Pioneered By: Andrew Sciacchitano, Jocele Shelton, Jacob Montgomery, Jordan McClure
Revised By: Molly Carter, Jennifer Moon, Jim Nordman, & Amanda VanderMeulen

Revised By: Trevor Goodman, Will Braschler

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According to Wolfram MathWorld, trigonometry is "the study of angles and of the angular relationships of planar and three-dimensional figures." For more in-depth information, see "What is Trigonometry?" courtesy of Dave's Short Trig Course.


Hipparchus of Rhodes

The first trigonometry problem may have been solved by Hipparchus of Rhodes (ca. 190-120). As a result of this, he is generally considered to be the father of trigonometry. His problem dealt with why the seasons are different lengths. Hipparchus used the observed lengths of each of the seasons to determine the arc length traveled by the sun in its orbit during each season. He then found the lengths of the chords that connect the sun’s position at the ends of the seasons. These lengths enabled him to determine how far the earth was from the center of the sun’s orbit. A diagram of this can be found below.

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In addition, Hipparchus was able to create a table to approximate the chord lengths of a circle. More can be found about Hipparchus of Rhodes HERE.

Finding Chord Lengths

Euclid may have been the first person to show how to calculate other chord lengths when he determined the lengths of the sides of regular inscribed pentagons and decagons. A diagram of this can be found below.

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Using the foundation that Euclid started, Ptolemy of Alexandria (90–168 CE) constructed a table of chord lengths for a circle of radius 60 in half-degree increments. Ptolemy’s table is equivalent to a table of sines in quarter-degree increments. Interestingly, his calculations were carried out to seven-digit accuracy.

Indian astronomers in the third fourth or fifth century CE started the shift to the half-cord or sine. The use of sine was then adopted by Arabic mathematicians, later translated into the Latin word Sinus, literally meaning half-cord. From the Latin root is where we get the English Sine!

chord of a circle.jpg

The Unit Circle

The development of the unit circle can be credited to Leonhard Euler (1707-83). It was Euler who decided that for the purposes of calculus, the radius should be fixed at 1. He realized that measuring the arc length and the line length in the same units would be beneficial; in addition, he concluded that if the radius of the circle is 1, then the circumference is 2 Pi. From this, it seems as though he was using radians. However, radians weren't used for almost a hundred years after Euler's death.


Triangle Trigonometry

Triangle trigonometry started with the simple problem of finding the length of a shadow cast by an object at certain angles of the sun. An example of this can be found below.

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Finding the answer to this question were mathematicians such as Ptolemy and Al-Khwarizmi of Baghdad (ca. 790–840). Interestingly, Al-Khwarizmi's name is the root for the word "algorithm" and whose book, al-jabr, is the root for the word algebra. The early triangle trigonometry problems led to the formation of trigonometric identities.

The name for tangent comes from the fact the length being solved for is tangent to the circle. The secant, arising from the Latin secantem, meaning “cutting,” is the length of the radial line segment cut off by the tangent. The cosine, cotangent, and cosecant are the corresponding line segments for the complementary angle.

Using trigonometry to solve for right triangles became prominent with the posthumous publishing of Johann Müller's De Triangulis Omnimodis (or, On Triangles of Every Kind). Müller, Georg Rheticus, and Bartholomew Pitiscus also used trigonometry and similar triangles to solve for an unknown side of any given right triangle for which one of the acute angles and one other side are given.

For more on the history of trigonometry: History of trig


The Unit Circle

Trigonometric functions are defined using the unit circle. The Unit Circle is centered at (0,0) and has a radius of 1 unit. Let Θ (theta) be an angle measured counterclockwise from the x-axis along an arc of the circle. Then cos Θ is the horizontal coorinate of the arc endpoint, and sin Θis the vertical coordinate, as shown below.



