What Is Trigonometry?
The word trigonometry comes from two Greek words: trigonon (triangle) and metria (measure).
Trigonometry is a branch of mathematics, and especially of geometry, that specializes in the measurement of triangles.
In this lesson, you will learn
- how to name the sides of a right triangle and write the six trigonometric functions
- how to use SOH CAH TOA to find the sine, cosine, and tangent of an angle
- who invented trigonometry, what it is used for, and when it is taught
The Six Trigonometric Functions
Suppose you have a right triangle like the one shown. The hypotenuse is the longest side, across from the right angle. The opposite side is across from the angle θ, and the adjacent side is next to it.
There are six basic trigonometric functions that are commonly used to solve problems: sine, cosine, tangent, cotangent, secant, and cosecant.
| Function | Short form | Formula |
|---|---|---|
| Sine | sin θ | oppositehypotenuse |
| Cosine | cos θ | adjacenthypotenuse |
| Tangent | tan θ | oppositeadjacent |
| Cotangent | cot θ | 1tan θ = adjacentopposite |
| Secant | sec θ | 1cos θ = hypotenuseadjacent |
| Cosecant | csc θ | 1sin θ = hypotenuseopposite |
Remember It With SOH CAH TOA
Teachers often use this memory trick for sine, cosine, and tangent.
- SOH: sin θ = oppositehypotenuse = 35
- CAH: cos θ = adjacenthypotenuse = 45
- TOA: tan θ = oppositeadjacent = 34
- The other three are the reciprocals, so flip each fraction: csc θ = 53, sec θ = 54, cot θ = 43
sin θ = 35, cos θ = 45, tan θ = 34, csc θ = 53, sec θ = 54, cot θ = 43
Opposite and adjacent depend on which angle you use. If you switch to the other acute angle, the two legs swap names. Only the hypotenuse always stays the same.
A right triangle has opposite side 5, adjacent side 12, and hypotenuse 13. What is tan θ?
Show answer
tan θ = 512. TOA: tangent = opposite / adjacent.
Trigonometry Lessons, in Order
My goal is to help you understand and master trigonometry in no time and with no tears. I designed these lessons carefully, backed by years of experience teaching mathematics in high school, and organized them to follow a logical pattern, so study them in the order shown. I am confident you will conquer trigonometry when you reach the end. Lessons are added regularly, so come back often to see what is new.
- Pythagorean theorem
- The tangent ratio
- The sine / cosine ratio
- The unit circle
- Using the unit circle to find sin(45°) and cos(45°)
- Using the 30-60-90 triangle to find sine and cosine
- Using the unit circle to find the sine and cosine of 30° and 60°
- Summary of lessons 1 through 6
- Angle in standard position
- Radian measure
- Relationship between degrees and radians
- Convert from radians to degrees
- Convert from degrees to radians
Additional Topics in Trigonometry
Who Invented Trigonometry?
Astronomers use trigonometry a lot to calculate the distances and positions of planets and stars. According to many scholars, Hipparchus of Nicaea, often considered the greatest astronomer of antiquity, invented trigonometry. He was the first to make a table of chords, the earliest example of a table of trigonometric values.
Trigonometry has been around for more than 3,000 years. The Babylonians, the Egyptians, and the Greeks all used it to find the lengths of the sides of triangles and the measures of their angles, mainly in astronomy and surveying.
Greek astronomers such as Aristarchus, Hipparchus, and Ptolemy made valuable contributions to astronomy. Aristarchus of Samos was the first to suggest that the Sun is at the center (the heliocentric hypothesis), but Hipparchus and Ptolemy kept the Earth at the center. Their view lasted until the 1500s, when the Polish astronomer Nicolaus Copernicus brought back the idea of a Sun-centered system.
Who made the first table of chords, the earliest table of trigonometric values?
Show answer
Hipparchus of Nicaea. That is why many scholars call him the inventor of trigonometry.
What Is Trigonometry Used For?
Today, trigonometry is used in electronics, engineering, calculus, and satellite navigation systems. It also describes many physical phenomena, including sound waves, light rays, pendulums, and the motion of atomic particles.
A study of trigonometry mainly involves angles, the right triangle, and the unit circle. Here are a few of the many problems you can solve with it:
- Angle of elevation
- Angle of depression
- Height of a very tall building
- Diameter of the Moon
- Area swept by a windshield wiper
- The height h of the tree is opposite the 35° angle, and the 20 m distance is adjacent to it.
- Opposite and adjacent mean TOA: tan 35° = h20
- Multiply both sides by 20: h = 20 × tan 35° ≈ 20 × 0.7002 ≈ 14.0
The tree is about 14 m tall.
When Is Trigonometry Taught?
In a 10th-grade geometry class, the teacher may cover some basic trigonometry. However, a complete study of trigonometry takes an entire school year.
Trigonometry uses many important ideas from algebra and geometry, which is why they are prerequisites. In general, trigonometry is taught after pre-algebra, algebra, and geometry. In the United States, it is usually taught in 11th grade.
Students do not always have to take trigonometry, though. Advanced students may take pre-calculus in 11th grade instead, and pre-calculus textbooks usually cover trigonometry in depth. Students can also take a combined course called algebra and trigonometry in 11th or 12th grade. It is not common to take trigonometry in middle school unless a student is especially dedicated or gifted.
