Trigonometric Equations And Roots

Trigonometry is the stream in mathematics that consists of the relationship between the angles and length of the sides of a triangle. Trigonometric equations play an important role in navigation, periodic functions, etc. Euclid and Archimedes worked on chords and inscribed angles in a circle which they expressed in a geometrical way and proved various theorems that are now a part of trigonometry. From the 18th century onwards, more development in trigonometry was observed.

Trigonometric Equations

The branch of trigonometry deals with the trigonometric ratios that are used to determine the incomplete sides and angles of the triangle. The fundamentals of trigonometry revolve around the main function of sine, cosine, and tangent. These functions can be expanded to include other ratios of trigonometry in the Cartesian plane. Trigonometry also deals with the identities which are commonly used for expressing the equation in a simpler form for solving the problems. The equations that involve the trigonometric functions are called trigonometric equations.  Trigonometric equations that hold true for all the values of variables are termed trigonometric identities.

Trigonometry consists of basic trigonometry and inverse trigonometry.

Basic Trigonometry– The basic trigonometry consists of three ratios like sine, cosine, and tangent in the Cartesian Plane.

Inverse Trigonometry– This consists of the inverted form of the fundamental trigonometric ratios like cosine, cosecant, and cotangent. These inversed ratios are also called anti-trigonometric functions or arc functions. The inversed ratios are used to calculate the angle for each trigonometric ratio.

Father of Trigonometry

Hipparchus (from Nicaea), a Greek astronomer, geographer, and mathematician, was born in 190 BC. He was a great astronomer in ancient times. He is known for his initial invention of the precision of equinoxes. He introduced the modern tables for the sine values. These sine values were used by him to solve trigonometry and spherical trigonometry. In trigonometry, he used all his solar and lunar techniques to develop a reliable method to predict a solar eclipse. He discovered the measurement of Earth’s possession, invented the astrolabe, and introduced the armillary sphere, which was used for the calculation of the star catalog.

From the 3rd century, geometrical applications were used in astronomical studies for studying the fixed relationship with the angles of the triangles. Trigonometric equations can be pervasive in pure and applied mathematics.

Roots of Trigonometric Functions

A trigonometric equation is solved with the help of the trigonometric functions that are sine, cosine, cotangent, secant, cosecant, and tangent. During finding the values for a given equation, there are two cases that need to be considered.

Case 1- The interval for the equation is given, and we need to calculate the angle satisfying the equation in that specific interval.

Case 2– When the interval is not given, we need to just calculate the general solution for the equation.

The root loss occurs in a trigonometric equation when similar variables or terms are canceled from the sides, and it refers to the missing roots during the application of algebraic functions. To obtain the solution correctly instead of losing them, the cancellation cannot be made arbitrarily. The extraneous roots play an important role in this function. So, extraneous roots are the roots that are obtained when a trigonometric function is squared to obtain an accurate result.

For example, if P and Q are trigonometric functions, then;

Sin θ [P]= sin θ [Q]

Here on canceling the sin θ, the equation and solution are lost, and sin θ becomes equal to zero.

Example for Root of Trigonometry-

 Question Find the solution to the following equation such that 

0 ≤ x < 2π, 2sin² (y) − sin (y) + 3 = 4

Answer

2 sin²(y) – sin y − 1 = 0

Next, we factor.

(2 sin (y) + 1) (sin (y) − 1) = 0

Set each factor equal to zero and solve.

2 sin (y) + 1 = 0 or sin (y) − 1 = 0

sin (y) = −1 / 2

 sin(y) = 1

The angles that satisfy the given equation is

 The first is satisfied by angles 7π / 6 and 11π / 6, and the second is satisfied by angles π / 2.

Conclusion

Trigonometry is a branch of mathematics that has a wide range of applications, and the trigonometric functions can be expressed to find the relationship between the triangles, their sides, angles, etc. The trigonometric functions are employed for calculating the distance of a structure and finding out the elevation angle to calculate the height of a structure. These trigonometric functions have led the modern world to develop various innovative equipment and come up with unique ideas like global positioning systems and artificial intelligence.

In Marine engineering, trigonometric functions are used to design the marine ramps, a sloping surface that connects the lower and higher levels areas. Sound engineers prefer trigonometry to study the pattern of sound waves being produced through an instrument.