# Trigonometry

Trigonometry, also known as Trig, is the study of shapes with three corners and angles. It involves the measurement of their sides and angles. Although the modern word for Trigonometry derives from the Greek
*trigōnon*, this word including all the associated words such as sin, have roots in ancient India. This article has a calculator solver, GCSE type questions, formulas, rules, and more. It might be useful as an online reference sheet for revision, or for tutorial purposes. If you have one angle and one side, then the formulas provided can help find the missing sides, and the missing angles.

## Calculator Solver

My on-line calculator solver can find any missing sides and angles, based on the information you
provide. If you are looking for a *calculator* that is good at finding sides or finding angles, then the
Trigonometry Calculator Solver might help.

## Vocabulary List

Name | . | Alternative | . |

sine | 𝓍 = sin y | ||

cosine | 𝓍 = cos y | ||

tangent | 𝓍 = tan y | ||

arcsine | y = arcsin 𝓍 | y = sin ^{-1}𝓍 | |

arccosine | y = arccos 𝓍 | y = cos ^{-1}𝓍 | |

arctangent | y = arctan 𝓍 | y = tan ^{-1}𝓍 |

## SOHCAHTOA Triangle

Pronounced *soh-cah-toa*, this mnemonic is a tool to remember the trigonometric formulas.

- sin 𝓍 = opposite / hypotenuse
- cos 𝓍 = adjacent / hypotenuse
- tan 𝓍 = opposite / adjacent

The basic rule is that trigonometric formulas work only for right-angled triangles. Therefore, the triangle must have one 90° angle.

### How to find a missing side

Calculating the length of the 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.

### How to find angles

Finding angles is easy. To find the angle A, you will need two sides, either, opposite and hypotenuse, adjacent and hypotenuse, or opposite and adjacent. You will also need to know how to use the inverse trig function on your calculator. The inverse trigonometric functions are arccosine, arcsine, and arctangent. Once you have the ratio of the sides, you use this inverse function to find the angle.

Some calculators have a button marked INV. Pressing the inverse button first usually sets the calculator mode to inverse trig mode. In addition, make sure your calculator mode is set to degree mode (DEG) if you want the angle in degrees. Alternatively, you can set it to radian mode (RAD) if you want the angle in radians.

## Sine Formula

## Cosine Formula

## Tangent Formula

## Question 1

Shown above is a right-angled triangle. Given the length of two sides find the unknown angle 𝓍.

### Answer

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 tangent formula is the one to use when you have the opposite and adjacent sides. This is a non-calculator question, because when the opposite and adjacent sides are the same, the angle is always 45°. If you get a question like this in the exams then you will know straight away.

## Question 2

The *triangle* above has an angle of 45°. The hypotenuse is 14.1421 units, and you have to find the opposite side 𝓍.

### Answer

You have an angle, and the length of the hypotenuse, and you have to find the opposite side. If you look at the formula section of this page, you can see that the sine formula is the best to use when the opposite and hypotenuse sides are involved. In particular, the formula “o = h × sin A” will provide the answer.

## Question 3

The triangle above has an angle of 45°. The hypotenuse is 14.1421 units, and you 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 cosine formula is the best to use when the hypotenuse and adjacent sides are involved. In particular, “a = h × cos A” is the formula that will provide the correct solution.

## Uses in Real Life

The History of Trigonometry is extremely interesting. In the real world, man first used it in ancient India for, temple construction, and astronomy. They had knowledge of navigation by stars one thousand years before it became common knowledge in Europe. The earliest written record of trigonometry is in a Sutra known as “Opening of the Universe”. I do not know if it opened up the Universe but it certainly opened up the world.

Arab and Sumerian traders used it for star navigation in the desert where you can use the stars for reference instead of sand dunes. The Greeks used it for building monuments and buildings. 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, missiles, and going to the moon.

Architects and builders use this everyday in daily life, because in architecture, and building construction, everything has angles and length. Builders often use a spirit level and a tape measure, whilst an architect will typically use a compass and a tape measure.

### Aviation

Almost every aspect of aviation uses trigonometry, therefore if you wish to become a pilot, then you will need to know this by heart. The pilots use it for navigation, the designers use it for jet engine simulation and aerofoil research, and
the *Air Traffic Control* (ATC) 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 the pilot understands trigonometry 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 have a distance between two stars and an angle, then trigonometry is the ideal math for calculating unknowns.

### Spacecraft and Rockets

SpaceShipTwo uses trig to calculate the re-entry angle from apogee so that the spaceship lands at the correct location on the runway.

The Indian Rocket PSLV Series uses trig to calculate the optimum entry angle for injection of the payload into the Earth’s orbit, and sending spacecraft to Mars.

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 goes to mission control so that they can decide whether it is wise to send the Rover in that direction.

If you want to become an engineer, then this is vital learning because every aspect of mechanical engineering revolves around this science.

## This Article Continues...

TrigonometryTrigonometry Calculator Solver

Trigonometry GCSE Style Exam Question 1

Trigonometry GCSE Style Exam Question 2

Trigonometry GCSE Style Exam Question 3

Trigonometry GCSE Style Exam Question 4