A circle is divided into 360 parts called degrees. There are many theories about the use of degrees as a measure of the angles around a circle. Some say that the use of the number 360 corresponds to the amount of days in a year rounded down to the nearest 10. (for more theories click here). While the background of degrees is debatable the fact that they exist and can be used to measure angles in many different forms has provided a great opportunities.

external image degrees-360.gif


We also can measure the angles around a circle in units called radians. Unlike degrees, radians have a more mathematical background.They are a pure measure based on the radius of a circle. To Demonstrate we will start with the unit circle, which has a radius of 1 unit. To show 1 radian, we will leave the radius connected to the outside edge of the circle and rotate it until it is standing straight up, perpendicular to the original radius line. Next we will lay the line down along the curve of the circle. The angle this creates is 1 radian. If we continue to add the same arc length all of the way around the circle, we will find that the circle is about 6.2831 radians. We can easily compare this to the circumference of the circle, which for the unit circle is 2π or 6.2831. We can see an illustration of this below:

Radians GIF.gif

Converting Radians to Degrees or Degrees to Radians

Knowing that the entire way around the circle is 2π radians, we also know that half way around the circle is π radians. We can compare this to the fact that half way around the circle is 180° and get π radians = 180°. We can rewrite this as 1 radian = 180°/π degrees and 1 degree = π/180° radians. Below is a graph of all of the radian values corresponding with their matching degree values.

external image 300px-Degree-Radian_Conversion.svg.png

Right Triangles

Right triangles are often used in trigonometry, as seen above. A right triangle has three sides, hypotenuse, adjacent to a given angle Θ and opposite to given angle Θ as shown below.

external image trig-right-triangle-names.gif

The angle can be found using the definitions of the trigonometric functions. There are many mnemonics used to help students remember these, such as the most commonly used "SOHCAHTOA" where sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Another mnemonic used is "Tommy On A Ship Of His Caught A Herring." The trigonometric functions are cosine (cos x), sine (sin x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x). Below are some of the trigonometric functions.

external image trid4.gif

Geogebra Example

(an example of what the values of sine, cosine, tangent, cosecant, secant, and cotangent relate to on a triangle on the unit circle can be found on this webpage)

(an example of how the values of sine, cosine, tangent, cosecant, secant, and cotangent change as theta changes can be found on the webpage

When finding the lengths of the sides of a right triangle, a simple formula that most students know is the Pythagorean theorem. However, most students do not understand how the formula actually works. Below is an illustration of how this holds:

This illustration demonstrates that the lengths of the two smaller sides, a and b, individually squared create two squares in which the area is equal to that of the the square created by the hypotenuse, or c, squared.

-links given from HERE.


A mathematical identity (as defined by the Merriam-Webster's Dictionary - is an equation that is satisfied by all values of the symbol. Here, we can think of the word symbol as the exact same thing as a variable. For example, though trivial, x = x is an identity because for every value you can substitute for 'x', the equation will always be true. In trigonometry, identities are very important and help us tremendously. They can allow us to put a function in terms of one trigonometric function into terms of another. The most basic, and likely most common, identity is the Pythagorean identity, which states:



While these three equations appear different, they are equal to one another. Divide the first equation by cos^2 and you get the second equation. Divide the first equation by sin^2 and you get the third equation. Other useful identities can be found at TrigIDs.
These identities are put into groups relevant to what they represent. For example, the half angle identities represent what happens when the variable of a trigonometric funciton is divided by two.
The most intriguing and useful of these identities is probably the cos(2x) identity (seen in the 'double angle identities' group in the list of identities). What makes this so interesting is that it takes a function of degree one - (cos(2x)) - and converts it into a function of degree two. The same can be thought of in reverse as well.
Trigonometric identities are very useful when changing the representation of a trigonometric function. For example, if you have the square root of (1 - cosx)/2, then you can simply write this as sin(x/2). Applications of this and similar processes can be seen using all of the other trigonometric identities.

Putting together the things that we now know about trigonometric functions and the Unit Circle, we can construct a very well-defined graphical image of the Unit Circle itself, shown below, which is considered by many to be the base from which all of trigonometry stems.
UnitCircle.gifThe Unit Circle with points labeled in degrees, radians, and cartesian coordinates.

Trigonometric Identities Proofs

Adding Angles

Trigonometry can explain not only the properties of angles but also the properties of multiple angles. Let's look at an angle defined as the addition of angle A and angle B. We can draw this by drawing a right triangle with angle A touching a right triangle with angle B as shown below.
Double Angles.png
Since we define the hypotenuse of the second triangle to have length 1, we know that the legs of that triangle are cos(B) and sin(B). Since our triangles share a side, we know that the hypotenuse of the bottom triangle is also cos(B). This tells us that the bottom triangle has legs of length cos(B)cos(A) and cos(B)sin(A).