Summary
- Trigonometry is the branch of mathematics that measures triangles. Its name comes from the Greek words trigonon (triangle) and metria (measure).
- In a right triangle, the hypotenuse is across from the right angle. The opposite and adjacent sides depend on the angle θ you use.
- SOH CAH TOA gives sine, cosine, and tangent. Cosecant, secant, and cotangent are their reciprocals.
- Hipparchus of Nicaea made the first table of chords, so many scholars call him the inventor of trigonometry.
- Trigonometry is used in engineering, navigation, and the study of waves, and to find heights and distances you cannot measure directly.
- In the United States, trigonometry is usually taught in 11th grade, after algebra and geometry.
Test Yourself: Interactive 10-Question Quiz
Can you use SOH CAH TOA, convert degrees to radians, or solve a triangle with the Law of Sines? Every answer is explained step by step, and you get 10 new questions each time you play.
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What is Trigonometry?
Recommended Lessons
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Check What You Know: Trigonometry Quiz
Can you use SOHCAHTOA, convert degrees to radians, or solve a triangle with the Law of Sines? Take the 10-question quiz to find out. Every answer is explained step by step, and you get 10 new questions each time you play.
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What Is Trigonometry?
The word trigonometry comes from two Greek words: trigonon (triangle) and metria (measure). Trigonometry is a branch of mathematics, and especially of geometry, that specializes in the measurement of triangles.
The Six Trigonometric Functions
Suppose you have a right triangle like the one shown. The hypotenuse is the longest side, across from the right angle. The opposite side is across from the angle θ, and the adjacent side is next to it.
There are six basic trigonometric functions that are commonly used to solve problems: sine, cosine, tangent, cotangent, secant, and cosecant.
| Function | Short form | Formula |
|---|---|---|
| Sine | sin θ | oppositehypotenuse |
| Cosine | cos θ | adjacenthypotenuse |
| Tangent | tan θ | oppositeadjacent |
| Cotangent | cot θ | 1tan θ = adjacentopposite |
| Secant | sec θ | 1cos θ = hypotenuseadjacent |
| Cosecant | csc θ | 1sin θ = hypotenuseopposite |
Remember It With SOH CAH TOA
Teachers often use this memory trick for sine, cosine, and tangent.
Trigonometry Lessons, in Order
My goal is to help you understand and master trigonometry in no time and with no tears. I designed these lessons carefully, backed by years of experience teaching mathematics in high school, and organized them to follow a logical pattern, so study them in the order shown. I am confident you will conquer trigonometry when you reach the end. Lessons are added regularly, so come back often to see what is new.
- Pythagorean theorem
- The tangent ratio
- The sine / cosine ratio
- The unit circle
- Using the unit circle to find sin(45°) and cos(45°)
- Using the 30-60-90 triangle to find sine and cosine
- Using the unit circle to find the sine and cosine of 30° and 60°
- Summary of lessons 1 through 6
- Angle in standard position
- Radian measure
- Relationship between degrees and radians
- Convert from radians to degrees
- Convert from degrees to radians
Additional Topics in Trigonometry
Who Invented Trigonometry?
Astronomers use trigonometry a lot to calculate the distances and positions of planets and stars. According to many scholars, Hipparchus of Nicaea, often considered the greatest astronomer of antiquity, invented trigonometry. He was the first to make a table of chords, the earliest example of a table of trigonometric values.
Trigonometry has been around for more than 3,000 years. The Babylonians, the Egyptians, and the Greeks all used it to find the lengths of the sides of triangles and the measures of their angles, mainly in astronomy and surveying.
Greek astronomers such as Aristarchus, Hipparchus, and Ptolemy made valuable contributions to astronomy. Aristarchus of Samos was the first to suggest that the Sun is at the center (the heliocentric hypothesis), but Hipparchus and Ptolemy kept the Earth at the center. Their view lasted until the 1500s, when the Polish astronomer Nicolaus Copernicus brought back the idea of a Sun-centered system.
What Is Trigonometry Used For?
Today, trigonometry is used in electronics, engineering, calculus, and satellite navigation systems. It also describes many physical phenomena, including sound waves, light rays, pendulums, and the orbits of atomic particles.
A study of trigonometry mainly involves angles, the right triangle, and the unit circle. Here are a few of the many problems you can solve with it:
- Angle of elevation
- Angle of depression
- Height of a very tall building
- Diameter of the Moon
- Area swept by a windshield wiper
When Is Trigonometry Taught?
In a 10th-grade geometry class, the teacher may cover some basic trigonometry. However, a complete study of trigonometry takes an entire school year.
Trigonometry uses many important ideas from algebra and geometry, which is why they are prerequisites. In general, trigonometry is taught after pre-algebra, algebra, and geometry. In the United States, it is usually taught in 11th grade.
Students do not always have to take trigonometry, though. Advanced students may take pre-calculus in 11th grade instead, and pre-calculus textbooks usually cover trigonometry in depth. Students can also take a combined course called algebra and trigonometry in 11th or 12th grade. It is not common to take trigonometry in middle school unless a student is especially dedicated or gifted.