Trigonometry is one of those math subjects that sounds intimidating before you understand what it actually is. However, it comes down to a surprisingly clean idea: given a triangle, how do the lengths of its sides relate to its angles? Answer that question, and you can calculate distances you cannot measure directly, navigate across oceans, design buildings, and write video game physics engines.
This guide covers everything a student needs as an introduction to trigonometry:
- the definition
- history
- the three core functions
- SOH-CAH-TOA
- the six trigonometric functions
- a reference table of common angle values
- key formulas
- and real-world applications.
If you have ever wondered what trigonometry is and why it matters, read on.
What Is Trigonometry?
Trigonometry is a branch of mathematics that focuses on the relationships between sides and angles of right-angled triangles. This branch of math helps you understand how sides and angles are connected in any right triangle.
The word itself comes from two Greek roots: trigonon, meaning triangle, and metron, meaning measure. So trigonometry literally means “triangle measurement.”
In simpler terms, trig is the math that lets you calculate unknown sides or angles of a triangle when you know enough about the others. If you know one angle and one side length of a right triangle, you can work out every other measurement without touching a ruler.
A Brief History of Trigonometry
The history of trigonometry dates back to ancient civilizations. The ancient Greeks were among the first to study trigonometric functions formally, and Greek mathematicians such as Hipparchus and Ptolemy produced foundational work that still shapes how the subject is taught today.
Hipparchus, working around 150 BC, developed early trigonometric tables to solve problems in astronomy, calculating the distances and positions of celestial bodies with remarkable precision. Ptolemy expanded this work in his famous Almagest, applying spherical trigonometry to model the movements of planets and stars.
Islamic scholars made equally significant contributions during the Islamic Golden Age, refining the trigonometric functions we use today and advancing mathematical astronomy far beyond the Greek foundations. This period saw the sine, cosine, and tangent ratios formalized into the tools that are integral to trigonometry as it is taught in schools today.
The Right Triangle: Sides and Angles
Before the functions make sense, you need to understand the right triangle. Every right triangle has three sides and three angles. One of the angles is always exactly 90 degrees, which is the right angle. The other two angles are acute, each less than 90 degrees.
The three sides are named relative to one of the acute angles, which we call theta (θ):
- Hypotenuse: The longest side of the right triangle, always opposite the right angle. This side never changes regardless of which acute angle you choose as θ.
- Opposite side: The side directly across from angle θ, not touching it. Also called the perpendicular in some textbooks.
- Adjacent side: The side next to angle θ that is not the hypotenuse.
The three primary trigonometric functions are each a ratio of two of these sides, so knowing which side is opposite, which is adjacent, and which is the hypotenuse is the first skill to master.
The Three Primary Trigonometric Functions
Trigonometry is built on three fundamental trigonometric ratios that compare pairs of sides in a right triangle. Each ratio is defined relative to angle θ and is given a name, an abbreviation, and a formula. Each is abbreviated as a two or three-letter code (sin, cos, and tan) that you will see throughout every trig problem.
Sine (sin)
The sine function (abbreviated sin) compares the length of the opposite side to the hypotenuse.
sin(θ) = opposite/hypotenuse
If you know an angle and the length of the hypotenuse, you can use sine to find the length of the opposite side. For a 30-degree angle in a triangle with a hypotenuse of 10 cm, sin(30°) = 0.5, so the opposite side is 5 cm.
Cosine (cos)
The cosine function (abbreviated cos) compares the length of the adjacent side to the hypotenuse.
cos(θ) = adjacent/hypotenuse
If you know an angle and the hypotenuse, cosine gives you the adjacent side. For the same 30-degree angle and a hypotenuse of 10 cm, cos(30°) ≈ 0.866, so the adjacent side is approximately 8.66 cm.
Tangent (tan)
The tangent function (abbreviated tan) compares the length of the opposite side to the adjacent side.
tan(θ) = opposite/adjacent
If you know an angle and one non-hypotenuse side, the tangent connects the opposite and adjacent sides directly. The tangent is particularly useful in height-and-distance problems, such as finding how tall a building is when you know how far away you are standing and the angle you are looking up at it.
