Trigonometry



This formula sheet contains all the basic formulas used in trig geometry. It might be useful as an online reference sheet for revision, or for tutorial purposes. If you have an angle and one side and there are missing sides then the equations in this sheet can help find the missing sides, and even the missing angles.

Trig has always been the bane of high school mathematics, but it need not be so, because it's actually fun. Everything is explained and made easy so that it might be useful for GCSE students, Year 9, Year 10, Year 11, Grade 10, Grade 11, Grade 12, Class 10, Class 11, for kids, for beginners and for dummies like me. :-)



Trigonometry SOHCAHTOA Triangle Diagram

The word trigonometry that we use today comes from the Sanskrit word trikonmiti, where tri means three and kon means corner. The study of shapes with corners. Much of language and words that we use today has a basis in ancient India.

Soh Cah Toa Triangle: It's so simple, you don't need really silly mnemonics to remember the trigonometric formulas. You only need to remember the sine formula and you can figure out the others. Sin A = Opposite / Hypotenuse, Cos A = Adjacent / Hypotenuse, and Tan A = Opposite / Adjacent.

Rules: These Trig functions work only for right angled triangles. They will not work for non right triangles.

How to find a missing side: Finding sides is easy. If you have one side and an angle, then you can easily calculate the missing side and all you would need to do is to transpose the formula for the missing side. As shown in the formula equations below I have already transposed the formulas for the missing sides, hypotenuse, opposite, and adjacent. So all the work has been done and you simply need to choose the formula that best fits the problem at hand.

How to find angles: Finding Angles is easy. To find the angle A, you will need a minimum of two sides, either opposite and hypotenuse, adjacent and hypotenuse, or opposite and adjacent sides. You will also need to know how to use the inverse trig function on your calculator.

The inverse functions are arccos, arcsin, and arctan. Once you have the ratio of the sides you use this function to find the angle.

Some calculators have a button marked INV. The inverse button must be pressed first so that the calculator knows that you want the inverse trig function. Also make sure your calculator is set to Degree mode (DEG) if you want the answer in Degrees. Alternatively you can set it to radian mode (RAD) if you want the angle in radians.


Vocabulary List



Sin  Sine Function Alternative Symbol
Cos Cosine Function
Tan Tangent Function
Adjacent Side closest and adjacent to the angle
Opposite Side Opposite to the angle
Hypotenuse Slant or Sloped Side
arcSin Inverse Sine function arcSin Symbol
arcCos Inverse Cosine function arcCos Symbol
arcTan Inverse Tangent function arcTan Symbol


Uses in Real Life



In the Real World: The Hindus discovered it and used it for altar construction, temple construction, and astronomy, because their gods wanted prayers to be conducted at certain times of the year when the stars were in a particular alignment as specified in their ancient sutras. The earliest written record of trig was in a Sutra called 'Opening of the Universe'. I don't know if it opened up the Universe but it certainly opened up the world.

The Arabs and Sumerian's used it and thought it was useful for star navigation in the desert where you can use the stars for reference instead of sand dunes.

We Brits used it for navigation on seafaring ships and charting the seas for the eventual colonisation of the major continents to build an empire. The Americans used it for navigation of aeroplanes, rockets, missiles, and going to the moon.

Everyday uses in daily life: Some professions use trig more than others. Architects and builders use this the most in daily life. In architecture, and building construction, everything has angles and length, and therefore it is heavily used daily. Builders often use a spirit level and a tape measure. An architect will typically use a compass and a tape measure.

Aviation: Almost every aspect of aviation uses trig. The pilots use it for navigation, the designers use it for jet engine simulation and aerofoil research, and Air Traffic Control uses it daily to give vector bearings to pilots.

The Primary Instrument Cluster within an aeroplane cockpit relies heavily on trig calculations. Its microprocessor is continuously making these calculations and displaying them in graphical format. The ILS Computer also performs these calculations every time a plane comes in to land on the runway. It calculates the optimum glide slope based on the current vector. The next time you go on holiday to Ibiza and the plane makes a smooth landing, then you can be sure that the pilot understands trig and was paying attention to the localiser display. :-)

Used in Astronomy: Almost any kind of space research will include angles and distances of astronomical bodies in space. If you've got distances between stars and an angle then trig is the ideal math to use to calculate any unknowns.

Spaceships & Rockets: SpaceShipTwo uses trig to calculate the re-entry angle from apogee so that the spaceship lands at the correct location.

The Indian Rocket PSLV Series uses trig to calculate the optimum entry angle for injection of the payload into the Earth's orbit.

The Mars Rover Curiosity performs trig calculations automatically when it needs to calculate distances between mountains or rocks, or to determine the height of a mountain. All of this information is sent back to mission control so that they can decide whether it's wise to send the Rover in that direction.

Engineering: Certainly, if you want to become an engineer then this is vital learning because it is heavily used in every aspect of mechanical engineering.


Sin Formula and Equations



Sin Formula Diagram



Sine Formula

How to find Hypotenuse using Sine Formula

How to find Opposite using Sine Formula

How to find Angle using Sine Formula

These formulas are for the sine function and show how to find the angle, the opposite side, and hypotenuse, by transposing the general sine formula.


