The term Trigonometry has been derived from two Greek words, trigonon and metron, which means triangle and to measure respectively. However, until the 16th century, it was mainly about computing the values of the missing parts of the triangle numerically when the other values of the parts were mentioned. Keeping this in mind, let's start with an example to make the concept more clear. The third and the remaining two angles can be calculated if only the length of the two sides of the triangle and the enclosed angle’s measure is given. This distinguishes Trigonometry from geometry as it investigates the qualitative relations. Nevertheless, Trigonometry was considered a part of geometry while it became a separate branch of math during the 16th century.
In a modern sense, Trigonometry started with the Greeks while the first one to make a table of the values of the Trigonometric function was Hipparchus. During his time, these things were expressed in geometric terms like relations between the angles that subtended them and the chords of various forms. Moreover, until the 17th century, the modern symbols of Trigonometric functions were not established.
However, during the 16th century, Trigonometry started to change its character to an algebraic-analytic subject from a geometric discipline as two developments were spurred by the Frenchmen. It was the rise of symbolic algebra and the invention of analytic geometry.
Another aspect of Trigonometry that received great attention was surveying, following which the triangulation’s method was first suggested in 1533 by Gemma Frisius, a Dutch Mathematician. This method was further carried for the first time on a large scale by Willebrord Snell, a Dutchman, who surveyed a stretch of 130 km using around 33 triangles. Keeping these developments in mind, the French government undertook the responsibility to triangulate the entire country under the leadership of Jean Picard, an astronomer. Following this, the survey of the entire continent of India was taken by the British as a means to have an ambitious task. Popularly known by the name of the Great Trigonometric Survey, this survey lasted from 1800 to 1913, which led to the discovery of Mount Everest, the tallest mountain peak on Earth.
Owing to these developments, the scientists from the 18th century diverted their attention to the various aspects of the functions of Trigonometry.
Origin of the Trigonometry
For the very first time, the idea of ‘sine’ was used in the work by Aryabhatta in A.D 500. He took the word ardha-jya, which was later shortened to jive or jya in the due course. However, when translated, the word jiva became sinus, which meant curve in Latin when translated from the Arabic version. Following this, the word sinus was used as sine in the coming era in mathematical terms throughout Europe. Besides, the first abbreviated notation, known as sin, was termed by an English Professor of astronomy, identified as Edmund Gunter.
The terms ‘tangent’ and ‘ cosine’ originated much later in the time while the need to compute the sine of the complementary angle gave birth to the function of cosine, which was called Kotijya by Aryabhatta. However, Edmund Gunter, the English Professor of astronomy, termed it Cosinus while an English Mathematician, known as Sir Jonas Moore, abbreviated it as ‘cos’ in the year 1674.
Moreover, during the middle ages, Jewish and Arab scholars also played an important role in contributing to the branch. Tangents and cotangents the first table was made by Habash al-Hasib, who also wrote on the topics like astronomy.
Moreover, the main functions based on a right-angled triangle in Trigonometry are Sin, Tan, and Cos.
There are around 6 Trigonometry ratios consisting of sine, cosine, tangent, cosecant, secant, and cotangent, which tells the different possible combinations in a right-angled triangle.
Therefore, the use of Trigonometry in solving a side in right-angled triangles is only possible when the length of the other two sides of the length and angle of a side is already known. For that, one has to choose a ratio that contains the unknown and the given side before using algebra to find the value for the unknown side.
We know in Trigonometry we are going to discuss the relationship between angle and length in right-angled triangles. Let’s first remind ourselves, what we call sides of the right angled triangle.
Observe that above the right-angled triangle ABC is right angled at C. If we consider angle∠B then side AC will be perpendicular, side BC will be called based and side AB is called hypotenuse. Similarly, If we consider angle ∠A then BC will be perpendicular, AC will be considered base and AB will hypotenuse. Among these three, different ratios will give sin, cos, tan, cot, sec, cosec ratios.
Sin Theta Formula
So now we’ll discuss what sin theta and sin formula are. As we have discussed above, sin is a Trigonometric ratio that is perpendicular/hypotenuse.
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Here x represents the angle which is under consideration.
Example on Sin x Formula
1. If Cos x = 35, then find the value of Sin x.
Then Find the Value of Sin x.
Solution: We know that, cos θ = BaseHypotenuse
On comparing the given ratio, Base = 3, Hypotenuse= 5.
Now we also know Pythagoras theorem, which says,
(Hypotenuse)² = (Base)² + (Perpendicular)²
⇒ (Perpendicular)² = (Hypotenuse)² - (Base)²
⇒ (Perpendicular)² = 5² - 3²
⇒ (Perpendicular)² = 25 - 9
⇒ (Perpendicular)² = 16
⇒ (Perpendicular)² = 4
Here we are considering only positive signs because the length of the side can’t be negative.
Sin x - 45
Hence this is the required answer.
2. If Cosec x = 67 then Find the Value of Sin x.
Solution: We know that sin x = 1cosec x
Using the above value we have, sin x = 16/7
Hence, sin x = 76
Brief Overview of Trigonometry
As the name might give you an idea, Trigonometry is all about triangles.
Trigonometry is the branch of mathematics that deals with the relationship between angles and the length (not arc length) of a line made by angle. Whenever we talk about angles, we need to have 2 intersecting lines, through which angle has been made. Adding to that we need one more line on which we’ll measure the length. So, in a nutshell, we need three lines. That’s what “tri” stands for. Now there are many types of triangles we know. Which type of triangle should be? So, the triangle should be right-angled.
If we look deep into it, the system of Trigonometry developed to get angles and distances computed in such fields as mapmaking, astronomy, artillery range finding, and surveying while plane Trigonometry covers problems involving distances and angles in a plane. However, in spherical Trigonometry, it becomes a three-dimensional space.
Nevertheless, Trigonometry is that branch of the system, which helps us determine the unknown or missing sides, lengths, and angles in a given triangle. Specifically, one of the internal angles in the triangle should be 90 degrees. Now we’ll see different types of Trigonometric ratios after taking a peek into its origin.