Differentiation of Inverse Trigonometric Functions

Inverse trigonometric functions also play important role in differentiation. In this article, you are going know the derivative of every inverse trigonometric function and can solve problems which are related to Inverse trigonometry by applying one of the listed fundamentals.

Consider x is an independent variable but denotes domain of the inverse trigonometric functions. By considering this variable, Inverse trigonometric functions Arc Sine, Arc Cosine, Arc Tangent, Arc Co-Tangent, Arc Secant and Arc Cosecant are written as Sin^-1 x, Cos^-1 x, Tan^-1 x, Cot^-1 x, Sec^-1 x and Cosec^-1 x respectively. Similarly, differentiate or derivate element in differentiation can be written as d/dx.

Trigonometric Ratio Cosine

Cosine is one of the trigonometric ratios in mathematical subject Trigonometry. The ratio between adjacent side and hypotenuse of a right angle triangle is defined Cosine. Cosine function is usually written in its short form Cos. In fact Cosine is a dependent function and depended on angle of the right angle triangle. In other words, expressing Cosine function with simply Cos is wrong mathematically. Consider a right angle triangle as shown in the below picture and its sides are opposite side, adjacent side (some people call it as base) and hypotenuse and angle of the right angle triangle is Theta which is denoted with a symbol θ. Now, Cosine function can be written in its short form by integrating the angle i.e Cos θ.

Sine Graph

Sin Graph is a graphical representation of a relation between angles of right angle triangle and sine values for the respective angles. Simply, it is a graph between various angles and sine values at those angles. In fact analyzing this graph helps us to know the properties of the trigonometric function Sine. Similarly, this graph is popularly used in physics and engineering applications most importantly signals and systems, electrical engineering and etc.

Consider a right angle triangle and its angle is Theta, denoted with θ symbol. The sides of the same right angle triangle are opposite side, adjacent side and hypotenuse as shown in the below image. By considering angle, trigonometric ratio sine can be written as Sin θ.

Range of Sine

Range of trigonometric function Sine is defined as the set of values which are determined from Sine function for its domain. In other words, by substituting all angles belong to domain of sine; we can get all corresponding values and these values known as range. Range of the sine denoted with [-1, 1] which means all values within -1 to 1 and the reason of showing it with [ and ] brackets is we should also consider values -1 and 1. In other words denoting range with [ and ] brackets is it gives all vales from -1 to 1.

Domain of Sine

The domain of trigonometric function sine is the set of angles for which the sine function is defined. The domain of sine is input for this function and the all values of this trigonometric ratio are determined by putting domain’s values in angle’s place. In few steps you are going to understand this concept clearly.Domain of sine function includes rational numbers, irrational numbers, positive numbers, negative numbers, zero and integers but does not work for undermined values such as imaginary values, complex numbers and infinity. The pros and cons of this function exactly meet the property of real numbers group. Therefore, the domain of Sine function is Real Number Group. In other words, for the set of all values of real numbers group, sine function is defined. It simply represents with English alphabet R symbol. In other words, the domain of a sine function is R.

Componendo and Dividendo

Componendo and Dividendo is an algebraic fundamental, mostly helps us to simplify the complicated functions and also math problems. It plays significant role in all mathematical concepts. In this article you are going to know the mathematical proof of Componendo and Dividendo rule.

Consider a, b, c and d are four numbers which belong to all numbers group including complex numbers. Assume that the ratio of a and b are equal to the ratio of c and d. it can be expressed in mathematical form. However the value of a shoulder not equal to value of b. Similarly, the value of c shoulder not equal to value of d.

Pythagoras Theorem

Pythagoras Theorem is a mathematical fundamental which tells a mathematical relationship between opposite side, adjacent side and hypotenuse of a right angle triangle. Pythagoras Theorem is also known as Pythagorean Theorem. This fundamental principle is mostly useful in trigonometry.

The square of a hypotenuse of a right angle triangle is equals to summation of squares of opposite and adjacent side of same right angle triangle is defined Pythagorean Theorem.

Consider a right angle triangle and its opposite side, adjacent side and hypotenuses are drawn as shown in the graphical image.

Trigonometric Functions

The ratio between any two sides of a right angle triangle is defined a trigonometric function and ratio between different sides of same right angle triangle can be taken in 3! ways i.e 6 ways. We name to each ratio for our convenience because expressing ratios of two sides and also developing trigonometric fundamentals with same ratios confuse us and also create so many problems in studying this mathematical concept Trigonometry. Hence, we name to each ratio. As per our definition we understood that we take ratio between two sides. Therefore we also call them, trigonometric ratios. In other words we can call those six ratios as trigonometric ratios and also trigonometric functions. The lists of ratios’ names are given below.

Trigonometry

Trigonometry is a scientific mathematical concept which reveals relationship between sides of a triangle and its angle. This math concept is actually used to measure heights, distances and areas and became essential in engineering and physics.

The word Trigonometry is derived from two Greek words. One word is Trigonon and the second word is Metron. The meaning of trigonon and metron are triangle and measure respectively. Hence the meaning of trigonometry is measuring triangle.

Trigonometry was developed with angle of a right angle triangle and six functions which are known Trigonometric functions or ratios.

Limit of ratio of Sinθ and Angle θ is Unit Value

Proving the ratio of Sine and respective angle as unit value i.e 1 mathematically is objective in this article. This limits’ function mostly helps in dealing differentiation. In other words, it is most important formula in calculus.

In order to prove it mathematically, consider some assumptions. Assume θ is an angle of right angle triangle and with respect to angle the trigonometric function Sine is Sin θ. The angle of right angle triangle is not zero but approximately zero which means θ → 0 or θ ≈ 0.

The ratio of Sine and its angle is (Sin θ)/θ. By applying the limits function with angle’s assumption, it can be expressed as written below mathematically.