Cos Formula and Equations



Cosin Formula Diagram



Cosine Formula

How to find Hypotenuse using Cosine Formula

How to find Adjacent using Cosine Formula

How to find Angle using Cosine Formula

Here are some useful cosine formulas that have been transposed to find hypotenuse, adjacent, and the angle.


Tan Formula and Equations



Tan Formula Diagram



Tangent Formula

How to find Adjacent using Tangent Formula

How to find Opposite using Tangent Formula


How to find Angle using Tangent Formula

Here are some tangent formula equations that have been transposed. They show how to find the opposite side, adjacent side, and the angle.


Tan in terms of Sine and Cosine



TanA = Opposite / Adjacent



Tan function in terms of Sine and Cosine

As you can see, if we use the earlier expressions for the opposite and adjacent sides - taken from the Sine and Cosine formulas - and substitute them here, we can get the tangent in terms of sine and cosine. The hypotenuse h cancels out obviously. This equation is useful in algebra for expression conversion.


Calculator Solver



My On-Line Calculator Solver can find any missing sides and the angle, based on any information you feed it. If you have a problem where you need to be finding sides or finding angles, then the Trigonometry Calculator Solver might help.


History



The most fundamental breakthroughs in the study of Trigonometry came from the Indians, and in particular the Hindus. The concept of measuring angles, Sine, Cosine, including their inverse. They were also the first to have a sine table that looks very much like what we use today. They were using this for astronomical study calculating the paths of planets and stars. They had a correct model of the solar system and knew that the planets revolved around the Sun, when the rest of the world thought it was the other way round.

The Indian history is the most interesting of all and shows a brilliance which many educated scientists and mathematicians still marvel upon even today. Unfortunately the Indians have never been very good at expressing their genius or taking credit for their work and have left it to Western writers and historians. Bollywood is All Dancing Girls & Parties... which they can do very well, but a serious documentary film about one of their most respected mathematician or war hero they cannot. Why? All answers by email to me please... :-)

Navigation: In the early days of sailing ships Europeans did not have a navigation system. They used the 'dead reckoning' method which was highly unreliable. You could end up anywhere in the world using that system.

When Vasco Da Gamma found India his main concern was that if he ever left he would never be able to return. However he noticed that the Indians had a very robust and reliable navigation system using the stars, trigonometry, and a measuring instrument called Kamal.

Being a new technology that he could not understand he decided to use an Indian navigator Kanha to pilot the ship and in the process chart the route to India.

Kanha used the kamal to measure the latitude of the pole star. An instrument consisting of a board which is held in front of the eye, and string to measure angles. This instrument later became known as the sextant and the credit going to a european for its 'discovery'...

The Inventor of Trigonometry was Brahmagupta. He was born (589 AD) in Bhillamala Gujarat. Brahma Gupta's works in the books Khandakhadyaka and Brahma-Sphuta-Siddhanta - Opening of the Universe -  took astronomy to the Arabs and Sumerian's who then brought it to the Greeks.

The Arab contribution is also a significant one. They had the best writing technology. They created wonderfully bound books with beautifully illustrated artistic writing. Their method of mathematical expression and algebra was over a hundred years ahead of the Babylonians who were still writing everything out as if it were cooking instructions.


Questions



Here are some simple example question exercises for starting out. For beginners they form a vital part of the learning process.

Question 1 Diagram

Question 1: Find the angle x in the right angled triangle shown above in the diagram.

Answer: Here is a nice and simple question. For the experienced it can be worked out without calculator.

We are given the adjacent and opposite sides. If you look at all the formulas shown above on this page, you will see that the tan function is the one to use when you have the opposite and adjacent sides.

Answer 1 Formulas

tan x = 10 / 10 = 1, therefore X = arcTan 1. The arcTan function is the inverse tan function, on your calculator you might have an INV button that you might have to press first before pressing the tan button. The inverse tan means that we want the calculator to give us the angle whose tan function is equal to 1.

non-calculator: When the opposite and adjacent sides are the same, the angle is always 45°. The next time you get a question like that you won't need to use a calculator.



Question 2 Diagram

Question 2: You are given an angle of 45°. The hypotenuse is 14.1421, and you have to find the opposite side x.

Answer: You've got an angle, hypotenuse, and opposite side to find. If you look at the equations in the formula section of this page you can see that the sin formula is the best to use when the opposite and hypotenuse sides are involved. In particular,  o = h x sin A will give the answer.

Answer 2 Formulas



Question 3 Diagram

Question 3: You are given an angle of 45°, the hypotenuse, and have to find the adjacent side x.

Answer: If you look at all the formulas given in the formula section, you can see that the cos function is the best to use when the hypotenuse and adjacent sides are involved. In particular  adjacent = h x cos A is the formula that will solve the problem.

Answer 3 Formulas


GCSE Style Exam Questions & Answers



Here are some GCSE 'style' exam questions and answers that might be useful. Just treat these example problems as a quiz because they are very simple.

Trigonometry GCSE Style Exam Question 1
Trigonometry GCSE Style Exam Question 2
Trigonometry GCSE Style Exam Question 3
Trigonometry GCSE Style Exam Question 4


The Beauty of Mathematics
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Trigonometry
Vocabulary List
Uses in Real Life
Sin Formula and Equations
Cos Formula and Equations
Tan Formula and Equations
Calculator Solver
History
Questions
GCSE Style Exam Questions & Answers

















































































































































































































































































































































































































































































































































 
Author: Peter J. Vis