Chapter 13

Trigonometric Functions: define angle-ratio relations as functions (sin, cos, tan)

Trigonometric functions are like a precise ruler that shows how side-length ratios change as an angle changes. They are essential for expressing periodic phenomena such as clock cycles and seasons. This regularity supports wave motion, signal analysis, and modern AI techniques such as positional encoding .

On the unit circle, rotating by angle $\theta$ moves the point as $(\cos\theta, \sin\theta)$ .

sin θ: vertical height

sin(θ) = -0.87

cos θ: horizontal length

cos(θ) = 0.50

tan θ: slope ratio

tan(θ) = -1.73

Current at θ:sinθ = -0.87cosθ = 0.50tanθ = -1.73

Core relation: tanθ = sinθ / cosθ (cosθ ≠ 0)

On the unit circle, the horizontal projection is $\cos\theta$ , the vertical projection is $\sin\theta$ , and the slope ratio is $\tan\theta=\frac{\sin\theta}{\cos\theta}$ .

Create angle $\theta$ and locate point $P$ on the circle.
The x-coordinate of $P$ is $\cos\theta$ , and the y-coordinate is $\sin\theta$ .
When $\cos\theta\neq0$ , read slope using $\tan\theta=\sin\theta/\cos\theta$ .

What are trigonometric functions?

Concept: In a right triangle, trigonometric functions describe side ratios as functions of angle

\theta

. In formulas:

\sin\theta=\frac{\text{height}}{\text{hypotenuse}}

\cos\theta=\frac{\text{base}}{\text{hypotenuse}}

\tan\theta=\frac{\text{height}}{\text{base}}

Intuition (Ferris wheel analogy): On the unit circle (radius 1), after rotating by

\theta

, your position is

(\cos\theta,\sin\theta)

. So

\cos\theta

is horizontal position and

\sin\theta

is vertical position.

Math explanation: The unit-circle equation is

x^2+y^2=1

. Substituting

(x,y)=(\cos\theta,\sin\theta)

gives the core identity

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

. Also, slope is rise over run, so

\tan\theta=\frac{\sin\theta}{\cos\theta}

Practical bridge: Many real datasets are cyclic (day/night, seasons, direction). Instead of feeding raw linear numbers, encode them with trigonometric features so models learn smooth circular structure. For beginners, convert time first using 24 hours = 360 degrees before connecting to

\theta=2\pi\cdot\frac{t}{T}

It resolves cyclic boundary problems. For example, 23:00 and 00:00 are close in time but far apart as raw numbers. Using

\sin/\cos

encoding preserves their circular closeness.

It is foundational in Transformer positional encoding, where mixtures of sinusoidal frequencies provide order-aware representations.

It is the core language of wave and frequency modeling, which underpins signal processing and time-series AI.

Convert angle/time to radians, then use paired features

(\sin\theta,\cos\theta)

as model input.

Apply trigonometric encoding to cyclic data such as weekdays, seasons, and direction to remove discontinuities.

Cosine similarity also uses the same trigonometric intuition. For vectors

\mathbf{a},\mathbf{b}

, similarity is

\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}

, where the key is direction over magnitude. If directions align,

\cos\theta\approx1

; if orthogonal,

0

; if opposite,

-1

. This is why recommendation systems, embedding search, and RAG retrieval use cosine similarity to find semantically close items even when raw lengths differ.

This set minimizes repeated question types, so identify type first. Solve in order: (1) type identification (coordinate/sign/period/identity/ML encoding) →

(2) choose unit-circle coordinate

(\cos\theta,\sin\theta)

or formula →

(3) calculate and verify sign.

As in Chapter 04 style reasoning, start from definitions and narrow down step by step. Typical order is easy (coordinate/sign) → medium (period/sums/identity) → hard (ML encoding/composite). For time questions, converting with 24 hours = 360 degrees is safer than starting from raw

\pi

arithmetic.

