An exponential function is a mathematical function that can be applied to a wide range of real-world situations. It’s usually used to compute investments, model populations, and other comparable tasks, as well as to determine exponential decay or exponential growth. This article covers exponential function formulas, rules, properties, graphs, derivatives, exponential series, and examples.

Inverse trigonometric functions are closely related to fundamental trigonometric functions as a learning topic. The domain and range of inverse trigonometric functions are translated from the domain and range of trigonometric functions. In trigonometry, we look at the relationships between angles and sides in a right-angled triangle. Inverse trigonometric functions exist as well. Inverse trigonometric functions are symbolized by sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, and sec-1 x, respectively.

Let us dive deep into the formulas and concepts of Inverse Trigonometric and Exponential Functions to understand them better.

Inverse Trigonometric Functions

The following formulas have been combined together to make a list of inverse trigonometric formulas. These formulas can be used to convert one function to another, determine the principal angle values of these inverse trigonometric functions, and perform a number of arithmetic operations on them. Furthermore, all basic trigonometric function formulas have been transformed to inverse trigonometric function formulas and are divided into four categories are provided below.

  • Arbitrary Values
  • Reciprocal and Complementary functions
  • Sum and difference of functions
  • Double and triple of a function

Inverse Trigonometric Function for Arbitrary Values

For arbitrary values, the inverse trigonometric function formula can be used to express all six trigonometric functions. For the inverse trigonometric functions of sine, tangent, and cosecant, the negatives of the values are translated as the negatives of the function. The negatives of the domain are interpreted as subtracting the function from the value for functions of cosine, secant, and cotangent

  • sin-1(-x) = -sin-1x,x ∈ [-1,1]
  • tan-1(-x) = -tan-1x, x ∈ R

Exponential Function Formula

The definition of a basic exponential function is f(x) = bx, where ‘b’ is a constant and ‘x’ is a variable. f(x) = ex, where ‘e’ is “Euler’s number” and e = 2.718, is a popular exponential function…. If we broaden the scope of distinct exponential functions, we can include a constant that is a multiple of the variable in its power. In other words, an exponential function can take the form f(x) = ekx. It can also have the form f(x) = p ekx, where ‘p’ represents a constant. As a result, an exponential function can take one of the following shapes as given.

  • f(x) = bx
  • f(x) = abx
  • f(x) = abcx
  • f(x) = ex
  • f(x) = ekx
  • f(x) = p ekx

All letters except ‘x’ are constants, ‘x’ is a variable, and f(x) is an exponential function in terms of x. It’s also worth noting that any exponential function’s base must be a positive number. i.e., b > 0 and e > 0 in the above functions. Also, b should not equal 1 (if b = 1, the function f(x) = bx becomes f(x) = 1, and the function is linear but NOT exponential in this case).

Concept of Inverse Trigonometric Function for Reciprocal Functions

The inverse trigonometric function converts the specified inverse trigonometric function into its reciprocal function for reciprocal values of x. This is because sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other in trigonometric functions.

Concept of Inverse Trigonometric Function for Complementary Function

For identical x values, the sum of complementary inverse trigonometric functions equals a right angle. As a result, the complementary functions sine-cosine, tangent-cotangent, and secant-cosecant add up to /2. Complementary functions include sine-cosine, tangent-cotangent, and secant-cosecant.

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