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Trigonometric Functions

Introduction

Trigonometric functions connect angles, circles, triangles, graphs, periodic motion, and later ideas in calculus and complex numbers. This aiMOOC is designed for learners in Grades 11–13. You will move from geometric definitions to functions on the real line, learn to transform and solve trigonometric equations, and apply the ideas to models of repeating phenomena.

By the end of the course, you should be able to explain trigonometric functions from the unit circle, work confidently in radians, sketch and transform graphs, prove and use core identities, solve equations over specified intervals, interpret inverse trigonometric functions, and construct mathematical models for periodic situations.

Datei:Radian Angle Definition.svg

A radian is defined through arc length: if an arc of length $s$ lies on a circle of radius $r$ , then the central angle is $θ = s / r$ radians. Radians make formulas in calculus and periodic modeling especially natural.

Foundations: Angles, Triangles, and the Unit Circle

Degrees and Radians

A complete turn is $36 0^{\circ} = 2 π$ radians, so $18 0^{\circ} = π$ radians. To convert degrees to radians, multiply by $π / 180$ . To convert radians to degrees, multiply by $180 / π$ .

Radians are dimensionless ratios, but writing "rad" can make your interpretation clearer. In higher mathematics, an angle given without a degree symbol is normally understood to be in radians.

Useful benchmark angles include $0$ , $π / 6$ , $π / 4$ , $π / 3$ , $π / 2$ , and their rotations around the circle. Rather than memorizing isolated values, connect each angle to special triangles and the coordinates on the unit circle.

Right-Triangle Ratios

For an acute angle $θ$ in a right triangle:

$\sin θ = \frac{opposite}{hypotenuse}$

$\cos θ = \frac{adjacent}{hypotenuse}$

$\tan θ = \frac{opposite}{adjacent}$

These ratios are often remembered as SOH-CAH-TOA. The triangle definition is useful for geometric measurement, but it does not by itself cover arbitrary positive and negative angles. The unit circle extends the definitions to all relevant real inputs.

The Unit Circle Definition

Datei:Unit circle trigonometry.svg

On the unit circle, an angle $θ$ measured from the positive horizontal axis determines a point $P = (x, y)$ . The central definitions are

$x = \cos θ$ and $y = \sin θ$ .

This single geometric idea explains signs in all four quadrants, exact values at special angles, periodicity, and the identity $\sin^{2} θ + \cos^{2} θ = 1$ . The tangent is the ratio $\tan θ = \sin θ / \cos θ$ whenever $\cos θ \neq 0$ .

The Six Trigonometric Functions

Function	Unit-circle form	Important restriction
Sine	$\sin θ = y$	Defined for every real $θ$
Cosine	$\cos θ = x$	Defined for every real $θ$
Tangent	$\tan θ = y / x$	Requires $x \neq 0$
Cosecant	$\csc θ = 1 / y$	Requires $y \neq 0$
Secant	$\sec θ = 1 / x$	Requires $x \neq 0$
Cotangent	$\cot θ = x / y$	Requires $y \neq 0$

The reciprocal relationships are $\csc θ = 1 / \sin θ$ , $\sec θ = 1 / \cos θ$ , and $\cot θ = 1 / \tan θ$ wherever the expressions are defined.

Graphs and Transformations

Basic Graphs and Periodicity

Datei:Mplwp sin cos tan piaxis.svg

The sine and cosine functions both have range $[- 1, 1]$ and period $2 π$ . Their graphs repeat because adding $2 π$ corresponds to one complete revolution around the unit circle.

The tangent function has period $π$ . It is undefined at $π / 2 + k π$ , where $k$ is an integer, because cosine is zero there. Its graph therefore has vertical asymptotes at those inputs.

The circle-to-graph viewpoint is powerful: as a point rotates around the unit circle, its vertical coordinate traces sine while its horizontal coordinate traces cosine.

Amplitude, Period, Phase Shift, and Midline

A general sinusoidal model can be written

$y = A \sin (B (x - C)) + D$

or

$y = A \cos (B (x - C)) + D$ .

For these forms, the amplitude is $| A |$ , the period is $2 π / | B |$ , the phase shift is $C$ , and the midline is $y = D$ . A negative value of $A$ reflects the graph across its midline. Horizontal scaling is controlled by $B$ .

Tangent transformations use similar horizontal and vertical parameters, but tangent has period $π / | B |$ and has no amplitude because it is unbounded.

Periodic Motion and Sinusoids

Datei:Simple harmonic motion.svg

Many idealized oscillations are modeled by $x (t) = A \cos (ω t + φ)$ or $x (t) = A \sin (ω t + φ)$ . Here $A$ is amplitude, $ω$ is angular frequency, and $φ$ is phase. The period is $T = 2 π / ω$ when $ω > 0$ . Examples include simplified spring motion, alternating electrical signals, sound components, and rotating mechanisms.

