Relating Different Inverse Trigonometric Functions

There are six standard trigonometric functions: Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent. Now, the inverse of these functions is called Inverse Trigonometric Functions. They are also termed arcus, anti-trigonometric, or cyclometric functions, and we can denote them as sin-1x, cos-1x, tan-1x, cot-1x, cosec-1x, and sec-1x. Inverse trig functions are also known as arcsine, arccosine, arcsecant, arctangent, arccotangent, and arccosecant, keeping ‘arc’ as a prefix.

We can use Inverse trigonometric functions to acquire the angle with any trigonometric ratios. Moreover, we can convert all of them to inverse trigonometric formulas.

If sin y = x,

Then y = sin-1x, which means sin-1x is the inverse trigonometry angle.

x can have value as a whole number, decimal, fraction, or exponent.

Inverse Trigonometric Formulas

Inverse trigonometric formulas help in conveniently converting one inverse trigonometric function to another. This conversion enables us to search the function’s principal angle values or effectively perform numerous arithmetic operations across functions. 

We can structure these formulas into four sets, which are:

• Arbitrary Values

• Reciprocal and Complementary functions

• Sum and difference of functions

• Double and triple of a function

Inverse Trigonometric Function Formulas for Arbitrary Values

These formulas are applicable for all the six inverse trigonometric functions.

Here, the negative of the values is evaluated and converted to the function’s negative. For cosecant, secant, and cotangent functions, the domain’s negatives are translated as the subtraction of the function from the π value.

Below is the list of inverse trigonometric formulas for arbitrary values:

1. sin-1(-x) = -sin-1x;x ∈ [-1,1]

2. tan-1(-x) = -tan-1x; x ∈ R

3. cosec-1(-x) = -cosec-1x; x ∈ R – (-1,1)

4. cos-1(-x) = π – cos-1x,; x ∈ [-1,1]

5. sec-1(-x) = π – sec-1x; x ∈ R – (-1,1)

6. cot-1(-x) = π – cot-1x; x ∈ R

Inverse Trigonometric Function Formulas for Reciprocal and Complementary Functions

In reciprocal functions, x converts the available inverse trigonometric function into its reciprocal. It comes from the fact that sin and cosecant are reciprocal of each other, tangent and cotangent are reciprocal; similarly, cos and secant are reciprocal to each other.

1. sin-1x = cosec-11/x, x ∈ R – (-1,1)

2. cos-1x = sec-11/x, x ∈ R – (-1,1)

3. tan-1x = cot-11/x, x > 0
   tan-1x = – π + cot-1 x, x < 0

The sum of two complementary inverse trigonometric functions is a right angle, i.e. 90 degrees. Hence, complementary functions of sine-cosine, tangent-cotangent, secant-cosecant add up to π/2. 

1. sin-1x + cos-1x = π/2, x ∈ [-1,1]

2. tan-1x + cot-1x = π/2, x ∈ R

3. sec-1x + cosec-1x = π/2, x ∈ R – [-1,1]

Formulas on Sum and Difference of Inverse Trigonometric Functions

Two inverse trigonometric functions’ sum and the difference can produce a single inverse function. 

1. sin-1x + sin-1y = sin-1(x.√(1 – y2) + y√(1 – x2))

2. sin-1x – sin-1y = sin-1(x.√(1 – y2) – y√(1 – x2))

3. cos-1x + cos-1y = cos-1(xy – √(1 – x2).√(1 – y2))

4. cos-1x – cos-1y = cos-1(xy + √(1 – x2).√(1 – y2))

5. tan-1x + tan-1y = tan-1(x + y)/(1 – xy), if xy < 1

6. tan-1x + tan-1y = tan-1(x – y)/(1 + xy), if xy > – 1

Formulas on Double and Triple of a Function 

Doubles and triples of the inverse trig functions are solved in a way to make it a single trigonometric inverse function. Formulas set are as follows:

Double:

1. 2sin-1x = sin-1(2x.√(1 – x2))

2. 2cos-1x = cos-1(2x2 – 1)

3. 2tan-1x = tan-1(2x/1 – x2)

Triple:

1. 3sin-1x = sin-1(3x – 4x3)

2. 3cos-1x = cos-1(4x3 – 3x)

3. 3tan-1x = tan-1(3x – x3/1 – 3x2)

Relating different Inverse Trigonometric Functions

There is the relation between all inverse trig functions in a reciprocal form, odd-even form or identities relation. Below explained are some forms of relation with proof: 

• Relation between Arcsine and Arccosecant:

sin-11x = cosec-1(1/x) 

Proof: 

Let y = sin-1x

Then, sin y = x 

1/sin y = 1/x (Since, cosecant is the reciprocal of sine)

Cosec y = 1/x

Y = cosec-1(1/x)

Sin-1x = cosec-1(1/x)

Hence proved, sin-1x = cosec-1(1/x) 

• Relation between Arccosine and Arcsecant:

cos-1x = sec-1(1/x)

Proof: 

Let y = cos-1x 

Then, cos y = x

1/cos y = 1/x (Since, secant is the reciprocal of cosine)

Sec y = 1/x

Y = sec-1(1/x)

Hence proved, cos-1x = sec-1(1/x)(since, y = cos-1x)

• Relation between Arcsine and Arccosine:

Sin-1x + cos-1x = 2

Proof: 

Let y = sin-1x

sin y = x

2-y = cos-1x

cos-1x + y = 2

Hence proved, sin-1x + cos-1x = 2

• Relation between Arctangent and Arccotangent:

tan-1x+cot-1x = 2, x belongs to R

Proof: 

Let y = tan-1x

Tany = x

2 – y = cot-1x

Cot-1x + y = 2

Hence proved, tan-1x + cot-1x = 2

• Relation between Arccosecant and Arcsecant

sec-1x+cosec-1x = 2

Proof: 

Let y = sec-1x

secy = x

2 – y = cosec-1x 

Cosec-1x + y = 2

Hence proved, sec-1x + cosec-1x = 2

Conclusion

Just like trigonometric functions are related to each other in a specific manner, inverse trigonometric functions have a relation. However, there is a particular range of variable ‘x’ in which inverse trigonometric functions relate to each other. 

Inverse trigonometric functions help find the right-angled triangle’s unknown angles. Moreover, we can use them to find the sun’s elevation angle. Also, these inverse trigonometric functions can help find the angles of the bridges to build scale models.