1. What trigonometry studies

Trigonometry is the study of relationships between angles and side lengths in triangles, and more generally the study of periodic behavior through the sine, cosine, and tangent functions.

At its core, trigonometry connects:

• Geometry and measurement

• Circular motion and periodic waves

• Algebraic identities and equation solving

The subject appears in:

• Surveying and navigation

• Physics and engineering

• Computer graphics

• Signal processing

• Calculus and differential equations

The most important idea is that trigonometric functions are not just triangle ratios. They are functions defined on angles, and the unit circle provides the cleanest definition.

2. Angles and the unit circle

Angle measure

An angle can be measured in degrees or radians.

The conversion between them is:

So:

Radians

Radians are the natural unit for trig and calculus. For a circle of radius $r$, an angle $\theta$ in radians satisfies:

where $s$ is arc length.

For the unit circle, $r = 1$, so the angle equals the arc length.

Unit circle definition

The unit circle is the circle centered at the origin with radius 1:

For an angle $\theta$ measured from the positive $x$-axis, the point on the unit circle is:

This gives the fundamental definitions:

provided $x \neq 0$.

Quadrants and signs

The signs of trig functions depend on the quadrant:

A common mnemonic is:

• All are positive in Quadrant I

• Sine is positive in Quadrants I and II

• Tangent is positive in Quadrants I and III

• Cosine is positive in Quadrants I and IV

Reference angles

A reference angle is the acute angle between the terminal side of an angle and the $x$-axis.

Reference angles let you determine trig values quickly:

1. Find the reference angle.

2. Compute the trig value using the acute angle.

3. Apply the sign from the quadrant.

Example:

because the reference angle is $\pi/6$ and sine is positive in Quadrant II.

3. Right-triangle ratios

For a right triangle, trig functions can be defined as ratios of side lengths relative to a chosen acute angle $\theta$.

Let:

• Opposite side = side across from $\theta$

• Adjacent side = side next to $\theta$

• Hypotenuse = longest side

Then:

The reciprocal functions are:

SOH-CAH-TOA

A compact memory aid:

• SOH: sine = opposite / hypotenuse

• CAH: cosine = adjacent / hypotenuse

• TOA: tangent = opposite / adjacent

Common exact values

The most common exact trig values come from special right triangles.

45-45-90 triangle

Sides are in the ratio:

So for a $45^\circ$ angle:

30-60-90 triangle

Sides are in the ratio:

So:

and:

4. Core trigonometric functions

Definitions

The primary trig functions are:

They are periodic functions, which means they repeat values after a fixed interval.

Periods

The basic periods are:

Domain and range

for any integer $k$.

Even and odd functions

Trigonometric symmetry is important:

So sine and tangent are odd, while cosine is even.

Pythagorean identity

The central identity is:

Dividing by $\cos^2 x$ gives:

Dividing by $\sin^2 x$ gives:

These identities are used constantly for simplification and proof.

5. Graphs and transformations

Standard graphs

The graphs of $y = \sin x$ and $y = \cos x$ oscillate between $-1$ and $1$ with period $2\pi$.

Key points:

The graph of $y = \tan x$ has vertical asymptotes at:

General forms

The common transformed forms are:

where:

• $|A|$ is the amplitude for sine and cosine

• Period for sine and cosine is $\frac{2\pi}{|B|}$

• Period for tangent is $\frac{\pi}{|B|}$

• Horizontal shift is $\frac{C}{B}$

• Vertical shift is $D$

Amplitude, period, and phase shift

For $y = A\sin(Bx - C) + D$:

For cosine, the same formulas apply.

Graphing workflow

To graph a transformed trig function:

1. Identify $A$, $B$, $C$, and $D$.

2. Compute amplitude, period, and midline.

3. Plot key points over one period.

4. Apply reflections or shifts.

5. Check asymptotes for tangent, secant, cotangent, and cosecant when relevant.

Example:

has:

• Amplitude $2$

• Period $\frac{2\pi}{3}$

• Phase shift $\frac{\pi/2}{3} = \frac{\pi}{6}$ to the right

• Midline $y = -1$

6. Key identities

Cofunction identities

These connect complementary angles:

Sum and difference identities

For sine:

For cosine:

For tangent:

Double-angle identities

Equivalent forms of cosine double angle:

Half-angle formulas

From the double-angle identity:

These are especially useful when reducing powers or evaluating exact values.

Product-to-sum and sum-to-product

These appear in more advanced algebra and signal analysis.

