SafekipediaTrigonometry
Adapted from Wikipedia · Discoverer experience
Trigonometry is a special part of mathematics that helps us understand how angles and the sides of triangles are connected. It started a long time ago, around the 3rd century BC, when people were studying the stars and needed math to help them. The word "trigonometry" comes from Ancient Greek words for "triangle" and "measure."
In trigonometry, we use special tools called trigonometric functions. These tools help us find out how the angles in a right triangle relate to the lengths of its sides. People have used trigonometry for many important things, like measuring the Earth, planning maps, and even moving spacecraft.
One cool thing about trigonometry is that it has many useful patterns, called trigonometric identities. These patterns help us make complicated math problems easier or find new ways to solve them.
History
Main article: History of trigonometry
Long ago, Sumerian astronomers studied how to measure angles, dividing circles into 360 parts. They and the Babylonians looked at the sides of similar triangles and found patterns, but they didn’t create a full system for finding sides and angles. Later, Hellenistic mathematicians like Euclid and Archimedes explored shapes inside circles and proved ideas that match what we now call trigonometry, though they used geometry instead of algebra.
In 140 BC, Hipparchus made the first tables to help solve triangle problems. In the 2nd century AD, Ptolemy made even better tables for use in astronomy. Over time, Indian and Islamic mathematicians added more ideas and tables. By the 10th century AD, all six main trigonometric tools were in use. Later, trigonometry became very important for navigation and mapping, and it kept growing as new ideas were added.
Trigonometric ratios
Main article: Trigonometric function
Trigonometric ratios are special numbers that help us understand triangles. They are found by comparing the lengths of the sides of a right triangle. These ratios stay the same no matter how big or small the triangle is, as long as one angle stays the same.
These ratios can be used to create functions that tell us more about angles. For example, the sine of an angle tells us how long the side opposite the angle is compared to the longest side, called the hypotenuse. The cosine tells us how long the side next to the angle is compared to the hypotenuse. The tangent tells us how long the side opposite the angle is compared to the side next to the angle.
Mnemonics
Main article: Mnemonics in trigonometry
Sometimes it’s hard to remember names and rules. A helpful way to remember the basic ratios is by using a fun phrase like “SOH CAH TOA”:
- Sine = Opposite ÷ Hypotenuse
- Cosine = Adjacent ÷ Hypotenuse
- Tangent = Opposite ÷ Adjacent
The unit circle and common trigonometric values
Main article: Unit circle
We can also picture these ratios using a special circle called the unit circle. This circle has a radius of 1 and is centered at a point. By placing an angle inside this circle, we can find points that help us calculate the sine and cosine of the angle.
| Input(Radians) | 0 | π / 6 {\displaystyle \pi /6} | π / 4 {\displaystyle \pi /4} | π / 3 {\displaystyle \pi /3} | π / 2 {\displaystyle \pi /2} | 2 π / 3 {\displaystyle 2\pi /3} | 3 π / 4 {\displaystyle 3\pi /4} | 5 π / 6 {\displaystyle 5\pi /6} | π {\displaystyle \pi } |
|---|---|---|---|---|---|---|---|---|---|
| Input(Degrees) | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