Now if we look at the highlighted triangle, we know that the hypotenuse is equal to sin(B). This tells us that the highlighted triangle has legs of length sin(B)cos(A) and sin(B)sin(A).

The sin(A+B) is going to equal the sum of the vertical components of the bottom triangle and the highlighted triangle. This means that

sin(A+B) = sin(A)cos(B) + cos(A)sin(B)

We also know that the cos(A+B) is going to equal the horizontal component of the bottom triangle minus the horizontal component of the highlighted triangle. This means that

cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
Since the tangent is defined as sin/cos, we know that

tan(A+B) = sin(A+B) / cos(A+B)
We can substitute our known values for the sin(A+B) and the cos(A+B) from above to find
Step 1.png
Since the sin(B) = cos(B)tan(B), we can factor out the cos(B) term from the top and the bottom to get the tan(A+B) is equal to
Step 2.png
Note again that sin(A) = cos(A)tan(A). We can therefore factor out the cos(A) term from the top and bottom to get

Step 3.png

We have some common terms in both the top and bottom. We can eliminate the cos(A) and cos(B) terms to find

Step 4.png

Therefore we know that the tan(A+B) = [tan(A) + tan(B)] \ [1 - tan(A)tan(B)]

Pythagorean Identity

First recall that the equation to the unit circle centered at the origin is x2+y2=1. Let P be a point on the unit circle with an angle
Ɵ from the x-axis.

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The x and y coordinates of P are given as x=cos Ɵ and y=sin Ɵ. By substituting these values of x and y into our equation for the unit circle we can see that

(A worksheet about the Pythagorean Identity can be found HERE.)

The Law of Sines & The Law of Cosines:

Pythagorean The Functions Applied to Non-Right Triangles

We use the Law of Sines and Cosines to solve oblique triangles, which are triangles that do not have a right angle. If you are given information about a triangle and asked to solve for the unknowns you can use these two formulas.

Law of Sines


You use two equivalent ratios at a time based of the information that you have been given and what you are being asked to find.

Law of Cosines


But which one do you use? Below shows the method you would use to solve the unknown values, depending on what you are given.
1. If you are given two angles and any side of a triangle, also known as AAS or ASA, you will use the Law of Sines to solve.
AAS Tri.png

2. If you are given two sides and an angle opposite one of the given sides, also know as SSA, you will you the Law of Sines to solve.
SSA Tri.png
*This is also know as the ambiguous case, since the situation could produce one of three possible solutions, depending on the specific values of the two sides and the opposite angle. You would start by solving for one of the unknown angles and if |sin A| > 1, then no solution is possible. If |sin A|< 1, then there are two angles to consider, A and A', which is equal to (180° - A). We have at least one solution with A, but we need to check A' by adding it to the angle that was given. If it is greater than 180°, then there is only the one solution, but if it is less than 180°, there is a second solution.

3. If you given all three sides, or SSS, you will use the Law of Cosines to solve.
SSS - Tri.png

4. If you are given two sides and the included angle, or SAS, you will use the Law of Cosines to solve.
SAS Tri.png

Pythagorean Proofs of the Law of Sines and Cosines

The Law of Sines asserts that a certain ratio holds true in any triangle, namely, that the ratio of the sine of an angle to the side opposite that angle is a constant. The proof is as follows.

Consider the triangle ABC.


Now we will drop a perpendicular from one of the vertices of the triangle and label the length of the perpendicular h. Let's consider the angle A. Now, sinA is defined as the side opposite angle A divided by the hypotenuse of ACD, or h/b. So using algebra, bsinAh. Now also consider angle B. similarly, sinBh/a. So then asinB = h. Now we have two equal expressions.

asinB = bsinA
(sinA)/a = (sinB)/b
Since this will work with any arbitrary perpendicular, we can say with certainty that the ratio will hold true for all sides and all angles of our triangle, therefore

(sinA)/a = (sinB)/b = (sinC)/c

The Law of Cosines is a little bit more complex of a claim for triangles. Consider the triangle ABC again, redrawn below.