Example problems and step-by-step solutions:

Problem $\theta=180^\circ$
Solution $(-1,0)$

Problem $y=\cos(6x)$
Solution $\frac{360}{k}$

Problem $hour=18$
Solution $24\text{h}=360^\circ$

Problem	Solution
Example 1 (Easy). On the unit circle, what is the y-coordinate at $\theta=180^\circ$ ?	Point is $(-1,0)$ , so y-coordinate is $0$ . Therefore $\sin\theta=0$ .
Example 2 (Medium). What is the period (degrees) of $y=\cos(6x)$ ?	Use $\frac{360}{k}$ with $k=6$ : $\frac{360}{6}=60$ .
Example 3 (Hard). For $hour=18$ , find $\sin(2\pi\cdot hour/24)$ .	$24\text{h}=360^\circ$ , so $18\text{h}=270^\circ$ , and $\sin270^\circ=-1$ .

How to solve by type

Type $Unit-circle coordinate$
Description $x+y$
How to get the answer $x=\cos\theta$

Type $Quadrant sign$
Description $Determine + / - of function values$
How to get the answer $\sin$

Type $Period calculation$
Description $y=\sin(kx), y=\cos(kx)$
How to get the answer $\frac{360}{k}$

Type $Identity/composite$
Description $\sin^2\theta+\cos^2\theta$
How to get the answer $Apply identity, then substitute$

Type $ML application (beginner)$
Description $Time/direction encoding$
How to get the answer $\theta=2\pi\cdot\frac{t}{T}$

Type	Description	How to get the answer
Unit-circle coordinate	Ask for x, y, or $x+y$	Locate standard angle, read $x=\cos\theta$ , $y=\sin\theta$
Quadrant sign	Determine + / - of function values	Check signs of x,y by quadrant, then infer $\sin$ , $\cos$ , $\tan=\frac{y}{x}$
Period calculation	$y=\sin(kx), y=\cos(kx)$	Period in degrees is $\frac{360}{k}$
Identity/composite	$\sin^2\theta+\cos^2\theta$ , sums/ratios	Apply identity, then substitute
ML application (beginner)	Time/direction encoding	Convert with 24h=360° first, then connect to $\theta=2\pi\cdot\frac{t}{T}$

Example (Unit-circle coordinate)

\theta=270^\circ

, find

x+y

Solution

1) Point at

270^\circ

(0,-1)

x+y=0+(-1)=-1

So the answer is -1.

Example (Quadrant sign)

In Quadrant II, what is the sign of

\tan\theta

Solution

1) In QII,

\sin\theta>0

\cos\theta<0

2) So

\tan\theta=\frac{\sin\theta}{\cos\theta}<0

So the answer is negative.

Example (Period calculation)

Find the period (degrees) of

y=\sin(8x)

Solution

1) Use period formula

\frac{360}{k}

2) With

k=8

\frac{360}{8}=45

So the answer is 45.

Example (ML application, without direct $\pi$ arithmetic)

For

hour=6

, when a day (24h) is mapped to 360°, what is the angle and

\sin\theta

Solution

1) 24h = 360°, so 1h = 15°

2) 6h =

6\times15=90^\circ

\sin90^\circ=1

So the answer is 1.

(Equivalent formula form:

\theta=2\pi\cdot\frac{6}{24}=\frac{\pi}{2}

What are trigonometric functions?

Concept: In a right triangle, trigonometric functions describe side ratios as functions of angle

\theta

. In formulas:

\sin\theta=\frac{\text{height}}{\text{hypotenuse}}

\cos\theta=\frac{\text{base}}{\text{hypotenuse}}

\tan\theta=\frac{\text{height}}{\text{base}}

Intuition (Ferris wheel analogy): On the unit circle (radius 1), after rotating by

\theta

, your position is

(\cos\theta,\sin\theta)

. So

\cos\theta

is horizontal position and

\sin\theta

is vertical position.

Math explanation: The unit-circle equation is

x^2+y^2=1

. Substituting

(x,y)=(\cos\theta,\sin\theta)

gives the core identity

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

. Also, slope is rise over run, so

\tan\theta=\frac{\sin\theta}{\cos\theta}

\theta=2\pi\cdot\frac{t}{T}

It resolves cyclic boundary problems. For example, 23:00 and 00:00 are close in time but far apart as raw numbers. Using

\sin/\cos

encoding preserves their circular closeness.

It is foundational in Transformer positional encoding, where mixtures of sinusoidal frequencies provide order-aware representations.

It is the core language of wave and frequency modeling, which underpins signal processing and time-series AI.

