Logarithmic Trigonometric Function

The logarithmic terms and the trigonometric functions are the building blocks for logarithmic equations and trigonometric equations, respectively. The logarithmic terms are used to build logarithmic equations. It is possible to combine the usage of logarithmic and trigonometric functions to generate strange equations that incorporate both functions as variables. Logarithmic functions are the inverses of exponential functions, and any exponential function can be represented in logarithmic form, and any exponential function can be expressed in logarithmic form. A similar transformation can be performed on any logarithmic functions to convert them to exponential form. We can work with very huge numbers while handling numbers that are much more manageable when we employ logarithms, which is a very helpful property of mathematics. To solve the equation x = 2y for y in order for it to be stated in function form, a new term or symbol would have to be introduced. If x = 2y, then y = (the power on base 2) to equal x is equal to (the power on base 2). In order to meet this requirement, the term logarithm, abbreviated log, is established.

y = (the power on base 2) multiplied by the number x

This equation can be written as y = log2 x, which is a more straightforward representation.

This can be written as “y equals the log of x, base 2” or “y equals the log of x, base 2” or “y equals the log, base 2 of x.”

It is the exponent by which another fixed value, the base, must be raised in order to produce a number equal to the logarithm of that number. Example: The logarithm of 1000 to base 10 is 3, which is due to the fact that 1000 is ten to the power three: 1000 = 10 10 10 = 103.

Known as the common logarithm, the logarithm to base b=10 can be used in a wide variety of scientific and technical applications. The natural logarithm has as its base the constant e (2.718), and it is widely used in pure mathematics, particularly calculus. The binary logarithm, which has base b=2, is widely used in computer science and has a long history.

The concept of logarithms is to perform the opposite of the action of exponentiation, which is to raise a number to a higher power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2, as follows: 23=8, and 8 is the product of three factors of 2. From this we may deduce that the logarithm of 8 with respect to base 2 is 3, and hence log28 = 3.

Several mathematicians utilised logarithms to convert multiplication and division issues into addition and subtraction questions before the discovery of calculus in the late nineteenth century. In logarithms, the power of some numbers (typically the base number) is increased in order to obtain another number. The exponential function’s inverse can be considered. In mathematics and science, huge powers of numbers abound, and the most important and useful of these powers of numbers are the logarithms, as we all know.

Formula of logarithmic Trigonometric Functions

The following table contains the formulas for some trigonometric functions. They are as follows:

The sine function of an angle is defined as the relationship between the lengths of the opposite sides and the lengths of the hypotenuse. The sin value should be Sin a= Opposite/Hypotenuse=CB/CA.

According to the cos function formula, the ratio of the length of the neighbouring side to the length of the hypotenuse is equal to one. The cos function can be deduced from the reference diagram shown above as follows:

In mathematics, Cos a = Adjacent/Hypotenuse = AB/CA.

The tan function formula is defined as the ratio of the length of the opposite side of a right-angled triangle to the length of the adjacent side of a right-angled triangle. The learner should be aware that the tan function can be represented in terms of the sine and cosine functions, expressed as their ratio. This leads us to deduce that the tan function is defined as Tan a = Opposite/Adjacent = CB/BA. tan can also be written as Tan a = sina/cosa, which is the sum of the sine and cos functions.

Additional functions are represented by formulas, and they are as follows:

BA and CB are adjacent and opposite in terms of cot a = 1/ (tan a).

Cosec a = 1/ (sin a) = Hypotenuse/Opposite = CA/CB = Cosec a = 1/ (sin a)

The hypotenuse and adjacent are represented by the cosine of sec a = 1/ (cosine a).

There are just a few inverse trigonometric functions in existence. In this context, the inverse of cosecant, secant, cotangent, tangent, cosine, and sine are referred to as arc cosecant, arc secant, arc cotangent, arc tangent, arc cosine, and arc sine, and the inverse of cosecant, secant, cotangent, tangent, cosine, and sine are referred to as arc cosecant, secant, arc cotangent.

The logarithmic function is an important tool in the realm of mathematical computation and analysis. 

Discovery of Logarithmic Function

Logarithms were discovered in the 16th century by John Napier, a Scottish mathematician, scientist, and astronomer who lived during the reign of King James I of England. It has a wide range of applications in astronomical and scientific computations that involve large numbers of variables. Graphs of logarithmic functions are closely connected to graphs of exponential functions, and they are sometimes referred to as the inverse of exponential functions. It is possible to turn the exponential function ax = N into a logarithmic function logaN = x.

If the logarithm of any number N is understood as an exponential form, the exponent to which the base of the logarithm should be raised in order to produce the number N is the logarithm of the exponent to which the base of the logarithm should be raised. We will learn more about logarithmic functions, logarithm types, the graph of a logarithmic function, and the properties of logarithms in this section of the course.

Example of Trigonometric Logarithmic Functions

•Calculate the angle of Sin 105°.

Solution: Sin 105° can be represented as sin (60° + 45°), which is the same as sin (A + B) in terms of notation.

We are aware that the formula for sin (A + B) = sin is as follows: the product of A cos B Plus the product of A is the sine of B.

As a result, sin 105° = sin (60° + 45°) = sin 60° ×cos 45° + sin 60°× cos 60° × sin 45° 

= √3/2 × 1/√2 + 1/2 × 1/√2

= √3/2√2 + 1/2√2

= (√3+1)/2√2

Conclusion

The logarithmic terms and the trigonometric functions are the building blocks for logarithmic equations and trigonometric equations, respectively. The logarithmic terms are used to build logarithmic equations. A few inverse trigonometric functions are known to be available. In this context, the inverse of cosecant, secant, cotangent, tangent, cosine, and sine are referred to as arc cosecant, arc secant, arc cotangent, arc tangent, arc cosine, and arc sine, and the inverse of cosecant, secant, cotangent, tangent, cosine, and sine are referred to as arc cosecant, secant, arc cotangent.