| sine | 0 {\displaystyle 0} | 1 / 2 {\displaystyle 1/2} | 2 / 2 {\displaystyle {\sqrt {2}}/2} | 3 / 2 {\displaystyle {\sqrt {3}}/2} | 1 {\displaystyle 1} | 3 / 2 {\displaystyle {\sqrt {3}}/2} | 2 / 2 {\displaystyle {\sqrt {2}}/2} | 1 / 2 {\displaystyle 1/2} | 0 {\displaystyle 0} |
| cosine | 1 {\displaystyle 1} | 3 / 2 {\displaystyle {\sqrt {3}}/2} | 2 / 2 {\displaystyle {\sqrt {2}}/2} | 1 / 2 {\displaystyle 1/2} | 0 {\displaystyle 0} | − 1 / 2 {\displaystyle -1/2} | − 2 / 2 {\displaystyle -{\sqrt {2}}/2} | − 3 / 2 {\displaystyle -{\sqrt {3}}/2} | − 1 {\displaystyle -1} |
| tangent | 0 {\displaystyle 0} | 3 / 3 {\displaystyle {\sqrt {3}}/3} | 1 {\displaystyle 1} | 3 {\displaystyle {\sqrt {3}}} | undefined | − 3 {\displaystyle -{\sqrt {3}}} | − 1 {\displaystyle -1} | − 3 / 3 {\displaystyle -{\sqrt {3}}/3} | 0 {\displaystyle 0} |
| secant | 1 {\displaystyle 1} | 2 3 / 3 {\displaystyle 2{\sqrt {3}}/3} | 2 {\displaystyle {\sqrt {2}}} | 2 {\displaystyle 2} | undefined | − 2 {\displaystyle -2} | − 2 {\displaystyle -{\sqrt {2}}} | − 2 3 / 3 {\displaystyle -2{\sqrt {3}}/3} | − 1 {\displaystyle -1} |
| cosecant | undefined | 2 {\displaystyle 2} | 2 {\displaystyle {\sqrt {2}}} | 2 3 / 3 {\displaystyle 2{\sqrt {3}}/3} | 1 {\displaystyle 1} | 2 3 / 3 {\displaystyle 2{\sqrt {3}}/3} | 2 {\displaystyle {\sqrt {2}}} | 2 {\displaystyle 2} | undefined |
| cotangent | undefined | 3 {\displaystyle {\sqrt {3}}} | 1 {\displaystyle 1} | 3 / 3 {\displaystyle {\sqrt {3}}/3} | 0 {\displaystyle 0} | − 3 / 3 {\displaystyle -{\sqrt {3}}/3} | − 1 {\displaystyle -1} | − 3 {\displaystyle -{\sqrt {3}}} | undefined |
Trigonometric functions of real or complex variables
Main article: Trigonometric function
Using the unit circle, we can expand the ideas about angles and triangle sides to include all positive and negative numbers. This helps us understand these special math ideas better.
Graphs of trigonometric functions
The following table shows important facts about the graphs of the six main trigonometric functions:
Inverse trigonometric functions
Main article: Inverse trigonometric functions
The six main trigonometric functions repeat their values, so they cannot be turned "backwards" in a simple way. However, by limiting the angles we look at, we can create inverse functions that help solve problems.
The names, ranges, and domains for these inverse functions are shown in the next table:
Power series representations
When we think of trigonometric functions as working with real numbers, we can describe them using special sums called series. For example, sine and cosine can be written as:
sin x = x − x3/3! + x5/5! − x7/7! + ⋯
cos x = 1 − x2/2! + x4/4! − x6/6! + ⋯
With these ideas, we can also define trigonometric functions for complex numbers. An important rule connects exponential functions with trigonometric ones:
ex + i y = ex ( cos y + i sin y )
Calculating trigonometric functions
Main article: Trigonometric tables
Trigonometric functions were some of the first things people used tables for in math. Students learned to find values in these tables and estimate between them. Special tools called slide rules had scales for these functions.
Today, scientific calculators have buttons for the main trigonometric functions like sine, cosine, and tangent. Computers also have built-in ways to calculate these functions quickly.
Other trigonometric functions
Main article: Trigonometric functions § History
Besides the six main functions, there are a few other ones that were important in the past. These include the chord, versine, coversine, haversine, and exsecant. You can find more about how they relate to each other in lists of trigonometric identities.