A model is only as good as its assumptions. Real systems may contain damping, noise, multiple frequencies, or changing amplitudes, so you should always compare the model with measured data.

Identities and Trigonometric Equations

Core Identities

The Pythagorean identity follows directly from the unit-circle equation $x^{2} + y^{2} = 1$ :

$\sin^{2} θ + \cos^{2} θ = 1$ .

Dividing this identity by $\cos^{2} θ$ gives $1 + \tan^{2} θ = \sec^{2} θ$ where cosine is nonzero. Dividing by $\sin^{2} θ$ gives $1 + \cot^{2} θ = \csc^{2} θ$ where sine is nonzero.

The angle-sum and angle-difference identities are

$\sin (α \pm β) = \sin α \cos β \pm \cos α \sin β$

and

$\cos (α \pm β) = \cos α \cos β \mp \sin α \sin β$ .

Setting $α = β$ produces the double-angle identities, including $\sin (2 α) = 2 \sin α \cos α$ and $\cos (2 α) = \cos^{2} α - \sin^{2} α$ .

Proving Identities Strategically

An identity is an equality that is true for every input where both sides are defined. To prove one, transform one side into the other using established identities and algebra. Good strategies include rewriting everything in sine and cosine, factoring, finding common denominators, and recognizing a Pythagorean expression.

Do not "prove" an identity by checking a few numerical values. Numerical checks can detect mistakes, but they do not establish a statement for every permitted input. Also track domain restrictions: an algebraic step involving division can exclude values.

Solving Trigonometric Equations

A trigonometric equation is true only for particular input values. Because the functions are periodic, equations often have infinitely many real solutions.

For example, $\sin x = 0$ has the general solution $x = k π$ , where $k$ is any integer. By contrast, if you are asked to solve on $[0, 2 π)$ , you list only the solutions in that interval.

A reliable process is to isolate a trigonometric expression, find reference-angle solutions, use quadrant information, add periodic repetitions when required, and verify that no algebraic transformation introduced invalid answers.

Inverse Trigonometric Functions

Datei:Mplwp inverse trigonometric functions piaxis.svg

Periodic functions are not one-to-one on their entire domains, so they cannot have global inverses without restrictions. The standard principal branches are:

Function	Domain	Principal range
$\arcsin x$	$[- 1, 1]$	$[- π / 2, π / 2]$
$\arccos x$	$[- 1, 1]$	$[0, π]$
$\arctan x$	All real numbers	$(- π / 2, π / 2)$

The expression $\sin^{- 1} x$ usually means inverse sine, not $1 / \sin x$ . The reciprocal of sine is cosecant. Writing $\arcsin x$ avoids this ambiguity.

Inverse trigonometric functions are useful when a ratio is known and an angle must be recovered. In applications, always check whether your calculator is in degree mode or radian mode.

Applications in Geometry and Modeling

Non-Right Triangles

The law of sines relates sides and opposite angles:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ .

Fehler beim Erstellen des Vorschaubildes:

The law of cosines generalizes the Pythagorean theorem:

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$ .

Use these relationships to solve non-right triangles in surveying, navigation, construction, and geometric modeling. Be careful with the ambiguous case of the law of sines: some side-angle data can produce zero, one, or two valid triangles.

Building a Sinusoidal Model

Suppose a quantity varies between a maximum $M$ and a minimum $m$ . Then a first estimate for amplitude is $(M - m) / 2$ and for the midline is $(M + m) / 2$ . Measure the time or input distance between repeating peaks to estimate the period. Convert the period $T$ to angular frequency with $B = 2 π / T$ , then choose a sine or cosine phase shift that matches a known reference point.

After fitting the model, interpret every parameter in context and test residuals or prediction errors. A graph that "looks sinusoidal" is not enough evidence by itself.

Grade 13 Extension: Calculus, Complex Numbers, and Fourier Ideas

Trigonometric Functions in Calculus

When the input is measured in radians,

$\frac{d}{d x} \sin x = \cos x$ and $\frac{d}{d x} \cos x = - \sin x$ .

The derivative of tangent is $\sec^{2} x$ wherever tangent is defined. These elegant derivative formulas are one reason radians are the natural angle measure in calculus. If an inner function is present, such as $\sin (3 x)$ , the chain rule contributes an additional factor.

Euler's Formula

For a real angle $x$ , Euler's formula connects exponential and trigonometric functions:

$e^{i x} = \cos x + i \sin x$ .

This identity links rotation in the complex plane with sine and cosine and helps explain why trigonometric functions appear throughout differential equations, waves, electrical engineering, and signal analysis.

Fourier Series as an Extension

A Fourier series represents many periodic functions as sums of sines and cosines with different frequencies. At this level, the key conceptual idea is that complicated periodic behavior can often be analyzed by decomposing it into simpler harmonic components.

