Trigonometry Step by Step
What is trigonometry?
Trigonometry is made of two words that originated from the Greek language. “Trigonon” means triangle, and “metron” means measure. Well, trigonometry is all about triangles. Trigonometry is introduced to students in Class 9th, and the subject broadens in each class up to 12th standard.
The application of trigonometry is vast, and it is one of the most used topics in real life by mathematicians, engineers, scientists, etc. In trigonometry, we learn the different relationships between sides and angles of a triangle.
Trigonometry has ratios and identities. They are used in mathematics to solve different triangles’ problems and apply them in real life to find out the estimated time and distances of natural and manmade objects.
Origin of Trigonometry
Trigonometry came into existence back in the 3rd century BC when people were studying geometry and astronomical studies. Indians and Greeks were the two developers of trigonometry. When the former was working on calculations of chords, the latter were creating tables for different kinds of trigonometric ratios.
Surya Siddhanta invented the first modern sine convention. Its properties were then documented by the 5th century’s Indian Mathematician and astronomer Aryabhatta.
As the inventions of different ratios and identities evolved, in the 10th century, Islamic mathematicians were using all six trigonometric functions and started applying it to spherical geometry.
With the constant need for navigation and accurate maps, the field of trigonometry advanced into a significant branch of mathematics. As the centuries passed, many scholars and mathematicians started writing books based on their findings in the field of trigonometry. Some authors and scholars were James Gregory, Gemma Frisius, Colin Mclaurin, Brook Taylor, and many more.
Trigonometric Ratios
As we read about trigonometry history, you will now learn six significant trigonometric ratios applied in triangles. Let’s see what those are.
Ø is the angle that we consider when we solve any trigonometric equation. All the sides correspond and are named according to this angle.
In a right-angled triangle, the 3 sides are called:
- Hypotenuse (the side opposite to the right angle)
- Perpendicular (the side opposite to ø angle)
- Base (the side adjacent to ø angle)
The six trigonometric ratios are called as:
- Sin ø
- Cos ø
- Tan ø
- Cosec ø
- Sec ø
- Cot ø
Now we will see the different sides that we have to consider for these angles:
- Sin ø = Perpendicular/Hypotenuse
- Cos ø = Base/Hypotenuse
- Tan ø = Perpendicular/Base
You can find tan ø by dividing the sin ø and cos ø that is tan ø = sin ø/cos ø. Similarly, for cot ø, the equation of sin ø and cos ø will be cos ø/sin ø.
Let us see a simple example of sin ø.
The question is:
- What is “sin 36” in the triangle with
- Hypotenuse – 4.9
- Perpendicular – 2.8
- Base – 4
Thus, to find out sin 36, we have to divide 2.8 by 4.9. It will answer 0.57.
You will find questions related to finding the sin or cosine angle or vice versa. These ratios are the basics of trigonometry to get started. The equations and identities are the next steps in this branch of mathematics.
Let us dig deep into trigonometry now.
Degrees and Radians
Degrees are also written as radians. Let us see some of the following facts:
- Right angle 90°= π/2 radians
- Straight angle 180° = π radians
- Full rotation 360° = 2π radians
Sin and Tan are used to find the height of a tree. Take, for example:
We know the angle is 45°, and the length between the tip of the tree and ground is 20 m, which is also the hypotenuse. We can find out the value of sin 45 through a calculator. The answer comes around 0.7071.
Hence the equation results to 0.7071 = perpendicular/hypotenuse,
i.e. 0.7071 = perpendicular/20
So, the height of the tree will be 14.14 m.
You can write the answer up to two decimal numbers. All evaluation systems universally accept it.
There are a lot of exercise questions available online to practice. You must also solve previous year question papers related to your board like ICSE, CBSE, etc. It prepares you for the type of questions you will face in the finals.
Trigonometric Identities
After understanding the trigonometric ratios, we will understand how the identities are derived from them.
