Trigonometric And Inverse Trigonometric Functions

Introduction

Inverse trigonometric functions are closely related to fundamental trigonometric functions. Sin, cos, tan, cosec, sec, and cot are the basic trigonometric functions. Inverse trigonometric functions have all the formulas for basic trigonometric functions, such as the sum, double, and triple of a function. 

Inverse Trigonometric Formulas

The inverse of the fundamental trigonometric functions are called inverse trigonometric functions. Sin-1 x = can be substituted for the basic trigonometric function sin = x. X can take the form of whole integers, decimals, fractions, or exponents in this case. We obtain = sin-1 x (1/2) for = 30°, where lies between 0° and 90°. Inverse trigonometric function formulas can be made from any trigonometric formula.

1. For Arbitrary Values

All six trigonometric functions can be expressed using the inverse trigonometric function formula for arbitrary values. The negatives of the values are translated as the negatives of the function for the inverse trigonometric functions of sine, tangent, and cosecant.

1. For Reciprocal Functions

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

1. For Complementary Functions

A right angle is obtained by adding the complementary inverse trigonometric functions. The addition of complementary inverse trigonometric functions equals a right angle for the identical values of x. As a result, the sine-cosine, tangent-cotangent, and secant-cosecant complementary functions add up to π/2. It’s important to refer to the study material notes on trigonometric and Inverse trigonometric functions to learn more about complementary functions.

1. Sum and Minus of Inverse Trigonometric Function Formulas

As shown in the formulas below, the sum and difference of 2 inverse trigonometric functions can come together to generate a single inverse function. From the trigonometric function formulas, the sum and difference of the inverse trigonometric functions have been derived as sin (A + B), cos(A + B), tan(A + B). 

1. Double of Inverse Trigonometric Function Formulas

The following formulas can be used to solve the double of an inverse trigonometric function to obtain a single trigonometric function.

• 2sin-1x is equal to sin-1(2x.√(1 – x2))
• 2cos-1x is equal to cos-1(2×2 – 1)
• 2tan-1x is equal to tan-1(2x/1 – x2)

6. Triple of Inverse Trigonometric Function Formulas

As shown in the formulas below, the triple of inverse trigonometric functions can be solved to yield a single inverse trigonometric function.

• 3sin-1x is equal to sin-1(3x – 4×3)
• 3cos-1x is equal to cos-1(4×3 – 3x)
• 3tan-1x is equal to tan-1(3x – x3/1 – 3×2)

Types of Inverse Trigonometric Functions

Here are the types of inverse trigonometric functions:

1. Arcsine Function

The arcsine function, also known as the inverse sine function or sin-1 x, is the inverse of the sine function.

sin-1(-x) = -sin-1(x), x ∈ [-1, 1]

1. Arccosine Function

The arccosine function, commonly known as the inverse cosine function or cos-1 x, is the inverse of the cosine function.

cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]

1. Arctangent Function

The arctangent function, also known as the inverse tangent function or tan-1 x, is the inverse of the tan function.

tan-1(-x) = -tan-1(x), x ∈ R

1. Arccotangent Function

The arccotangent function, also known as the inverse cotangent function or cot-1 x, is the inverse of the cotangent function.

cot-1(-x) = π – cot-1(x), x ∈ R

1. Arcsecant Function

The arcsecant function, also known as the inverse secant function or sec-1 x, is the inverse of the secant function.

sec-1(-x) = π -sec-1(x), |x| ≥ 1

1. Arccosecant Function

The arccosecant function, also known as the inverse cosecant function or cosec-1 x, is the inverse of the cosecant function.

cosec-1(-x) = -cosec-1(x), |x| ≥ 1

 Tricks to Know About Inverse Trigonometric Functions

Some of the following hints will assist you in solving and using inverse trigonometric function formulas.

• sin-1(sin x) is equal to sin(sin-1x) = x, -π/2 ≤ x ≤π/2.
• sin-1x is different from (sin x)-1. Also, (sin x)-1 is equal to 1/sinx
• sin-1x is equal to θ, and θ refers to the angle, which is the principal value of this inverse trigonometric function.

Conclusion

Inverse trigonometric functions are also known as “Arc Functions” since they produce the length of arc required to acquire a given value of trigonometric functions. Inverse trigonometric functions, such as sine, cosecant, cosine, tangent, secant, and cotangent, conduct the opposite operation of trigonometric functions.