This extension is not required for every Grade 11–13 curriculum, but it shows why fluency with amplitude, phase, frequency, and the unit-circle meaning of sine and cosine is so valuable in later mathematics and science.

Worked Examples

Example: Exact Values from the Unit Circle

Evaluate $\sin (7 π / 6)$ , $\cos (7 π / 6)$ , and $\tan (7 π / 6)$ .

The angle $7 π / 6$ is $π + π / 6$ , so its reference angle is $π / 6$ and it lies in the third quadrant. Both sine and cosine are negative there. Therefore $\sin (7 π / 6) = - 1 / 2$ and $\cos (7 π / 6) = - \sqrt{3} / 2$ . Their ratio gives $\tan (7 π / 6) = 1 / \sqrt{3} = \sqrt{3} / 3$ .

Example: Reading a Transformed Cosine

For $y = - 2 \cos (3 (x - π / 4)) + 1$ , the amplitude is $2$ , the period is $2 π / 3$ , the phase shift is $π / 4$ to the right, and the midline is $y = 1$ . The negative leading coefficient reflects the cosine curve across the midline.

Example: Solving on a Restricted Interval

Solve $2 \sin x = \sqrt{3}$ for $0 \leq x < 2 π$ .

First divide by two: $\sin x = \sqrt{3} / 2$ . The reference angle is $π / 3$ . Sine is positive in quadrants I and II, so the solutions are $x = π / 3$ and $x = 2 π / 3$ .

Example: Modeling a Ferris Wheel

A Ferris wheel has radius $12$ m, its center is $14$ m above the ground, and one revolution takes $30$ s. A rider starts at the lowest point. One model is $h (t) = 14 - 12 \cos (π t / 15)$ . The amplitude $12$ represents the radius, the midline $14$ represents the center height, and the angular frequency $π / 15$ gives period $30$ s. At $t = 10$ s, the model predicts $h (10) = 20$ m.

Common Errors and Productive Habits

Mixing degrees and radians is one of the most frequent errors. Write the unit during setup, and check calculator mode before evaluating. Another common error is confusing $\sin^{- 1} x$ with $1 / \sin x$ ; use $\arcsin x$ for the inverse function and $\csc x$ for the reciprocal.

When solving equations, do not stop after finding one reference angle. Use symmetry, quadrant signs, and periodicity to find every solution requested. When proving identities, work with exact expressions as long as possible rather than rounding early.

A strong study routine connects four representations: a geometric picture, an exact symbolic expression, a graph, and a numerical value. Moving between these views helps you detect errors and understand why formulas work.

Interactive Tasks

Quiz: Test Your Knowledge

What does one radian measure geometrically? (The angle subtending an arc equal in length to the radius) (!An angle of exactly one degree) (!The angle in every equilateral triangle) (!A complete revolution)

On the unit circle, which coordinate equals cosine of the angle? (The horizontal coordinate) (!The vertical coordinate) (!The radius squared) (!The arc length)

Which identity is valid for every real angle? (Sine squared plus cosine squared equals one) (!Sine plus cosine equals one) (!Tangent squared plus cosine squared equals one) (!Sine squared minus cosine squared equals one)

What is the period of the basic sine function? (Two pi) (!Pi) (!Half pi) (!Four pi)

What is the amplitude of y equals three sine x? (Three) (!One) (!Two) (!Six)

When is tangent undefined? (When cosine is zero) (!When sine is zero) (!When sine equals cosine) (!When the angle is negative)

What is the principal range of inverse sine? (Negative half pi to positive half pi) (!Zero to two pi) (!Zero to pi) (!All real numbers)

Which statement describes the solutions of sine x equals zero? (Integer multiples of pi) (!Odd multiples of half pi) (!Only zero) (!All real numbers)

In a sinusoidal model, what does the vertical shift determine? (The midline) (!The period) (!The horizontal scale) (!The domain)

When angles are measured in radians, what is the derivative of sine x? (Cosine x) (!Negative cosine x) (!Sine x) (!Tangent x)

Memory Game

Radian	Angle measure defined by arc length divided by radius
Amplitude	Maximum vertical distance from a sinusoid to its midline
Period	Input length required for one complete repetition
PhaseShift	Horizontal displacement of a periodic graph
UnitCircle	Circle of radius one centered at the origin
Identity	Equation true for every permitted input
Arcsine	Principal inverse function of sine
Asymptote	Line that a graph approaches in a limiting sense

Drag and Drop

Match the correct terms.	Topic
Horizontal unit-circle coordinate	Cosine
Vertical unit-circle coordinate	Sine
Ratio of sine to cosine	Tangent
Reciprocal of cosine	Secant
Reciprocal of sine	Cosecant

...

