Radians to degrees: multiply by $\frac{180^\circ}{\pi}$ .

15.8 Differentiation, rates, and small-angle reasoning

The derivative formula $(\sin x)'=\cos x$ depends on radian measure. If an angle is measured in degrees, an extra constant conversion factor appears. This is why calculus and engineering signal models conventionally use radians.

Example — differentiate an AC waveform

v(t)=12\cos(500t+\pi/4)\ \mathrm V,

then the chain rule gives

\frac{dv}{dt}=-12(500)\sin(500t+\pi/4) =-6000\sin(500t+\pi/4)\ \mathrm{V/s}.

The factor $$500$$ is not optional; it records how rapidly the input angle changes. At a voltage peak, the sine factor is zero, so the graph has horizontal tangent. At a midline crossing, the magnitude of the rate is largest.

Engineering Example — capacitor current leads voltage rate

For an ideal capacitor, $i_C=C\,dv_C/dt$ . If $v_C(t)=V_m\cos(\omega t)$ , then

i_C(t)=-\omega CV_m\sin(\omega t) =\omega CV_m\cos(\omega t+\pi/2).

The current amplitude is $\omega CV_m$ and the current is one quarter-cycle ahead of voltage in this cosine convention. This conclusion comes from differentiation; it should not be extended blindly to non-sinusoidal sources or non-ideal components.

Common Mistakes

Confusing angular frequency $\omega$ (rad/s) with ordinary frequency $$f$$ (Hz).
Applying an identity after division without retaining its domain restriction.
Forgetting the chain-rule multiplier when differentiating $\sin(\omega t+\phi)$ .
Treating a phase angle as a time interval without using $\omega$ .

Exercises

Find amplitude, period, and phase of $3\sin(4t-\pi/2)$ .
Verify $\sin(a+b)$ using Euler’s formula.
Find all fourth roots of unity and plot their locations conceptually.
Differentiate $5\cos(3t+\pi/4)$ .
Explain why a phase angle must be stated with a reference convention.
Find $\frac{d}{dt}\tan(2t)$ and state the values of $$t$$ at which the original function is undefined.
A camera is $$10$$ m from a road. At an instant $$x=10$$ m and $$dx/dt=15$$ m/s. Use $\tan\theta=x/10$ to find $d\theta/dt$ at that instant.
For $v(t)=4-2\cos(20t)$ V, identify offset, amplitude, frequency, and range.
Prove $1+\tan^2x=\sec^2x$ and state where the proof applies.
Differentiate $7\sin(3t-\pi/5)$ and interpret the maximum possible rate.
A capacitor of $10\ \mu$ F has $v(t)=5\cos(1000t)$ V. Find $$i_C(t)$$ .