Trigonometric Identities

Trigonometric relations are used in diverse domains such as astronomy, artillery, mapmaking, etc. There are six major functions in trigonometry which are commonly used in today’s world. Names of these functions are – sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec) and cotangent (cot). We can remember the values of these functions easily through some tricks in the trigonometric table. It is very helpful as this table can avoid so many long calculations for which we had to carry calculators. This table also plays an essential role in various places like algorithms, the growth of mechanical computing devices, and whatnot.

Trigonometric identities

Equations that show equality, trigonometric functions, and are true for every value are trigonometric identities. Or we can say, equations that are true for right-angle triangles are trigonometric identities. Each side has its name in a right-angled triangle: the hypotenuse, adjacent, and the opposite. People usually find trigonometry quite difficult, but it is an important field of mathematics that many people use in various occupations. To make this easy for the people who find it difficult, many softwares like Trigonometric Identity Calculator solve long equations and make it quick and help us save time. 

List of trigonometric identities: 

1. Pythagorean identities 

~ 1 + cot²θ  = cosec²θ

~ 1 + tan²θ = sec²θ

~ sin²θ + cos²θ = 1

2. Angle sum and difference identities

cos(a+b) = cosa.cosb – sina.sinb

~ cos(a-b) = cosa.cosb + sin a.sin b

~ tan(a+b) =  tan a + tan b 

 1 – tan a.tan b

~ tan(a-b) =   tan a – tan b

1 + tan a.tan b

~ sin(a+b) = sina.cosb + cosa.sinb

~ sin(a-b) = sina.cosb – cosa.sinb

3. Double angle formule

~ sin(2θ) = 2.sinθ.cosθ

~tan(2θ) =   2.tanθ

1 – tan²θ

~ cos(2θ) = cos²θ – sin²θ

= 2.cos²θ – 1

= 1 – 2.sin²θ

4. Triple angle formule

~ sin(3θ) = 3sinθ – 4sin³θ

~ tan(3θ) = 3tanθ – tan³θ

1 – 3tan²θ

~ cos(3θ) = 4cos³θ – 3cosθ

5. Negative angle identities

~ tan(-θ) = -tanθ

~ cosec(-θ) = -cosecθ

~ sec(-θ) = secθ

~ cot(-θ) = -cotθ

~ sin(-θ) = -sinθ

~ cos(-θ) = cosθ

The Pythagorean Theorem

The Pythagorean theorem is also known as the Pythagoras theorem. It shows the relation between the three sides of a right-angled triangle. A triangle that has a 90 degree between two sides is known as the right-angle triangle. The sides adjacent to the angle are the legs, and the third side is known as the hypotenuse. The theorem proves that the sum of the square of the adjacent sides is equal to the square of the third side. The Pythagoras theorem has a significant role in trigonometric relations. Many people also find trigonometry because it has a comprehensive list of trigonometric identities.

The Pythagoras theorem can be used in many situations in our lives; for instance, if we know one side and one angle in a triangle, we can easily find the rest of the sides and angles by using the Pythagoras theorem. 

Why do we need Trigonometry?

Trigonometric relations are considered the most important branch of mathematics ever discovered. Triangles are one of the easiest forms found in nature, and their mathematics is vital, especially where precision is needed in the measurement of distances. There are dozens of applications where accurate distances are required, like astrology, navigation in naval, structural engineering., graphic designing, geographical surveys, etc. 

With the help of trigonometric relations, we can take precautionary steps from disasters like high tides as trigonometry is used in the calculator of heights and the impacts of tides. Architects also use it to make the necessary calculations required to make a building, like ground surfaces, roof slopes, and other aspects.

Graphs of Trigonometric Functions

Trigonometric relations can also be explained with the help of diagrams. Each of the six functions discussed above can be explained on a graph plane. These trigonometric graphs are used in many areas of engineering and science. When we plot the graph, on the x-axis, angles are taken in radians, and on the y-axis, values of the function on the degree radian are taken.

Conclusion: 

Trigonometry simply means maths from triangles. The ‘tri’ in trigonometry comes from triangles. Trigonometric relations are also considered the most important discovery in the mathematics field. Trigonometry may not have everyday applications, but it does help us to work with triangles with ease. Without trigonometry, life would be much more difficult. Without going through the trouble, we can easily find the height of a building. Hence, I think that it was a good invention by Hipparchus and thanks to this, many architects need not go through the trouble to calculate the unknown side of a specific triangle, so it helps in real life. It is very helpful in engineering careers.