SOH-CAH-TOA: How to Remember the Trig Ratios
SOH-CAH-TOA is the mnemonic every trigonometry student learns because the three ratios are hard to remember without it. It is the standard tool for solving right triangles, covering both finding missing sides and finding missing angles. Each group of three letters encodes one function:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
The letters S, C, and T stand for the function name. O, A, and H stand for opposite, adjacent, and hypotenuse. Reading each triplet tells you exactly which two sides to divide.
Worked example using SOH-CAH-TOA
A right triangle has an angle θ = 40° and a hypotenuse of 12 cm. Find the length of the opposite side.
- You need opposite; you have hypotenuse. That is Sine (SOH).
- Set up: sin(40°) = opposite / 12
- sin(40°) ≈ 0.643
- Opposite = 0.643 × 12 ≈ 7.72 cm
To find a missing angle instead of a missing side, use inverse trigonometric functions: arcsin, arccos, or arctan. If sin(θ) = 0.5, then θ = arcsin(0.5) = 30°.
If SOH-CAH-TOA feels slippery at first, that is completely normal. Working through a few problems step by step with private tutoring can solidify the process faster than re-reading explanations alone.
The Six Trigonometric Functions
The three primary functions (sin, cos, tan) each have a reciprocal function. Together, all six trigonometric functions describe every ratio relationship between the sides of a right triangle.
| Function | Abbreviation | Ratio |
|---|---|---|
| Sine | sin | opposite/hypotenuse |
| Cosine | cos | adjacent/hypotenuse |
| Tangent | tan | opposite/adjacent |
| Cosecant | cosec (csc) | hypotenuse/opposite |
| Secant | sec | hypotenuse/adjacent |
| Cotangent | cot | adjacent/opposite |
The reciprocal functions are simple: cosecant is 1/sin, secant is 1/cos, and cotangent is 1/tan.
Trigonometry Table: Values for Common Angles
Certain angles appear so frequently in trigonometry problems that their exact values are worth memorizing. The most common angles in degrees are 0°, 30°, 45°, 60°, and 90°. Here are the exact values of sin, cos, and tan for each:
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
A few observations worth noting:
- At 45°, sin and cos are equal (both 1/√2 ≈ 0.707).
- tan(90°) is undefined because it would require dividing by zero (the adjacent side has length zero at 90°).
- The values in the sin column for 0° through 90° read 0, 1/2, 1/√2, √3/2, 1. This pattern is easy to learn.
Note that angles can also be measured in radians rather than degrees. Radians are the standard unit in calculus and higher mathematics: 360° equals 2π radians, and 90° equals π/2 radians.
At the high school level, most problems use degrees, but being aware of radians prepares you for what comes next.
Key Trigonometric Formulas and Identities
Trigonometric formulas are essential tools in mathematics, offering crucial insights into the relationships between angles and sides in triangles. The most important formulas for high school students are the three ratio formulas already covered (sin, cos, tan), the Pythagorean identity, and the laws of sines and cosines for non-right triangles.
The Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This fundamental relationship underlies all of trigonometry.
Note that calculating square roots is an essential skill here, since finding a side length from the Pythagorean theorem almost always requires taking a square root.
The Pythagorean identity
The Pythagorean identity expresses the same relationship in terms of trig functions:
sin²θ + cos²θ = 1
This identity is true for any angle θ and is one of the most frequently used trigonometric identities. It follows directly from the Pythagorean theorem applied to the unit circle.
The law of sines and law of cosines
The law of sines and the law of cosines extend trigonometry beyond right triangles to any triangle:
- Law of sines: a/sin A = b/sin B = c/sin C
- Law of cosines: c² = a² + b² – 2ab cos C
These become important when solving triangles where no right angle is present.
What Is Trigonometry Used For?