So, you know about the Pythagoras theorem i.e.
A2 + B2 = C2
Dividing the equation with C2 we get the equation as (A/C)2 + (B/C)2 = 1,
A/C is perpendicular/hypotenuse (sin ø) and B/C is base/hypotenuse (cos ø),
so, voila, your formula is derived as sin2 ø + cos2 ø = 1.
Reciprocals of sin ø, cos ø, and tan ø
If you notice the formulas for sec, cosec, and cot, they are the reciprocal of sin, cosine, and tan, respectively.
You can write the ratios as:
- cosec ø = 1/sin ø
- sec ø = 1/cos ø
- cot ø = 1/tan ø
You can also write cot ø as cos ø/sin ø.
The other important identities are:
- Sin2 ø + cos2ø = 1
- Tan2 ø + 1 = sec2 ø
- Cot2 ø + 1 = cosec2 ø
Double Angle Trigonometric Identities
There are a set of trigonometric identities that are significant while solving questions related to it.
Let’s check them out:
- Sin (2ø) = 2sin ø cos ø = 2tan ø/1 + tan2 ø
- Cos (2ø) = cos2ø – sin2ø = 2cos2ø – 1 = 1 – 2sin2ø = 1 – tan2ø /1 + tan2ø
- Tan (2ø) = 2tan ø/1 – tan2ø
This was all about a few general identities and ratios used by class 9th and 10th students to solve different application-based problems and identity problems.
Applications of Trigonometry
Trigonometry in Music
This field of mathematics doesn’t have a direct hand in solving practical issues, but still, it is used in various things that we enjoy. Example: Music – you know how sound travels in the form of waves. It is not a regular sine or cosine function. You can still use it in creating technical music or computer music.
A digital device can’t comprehend music as humans do, so they comprehend it mathematically by its waves. So, sound engineers know the basics of trigonometry, and it is also used in physics for higher studies. The pleasant music the sound engineers create that takes our minds away from stress – thanks to geometry for that.
Trigonometry Is Applied to Measure Heights of Buildings, Mountains, And Hills
As we studied with examples above, trigonometry is applied to determine the height of the buildings, lampposts, lighthouses, mountains, etc. If you know the distance from where you see the building and the angle of elevation, you can simply solve the equation for the building’s height.
Similarly, if you know the value of one side and angle of depression from the top of the skyscraper or building, you can find the other side in the triangle. You must know one side and angle of it, and the solution for the other comes out easily.
Trigonometry in Construction
Construction uses a lot of trigonometry identities and ratios. They calculate the following things:
- The size of the fields, lots, and areas
- Make the walls parallel and perpendicular
- Install ceramic tiles (measurements)
- Roof inclination angles
- The building or tower’s total height, the width of the building, etc. and many other elements that require trigonometry to solve and find the distances and heights.
Architects also use trigonometry to measure roof slopes, structural load, ground surfaces, and other elements like sun shading and light angles.
Trigonometry Usage in Navigation
With trigonometry, you can set directions like North, South, East, and West. It tells you which direction to go when the compass shows you to get in a straight direction. This branch of mathematics is used in navigation to pinpoint a location. It is also utilized to find the distance of the shore from the middle or any other point in the sea. You can also use it to see the horizon.
Trigonometry in Electrical Engineering
The current power companies use AC (alternating current) to transmit electricity across long-distance wires. As you know, the electrical charge is reversed in regular intervals to deliver power carefully to homes and businesses in an AC. Electrical engineers make use of trigonometry to create the flow and the shift of direction. By using the sine function, they modify the voltage. Each time you switch on your television — you are applying trigonometry in your real life.
There are a lot of quizzes and worksheets online on trigonometry, which you can solve. These will help you in clearing your concepts at your pace. These are some of the benefits of online learning where you can assess your strengths and weaknesses in real-time, especially when it comes to tedious topics like trigonometry.