Example product-to-sum identity:

These identities are helpful for simplifying expressions with multiple angles.

Verifying identities

When proving an identity:

• Start with one side only

• Rewrite everything in sine and cosine if needed

• Use standard identities, algebra, and factoring

• Do not manipulate both sides simultaneously unless the steps are reversible

Common mistakes:

• Canceling terms across addition

• Using an identity in the wrong direction

• Forgetting domain restrictions

7. Solving trigonometric equations

Trig equations usually have infinitely many solutions because trig functions are periodic.

Basic strategy

1. Isolate a trig function if possible.

2. Solve the corresponding reference-angle equation.

3. Use periodicity to write all solutions.

4. Check for excluded values if the equation came from a rational expression.

Example: sine equation

Solve:

On $[0, 2\pi)$, the solutions are:

The full solution set is:

for any integer $k$.

Example: tangent equation

Solve:

One solution is:

Since tangent has period $\pi$:

Factoring approach

Many trig equations reduce to algebraic factorizations.

Example:

Factor:

So either:

or

This gives the full solution set.

Square-root and inverse issues

When squaring both sides or using inverse functions, extraneous solutions can appear.

Always check solutions in the original equation.

8. Inverse trigonometric functions

Inverse trig functions recover angles from trig values.

The principal inverses are:

Principal ranges

To make inverses functions, each is restricted to a principal range.

Important distinction

Inverse trig notation does not mean reciprocal.

For example:

not

The reciprocal of sine is $\csc x$.

Composition rules

On their principal domains:

The same caution applies to arccos and arctan.

Example

If:

then:

because $\tan(\pi/6) = \sqrt{3}/3$ and $\pi/6$ lies in the arctan principal range.

9. Laws for non-right triangles

Trigonometry also solves oblique triangles, which do not contain a right angle.

Law of sines

For a triangle with sides $a, b, c$ opposite angles $A, B, C$:

This is useful when you know:

• Two angles and a side, or

• Two sides and a non-included angle

Law of cosines

and cyclic variants:

This is useful when you know:

• Two sides and the included angle, or

• All three sides

Triangle area formulas

If two sides and the included angle are known:

This can be combined with the law of sines or cosines.

Ambiguous case

With the law of sines, the SSA case may produce:

• No triangle

• One triangle

• Two triangles

This is a real geometric ambiguity, not a calculator mistake.

Always check whether the given data actually forms a valid triangle.

10. Applications

Navigation and surveying

Trig is used to determine:

• Heights from angles of elevation

• Distances across inaccessible regions

• Bearing and direction

Example setup:

• Measure an angle of elevation $\theta$

• Measure a horizontal distance $d$

• Compute height using:

if the observer and target are level horizontally.

Physics

Trigonometry decomposes vectors into components:

It also models oscillations and waves:

where:

• $A$ is amplitude

• $\omega$ is angular frequency

• $\phi$ is phase shift

Engineering and signal processing

Periodic signals are often represented as sums of sines and cosines.

This leads to:

• Fourier series

• Harmonic analysis

• AC circuit analysis

Coordinate geometry

Trig helps describe circles and rotations:

and in 3D, rotations and projections rely on the same ideas.

11. Problem-solving workflow

Choosing the right tool

Use the following decision process:

1. If the triangle is right, start with SOH-CAH-TOA.

2. If the triangle is not right, use the law of sines or law of cosines.

3. If the problem involves a graph, identify amplitude, period, shift, and midline.

4. If the problem asks for an exact value, look for special angles or identities.

5. If the problem is an equation, isolate one trig function and use periodicity.

Common pitfalls

• Mixing degrees and radians

• Forgetting that inverse trig functions have restricted ranges

• Using the wrong reference angle sign

• Assuming SSA gives one triangle

• Forgetting excluded values when dividing by trig expressions

• Confusing $\sin^{-1}x$ with $1/\sin x$

Checking answers

Good checks include:

• Substitute the result back into the original equation

• Confirm quadrant and sign

• Confirm units are consistent

• Verify the angle lies in the requested interval

• For triangle problems, check that the side lengths satisfy triangle inequalities

12. Formula sheet

Definitions

Identities

Triangle laws

Exact values

End of note

Sources

• OpenStax Mathematics

• Mathematics LibreTexts

• Stewart, Calculus: Early Transcendentals

• Lay, Linear Algebra and Its Applications

• Rosen, Discrete Mathematics and Its Applications

• Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems

• Blitzstein and Hwang, Introduction to Probability

• Parell GitHub repository