| Function | Period | Domain | Range | Graph |
|---|---|---|---|---|
| sine | 2 π {\displaystyle 2\pi } | ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} | [ − 1 , 1 ] {\displaystyle [-1,1]} | |
| cosine | 2 π {\displaystyle 2\pi } | ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} | [ − 1 , 1 ] {\displaystyle [-1,1]} | |
| tangent | π {\displaystyle \pi } | x ≠ π / 2 + n π {\displaystyle x\neq \pi /2+n\pi } | ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} | |
| secant | 2 π {\displaystyle 2\pi } | x ≠ π / 2 + n π {\displaystyle x\neq \pi /2+n\pi } | ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle (-\infty ,-1]\cup [1,\infty )} | |
| cosecant | 2 π {\displaystyle 2\pi } | x ≠ n π {\displaystyle x\neq n\pi } | ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle (-\infty ,-1]\cup [1,\infty )} | |
| cotangent | π {\displaystyle \pi } | x ≠ n π {\displaystyle x\neq n\pi } | ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} |
| Name | Usual notation | Definition | Domain of x for real result | Range of usual principal value (radians) | Range of usual principal value (degrees) |
|---|---|---|---|---|---|
| arcsine | y = arcsin(x) | x = sin(y) | −1 ≤ x ≤ 1 | −π/2 ≤ y ≤ π/2 | −90° ≤ y ≤ 90° |
| arccosine | y = arccos(x) | x = cos(y) | −1 ≤ x ≤ 1 | 0 ≤ y ≤ π | 0° ≤ y ≤ 180° |
| arctangent | y = arctan(x) | x = tan(y) | all real numbers | −π/2 y | −90° y |
| arccotangent | y = arccot(x) | x = cot(y) | all real numbers | 0 y | 0° y |
| arcsecant | y = arcsec(x) | x = sec(y) | x ≤ −1 or 1 ≤ x | 0 ≤ y y ≤ π | 0° ≤ y y ≤ 180° |
| arccosecant | y = arccsc(x) | x = csc(y) | x ≤ −1 or 1 ≤ x | −π/2 ≤ y y ≤ π/2 | −90° ≤ y y ≤ 90° |
Applications
Main article: Uses of trigonometry
Trigonometry helps us understand angles and shapes in many areas. It has been used for a long time to study the stars and predict events like eclipses. Today, we still use it to find out how far away stars are and to help with satellite navigation.
People have used trigonometry for travel and mapping for years. It helps ships know where they are and plan their routes. Even today, tools like the Global Positioning System still use trigonometry.
In land measuring, trigonometry helps figure out distances and angles between places. It is also used in geography to measure big distances on Earth.
Trigonometry is important for studying waves, like sound and light. It helps describe how these waves behave and is used in many sciences and technologies, such as music, medical imaging, and computer graphics.
Astronomy
Main article: Astronomy
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, predicting eclipses, and describing the orbits of the planets.
In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, as well as in satellite navigation systems.
Navigation
Main article: Navigation
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.
Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.
Surveying
Main article: Surveying
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.
On a larger scale, trigonometry is used in geography to measure distances between landmarks.
Periodic functions
Main articles: Fourier series and Fourier transform
The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics and communications, among other fields.
Optics and acoustics
Main articles: optics and acoustics
Trigonometry is useful in many physical sciences, including acoustics, and optics. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.
Other applications
Other fields that use trigonometry or trigonometric functions include music theory, geodesy, audio synthesis, architecture, electronics, biology, medical imaging (CT scans and ultrasound), chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, image compression, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Identities
Main article: List of trigonometric identities
Trigonometry includes many special equations called identities that are always true, no matter what numbers you use.
These identities can be about angles alone or they can connect the sides and angles of triangles.
Triangle identities
For any triangle, there are special rules that connect the lengths of its sides to its angles.
One important rule is the law of sines, which says that the ratio of a side to the sine of its opposite angle is the same for all three sides.
Another rule is the law of cosines, which helps find missing side lengths or angles when you know some of the triangle’s parts.
There is also a law of tangents and a way to find the area of a triangle using two sides and the angle between them.
Trigonometric identities
Some identities come from the Pythagorean theorem and work for any angle.
For example, the square of the sine of an angle plus the square of the cosine of that angle always equals one.
Euler’s formula gives another way to express sine, cosine, and tangent using special numbers.
Other useful identities include rules for adding or subtracting angles and turning products into sums.
Related articles
This article is a child-friendly adaptation of the Wikipedia article on Trigonometry, available under CC BY-SA 4.0.
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