Crossword Puzzle

Radian	Which angle unit is defined using arc length and radius?
Periodic	What word describes a function that repeats after a fixed interval?
Tangent	Which trigonometric function equals sine divided by cosine?
Amplitude	What is the maximum displacement of a sinusoid from its midline called?
Identity	What is an equation called when it is true for every permitted input?
Sinusoid	What is a sine-shaped or cosine-shaped periodic curve called?

LearningApps

Cloze Text

A complete revolution measures

. On the unit circle, cosine gives the

. The sine value is the point's

. The identity linking sine and cosine is called the

. A sinusoid's maximum distance from its midline is its

. The length of one complete repetition is the

. A horizontal translation is often described as a

. The tangent function is undefined where

. Inverse trigonometric functions return a selected

. In calculus, the derivative formulas take their simplest form when angles use

.

Open-Ended Tasks

Easy

Unit circle poster: Create a clear image of the unit circle showing key angles in degrees and radians, coordinates, and the signs of sine and cosine in each quadrant.
Graph comparison: Produce a one-page visual comparison of sine, cosine, and tangent, labeling intercepts, extrema where relevant, periods, and tangent asymptotes.
Trigonometry interview: Interview a teacher, technician, engineer, craftsperson, musician, or scientist about a real situation where angles or periodic measurements matter, then summarize what you learned.
Calculator mode experiment: Evaluate the same angle expressions in degree mode and radian mode, record the different outputs, and explain why calculator mode changes the result.

Standard

Sinusoidal data project: Collect or find a small dataset with repeating behavior, fit a sine or cosine model, interpret its parameters, and compare predictions with the data.
Identity proof gallery: Write and illustrate three different trigonometric identity proofs, explaining the purpose of every algebraic step and any domain restrictions.
Trigonometry explainer video: Produce a short video that shows how rotating motion on the unit circle generates a sine or cosine graph.
Field measurement investigation: Visit a safe location such as a school courtyard, park, or public square and estimate an inaccessible height or distance using angle measurements and trigonometry; document assumptions and uncertainty.

Advanced

Inverse function analysis: Create a report explaining why sine, cosine, and tangent need restricted domains to have inverses, and justify the standard principal ranges using graphs.
Ferris wheel model: Design a realistic rotating-wheel scenario, derive a sinusoidal height function from physical parameters, and analyze at least three time-based questions.
Complex rotation project: Build a diagram or animation showing how multiplication by a complex number of modulus one represents rotation and connect it to Euler's formula.
Fourier investigation: Use a digital graphing or coding tool to approximate a non-sinusoidal periodic signal with several sine and cosine components, then explain how adding harmonics improves the approximation.

Learning Assessment

Model selection assessment: Given several periodic datasets, decide which can reasonably be modeled by a single sinusoid, justify your choices, and explain when a more complex model is needed.
Identity reasoning assessment: Prove a nontrivial trigonometric identity from established identities and annotate each transformation with its algebraic or trigonometric justification.
Equation solution assessment: Solve a trigonometric equation both on a restricted interval and in general form, then explain how periodicity changes the form of the answer.
Representation transfer assessment: Starting from a transformed trigonometric graph, reconstruct a possible equation and defend your parameter choices using amplitude, period, phase, and midline.
Application critique assessment: Evaluate a published or teacher-provided trigonometric model for a real phenomenon, identify its assumptions, and discuss the effect of measurement error or non-periodic influences.
Extension assessment: Explain how the unit-circle meaning of sine and cosine connects either to derivatives in calculus, complex rotation through Euler's formula, or harmonic decomposition in Fourier analysis.

Evidence of Learning

Evidence of learning should show more than correct answers. Strong evidence includes the following:

Knowledge: You can define the six trigonometric functions, explain radians, identify principal inverse-function ranges, and state core identities with their domain conditions.
Skills: You can convert angle measures, use the unit circle, sketch and transform graphs, solve equations, prove identities, use inverse functions, and select suitable triangle laws.
Mathematical communication: You can move among diagrams, formulas, graphs, tables, and verbal explanations while using precise notation and units.
Products: You can produce accurate graphs, models, proofs, investigations, visualizations, or videos that make the mathematics understandable to another learner.
Transfer: You can apply trigonometric reasoning to unfamiliar periodic or geometric situations, test assumptions, interpret parameters, and evaluate whether a model is credible.

OERs on the Topic

Linked Learning Areas

Trigonometric Functions

Trigonometric functions link several learning areas. In Geometry, they connect angles with lengths and circles. In Algebra, they are families of functions with transformations, equations, and identities. In Precalculus, they develop function reasoning and modeling. In Calculus, radians make derivatives and integrals natural. In Physics, Engineering, and Signal processing, sine and cosine describe rotation, oscillation, and harmonic components.