Convert angle/time to radians, then use paired features

(\sin\theta,\cos\theta)

as model input.

Apply trigonometric encoding to cyclic data such as weekdays, seasons, and direction to remove discontinuities.

Cosine similarity also uses the same trigonometric intuition. For vectors

\mathbf{a},\mathbf{b}

, similarity is

\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}

, where the key is direction over magnitude. If directions align,

\cos\theta\approx1

; if orthogonal,

0

; if opposite,

-1

. This is why recommendation systems, embedding search, and RAG retrieval use cosine similarity to find semantically close items even when raw lengths differ.

This set minimizes repeated question types, so identify type first. Solve in order: (1) type identification (coordinate/sign/period/identity/ML encoding) →

(2) choose unit-circle coordinate

(\cos\theta,\sin\theta)

or formula →

(3) calculate and verify sign.

\pi

arithmetic.

Example problems and step-by-step solutions:

Problem $\theta=180^\circ$
Solution $(-1,0)$

Problem $y=\cos(6x)$
Solution $\frac{360}{k}$

Problem $hour=18$
Solution $24\text{h}=360^\circ$

Problem	Solution
Example 1 (Easy). On the unit circle, what is the y-coordinate at $\theta=180^\circ$ ?	Point is $(-1,0)$ , so y-coordinate is $0$ . Therefore $\sin\theta=0$ .
Example 2 (Medium). What is the period (degrees) of $y=\cos(6x)$ ?	Use $\frac{360}{k}$ with $k=6$ : $\frac{360}{6}=60$ .
Example 3 (Hard). For $hour=18$ , find $\sin(2\pi\cdot hour/24)$ .	$24\text{h}=360^\circ$ , so $18\text{h}=270^\circ$ , and $\sin270^\circ=-1$ .

How to solve by type

Type $Unit-circle coordinate$
Description $x+y$
How to get the answer $x=\cos\theta$

Type $Quadrant sign$
Description $Determine + / - of function values$
How to get the answer $\sin$

Type $Period calculation$
Description $y=\sin(kx), y=\cos(kx)$
How to get the answer $\frac{360}{k}$

Type $Identity/composite$
Description $\sin^2\theta+\cos^2\theta$
How to get the answer $Apply identity, then substitute$

Type $ML application (beginner)$
Description $Time/direction encoding$
How to get the answer $\theta=2\pi\cdot\frac{t}{T}$

Type	Description	How to get the answer
Unit-circle coordinate	Ask for x, y, or $x+y$	Locate standard angle, read $x=\cos\theta$ , $y=\sin\theta$
Quadrant sign	Determine + / - of function values	Check signs of x,y by quadrant, then infer $\sin$ , $\cos$ , $\tan=\frac{y}{x}$
Period calculation	$y=\sin(kx), y=\cos(kx)$	Period in degrees is $\frac{360}{k}$
Identity/composite	$\sin^2\theta+\cos^2\theta$ , sums/ratios	Apply identity, then substitute
ML application (beginner)	Time/direction encoding	Convert with 24h=360° first, then connect to $\theta=2\pi\cdot\frac{t}{T}$

Example (Unit-circle coordinate)

\theta=270^\circ

, find

x+y

Solution

1) Point at

270^\circ

(0,-1)

x+y=0+(-1)=-1

So the answer is -1.

Example (Quadrant sign)

In Quadrant II, what is the sign of

\tan\theta

Solution

1) In QII,

\sin\theta>0

\cos\theta<0

2) So

\tan\theta=\frac{\sin\theta}{\cos\theta}<0

So the answer is negative.

Example (Period calculation)

Find the period (degrees) of

y=\sin(8x)

Solution

1) Use period formula

\frac{360}{k}

2) With

k=8

\frac{360}{8}=45

So the answer is 45.

Example (ML application, without direct $\pi$ arithmetic)

For

hour=6

, when a day (24h) is mapped to 360°, what is the angle and

\sin\theta

Solution

1) 24h = 360°, so 1h = 15°

2) 6h =

6\times15=90^\circ

\sin90^\circ=1

So the answer is 1.

(Equivalent formula form:

\theta=2\pi\cdot\frac{6}{24}=\frac{\pi}{2}