Looking at the triangle ACD and using the Pythagorean Theorem, we can see that

(x^2) + (h^2) = (b^2)
and we know from our Trigonometric Identities that

cosA = x/bbcosA = x
We will next examine triangle BCD and using the Pythagorean Theorem, we can see that

((c - x)^2)+(h^2) = (a^2)(c^2) - 2cx + (x^2) + (h^2) = (a^2)(c^2) - 2c(bcosA) + (b^2) = (a^2)(b^2) + (c^2) - 2bccosA = (a^2)
The last line of the equation we have derived is known as the Law of Cosines, and since this property will hold true for any perpendicular, we can conclude that

(a^2) = (b^2) + (c^2) - 2bccosA(b^2) = (a^2) + (c^2) - 2accosB(c^2) = (a^2) + (b^2) - 2abcosC

Making Connections with a Model: Great for Visual Learners

There are many connections that can be made with this picture. To get your students started you may show them that sin^2(x) + cos^2(x) = 1. You can see in this picture that all these can be connected and help the students start to formulate their own connections.

Example Problem:

An example can be found here HERE in the form of a PDF.

pictures from lawofsinespicture, lawofcosinespicture

Educational Trigonometry Links:
Trig In Nature
Trig At Work
Touch Trigonometry



In architecture, trigonometry plays a massive role, especially in the compilation of building plans. Architects have to calculate exact angles of intersection for the components of their structure to ensure stability and safety. Some instances of trigonometric use in architecture include: arches, domes, support beams, and suspension bridges. More information on how trigonometry is used in facades can be found HERE. Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles. Architecture remains one of the most important sectors of our society. As architects plan the design of buildings trigonometry is used to insure its longevity. In addition, trigonometry can be used for aesthetic purposes, as seen architecturally below.

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Astronomy has been studied by civilizations in all regions of the world from ancient times onward. In our modern age, trigonometry helps calculate distances between stars and learn more about the universe. Menelaus' Theorem helps astronomers gather information by providing a back drop in spherical triangle calculation.


Trigonometry is used in geology to estimate the true dip of bedding angles. An example of this can be found below. Calculating the true dip allows geologist to determine slope stability.

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Trigonometry is used in navigation to find distances of the shore from a point in the sea. It is also used in oceanography in calculating the height of the tides in the oceans. One day, it can be used in space travel, specifically finding the distance between celestial bodies in space. In addition, people and places can be found using triangulation. Triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline. A diagram of this can be found below.

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Trigonometry plays a huge role in all varieties of physics. For example, radio, microwave, and electromagnetic waves are all measured and graphed with trigonometric functions such as sine and cosine. Physics concerns itself greatly with three-dimensional space. Mathematics, specifically trigonometry, is critical to being able to determine values and help identify the nature of something in the space it exists in. Trigonometry is used so much in physics that it is as prevalent as simple addition and subtraction for the average person. A worksheet using trigonometry and physics can be found HERE.


For jobs that use trigonometry, see XP Math - Jobs That Use Right Triangle Trigonometry. In addition, each individual career reveals math content areas necessary for employment.


Common Core State Standards: Trigonometric Functions Standards
(The links below certain standards provide activities or resources related to those standards)

Extend the Domain of Trigonometric Functions Using the Unit Circle
Model Periodic Phenomena with Trigonometric Functions
  • Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
  • Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
  • Use inverse functions to solve trigonometric equations that arise in modeling contexts, evaluate the solutions using technology, and interpret them in terms of the context.
Prove and Apply Trigonometric Identities

Define Trigonometric Ratios and Solve Problems Involving Right Triangles
  • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle. leading to definitions of trigonometric ratios for acute angles.
  • Explain and use the relationship between the sine and cosine of complementary angles.
  • Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply Trigonometry to General Triangles
  • Derive the formula A=1/2ab*sinC for the area of the triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  • Prove the Laws of Sines and Cosines and use them to solve problems.
  • Understand and apply the Law of Sines and Law Cosines to find unknown measurements in right and non-right triangle (eg. surveying problems, resultant forces).

National Council of Teachers of Mathematics

Expectations: In grades 9–12 all students should
  • Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
  • Use trigonometric relationships to determine lengths and angle measures.
State Standards (As set by the CCSS institution):
Define trigonometric ratios and solve problems involving right triangles
  • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
  • Explain and use the relationship between the sine and cosine of complementary angles.
  • Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply trigonometry to general triangles
  • Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  • Prove the Laws of Sines and Cosines and use them to solve problems.
  • Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Note: No reference needed because of non-commercial use which is allowable under the CCSS institution terms of use