Trigonometry lets students solve problems involving angles and distances that would be impossible to measure directly, such as calculating height and distance, orbital paths, structural forces, and wave frequencies.
Here are the fields where it shows up most concretely.
Engineering and Architecture
Engineering and architecture rely on trigonometry constantly. For example:
- Bridge designers use sine and cosine to calculate forces in triangular trusses so that the weight is distributed evenly.
- Architects apply tangent ratios to find roof angles that drain water efficiently.
The classic angle of elevation problem (how tall is a building if you stand 30 metres away and look up at a 60-degree angle) is textbook trigonometry applied to construction every day.
Astronomy and Navigation
Hipparchus used trigonometry to estimate the distance to the Moon over two thousand years ago, and the same principles still power modern astronomy. Today, spherical trigonometry helps astronomers track orbits of planets and satellites.
GPS and satellite navigation systems use trigonometric calculations continuously to determine exact positions on Earth. In other words, every time you use a navigation app, trigonometry is working behind the scenes. Surveyors also rely on trigonometry to measure land boundaries, map terrain, and place infrastructure with precision across long distances.
Physics, Music, and Technology
- In physics, trigonometry models projectile motion (the angle and speed of a thrown object determine its path), wave behavior (light and sound waves are described using sine and cosine functions), and optics.
- In music, sound waves travel as sinusoidal patterns, and audio engineers use trig functions to analyze and manipulate pitch and frequency.
- In video game design, trigonometric functions control how characters move, rotate, and interact with objects in both 2D and 3D environments.
For students considering a career in any science or STEM field, a solid grasp of trigonometry is not optional.
Struggling With Trigonometry? Tutorax Can Help
Trigonometry introduces a level of abstraction that trips up many students, even those who did well in algebra and geometry. If you or your child is struggling with trigonometry or would like to build confidence in applying these functions under exam pressure, personalized support makes a real difference.
At Tutorax, 95% of our students see measurable improvement in academic performance. We have helped more than 20,000 students across Canada improve math skills and pass trigonometry and other high school math courses, including dedicated support for trigonometry for high school students at every level.
Here is what working with a Tutorax tutor looks like:
- One-on-one sessions targeting specific gaps, whether that is SOH-CAH-TOA, the trig table, identities, or applying functions to word problems
- Available as in-home tutoring or online tutoring, 7 days a week, with flexible scheduling
- No contract required: pay as you go with no minimum hours
- Tutor placement guaranteed in less than 7 days for in-home sessions, or within 24-48 hours online
Whether you are looking for tutoring in Toronto, math tutors in Montreal, or anywhere else across Canada, we’ll help you find a math tutor near you today.
Frequently Asked Questions About Trigonometry
What is trigonometry in simple terms?
Trigonometry is a branch of mathematics that uses the relationships between sides and angles of right triangles to calculate unknown measurements. Using sine, cosine, and tangent, you can find any missing side or angle in a right triangle as long as you know enough about the others.
What is SOH-CAH-TOA?
SOH-CAH-TOA is a mnemonic device for remembering the three primary trigonometric ratios.
- SOH stands for Sine equals Opposite over Hypotenuse
- CAH stands for Cosine equals Adjacent over Hypotenuse
- TOA stands for Tangent equals Opposite over Adjacent
It is the most widely used memory aid in high school trigonometry and is worth committing to memory before your first trig test.
Is trigonometry hard to learn?
Trigonometry is a confusing subject for many students at first, particularly the jump from labeled formulas to applying them in actual problems. Students who build a strong grasp of SOH-CAH-TOA and practice with real examples, ideally with a trigonometry tutor for step-by-step guidance, typically find the subject becomes manageable quickly.
Where is trigonometry used in real life?
Trigonometry has real-world applications in engineering, construction, astronomy, GPS and satellite navigation, physics, music production, and video game design. It is also the mathematical foundation of calculus, meaning it appears directly or indirectly in virtually every scientific and technical career.