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Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions : In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

There are several notations used for the inverse trigonometric functions.

The most common convention is to name inverse trigonometric functions using an arc- prefix, e.g., arcsin(x)arccos(x)arctan(x), etc. This convention is used throughout the article. When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, “the arc whose cosine is x” is the same as “the angle whose cosine is x”, because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. Similarly, in computer programming languages the inverse trigonometric functions are usually called asin, acos, atan.

The notations sin−1(x)cos−1(x)tan−1(x), etc., as introduced by John Herschel in 1813, are often used as well, but this convention logically conflicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, (cos(x))−1 = sec(x). Nevertheless, certain authors advise against using it for its ambiguity.

Another convention used by a few authors is to use a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1(x)Cos−1(x)Tan−1(x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin−1(x)cos−1(x), etc.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

In Calculus, a function is called a one-to-one function if it never takes on the same value twice; that is f(x1)~= f(x2) whenever x1~=x2. • Following that, if f is a one-to-one function with domain A and range B. Then its inverse function f-1 has domain B and range A and is defined by f^(-1)y=x => f(x)=y.

The domains of the trigonometric functions are restricted so that they become one-to-one and their inverse can be determined. • Since the definition of an inverse function says that f -1 (x)=y => f(y)=x.

Maths is a subject where it is possible to easily score 100% marks. In addition to providing a solid foundation for further studies, studying mathematics will help boost your overall percentage. Here are some Tips and Tricks to help you succeed.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

SYLLABUS:

 1. Relations and Functions: 10 marks
– Relations and Functions: 5 marks
– Inverse Trigonometric Functions: 5 marks

2. Algebra: 13 marks
– Matrices: 8 marks
– Determinants: 5 marks

3. Calculus: 44 marks
– Continuity and Differentiability: 8 marks
– Applications of Derivatives: 10 marks
– Integration: 12 marks
– Applications of Integrals: 6 marks
– Differential Equations: 8 marks

4. Vectors and Three-dimensional Geometry: 17 marks
– Vectors: 6 marks
– Three dimensional Geometry: 11 marks

5. Linear Programming: 6 marks

6. Probability: 10 marks

Total: 100 marks

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

TIPS:
–    Keep a separate list of important concepts and formulae to revise as often as possible.
–    Master the applications of the formulae.
–     If there are any doubts clear them with your teacher as soon as possible;don’t leave clearing of doubts till the last day.
–    Study basic concepts from NCERT books and practice both examples and questions given in NCERT.
–    Give extra attention to topics with higher marks weight age.
–    Practice long answer questions.
–    Practice questions from previous year papers, sample papers and model papers within the time frame you will have at the final exam.
–    Create a study schedule, focusing on your weak areas but still giving time to brush up on already completed topics.
–    Write out answers to get into the habit of answering questions in the required format.
–    Make an effort to write answers neatly, and pay attention to details.
–    Try to practice a variety of questions from different topics to gain speed.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

Maths plays an important role in many fields; Physics, Accounts, Commerce and in professional courses such as Engineering. Studying Maths also helps improve logical and analytical thinking. The key to success is practice, practice and practice.

We have studied that the inverse of a function f, denoted by f –1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, we studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist.

We shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

The inverse trigonometric functions play an important role in calculus for they serve to define many integrals. The concepts of inverse trigonometric functions is also used in science and engineering.

We have studied trigonometric functions, which are defined as follows:

  • sine function, i.e., sin : R → [– 1, 1]
  • cosine function, i.e., cos : R → [– 1, 1]
  • tangent function, i.e., tan : R – { x : x = (2n + 1) 2 π , n ∈ Z} → R
  • cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R
  • secant function, i.e., sec : R – { x : x = (2n + 1) 2 π , n ∈ Z} → R – (– 1, 1)
  • cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

We have also learnt that if f : X→Y such that f(x) = y is one-one and onto, then we can define a unique function g : Y→X such that g (y) = x, where x ∈ X and y = f (x), y ∈ Y.

  • Here, the domain of g = range of f and the range of g = domain of f.
  • The function g is called the inverse of f and is denoted by f –1 .
  • Further, g is also one-one and onto and inverse of g is f.
  • Thus, g –1 = (f –1) –1 = f.
  • We also have (f –1 o f ) (x) = f –1 (f (x)) = f –1 (y) = x and (f o f –1) (y) = f (f –1(y)) = f (x) = y.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.

sin–1  [–1, 1] [-π/2, π/2]
cos –1 [–1, 1] [0, π]
cosec–1 R – (–1,1) [-π/2, π/2] – {0}
 sec –1 R – (–1,1) [0, π] – {π/2}
 tan–1  R [-π/2, π/2]
 cot–1 R [0, π]

Properties of Inverse Trigonometric Functions

In this section, we shall prove some important properties of inverse trigonometric functions. It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined. Some results may not be valid for all values of the domains of inverse trigonometric functions.

In fact, they will be valid only for some values of x for which inverse trigonometric functions are defined. We will not go into the details of these values of x in the domain as this discussion goes beyond the scope of this text book.

Let us recall that if y = sin–1 x, then x = sin y and if x = sin y, then y = sin–1  x.

This is equivalent to sin (sin–1  x) = x, x ∈ [– 1, 1] and sin–1  (sin x) = x, x ∈  [-π/2, π/2]

Same is true for other five inverse trigonometric functions as well.

We now prove some properties of inverse trigonometric functions.

1. (i) sin–1  1/ x = cosec–1 x, x ≥ 1 or x ≤ – 1

(ii) cos–1 1 /x = sec –1x, x ≥ 1 or x ≤ – 1

(iii) tan–1 1 /x = cot–1 x, x > 0

To prove the first result, we put cosec–1 x = y, i.e., x = cosec y

Therefore 1 x = sin y

Hence sin–1  1/ x = y or sin–1  1/ x = cosec–1 x

Similarly, we can prove the other parts.

2. (i) sin–1  (–x) = – sin–1  x, x ∈ [– 1, 1]

(ii) tan–1 (–x) = – tan–1 x, x ∈ R

(iii) cosec–1 (–x) = – cosec–1 x, | x | ≥ 1

Let sin–1 (–x) = y, i.e., –x = sin y so that x = – sin y, i.e., x = sin (–y).

Hence sin–1  x = – y = –sin–1  (–x)

Therefore sin–1 (–x) = – sin–1 x

Similarly, we can prove the other parts.

3. (i) cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1]

(ii) sec–1 (–x) = π – sec–1 x, | x | ≥ 1

(iii) cot–1 (–x) = π – cot–1 x, x ∈ R

Let cos–1 (–x) = y i.e., – x = cos y so that x = – cos y = cos (π – y)

Therefore cos–1 x = π – y = π – cos–1 (–x)

Hence cos–1 (–x) = π – cos–1 x

Similarly, we can prove the other parts.

4. (i) sin–1  x + cos–1 x = 2 π , x ∈ [– 1, 1]

(ii) tan–1 x + cot–1 x = 2 π , x ∈ R

(iii) cosec–1 x + sec–1 x = 2 π , | x | ≥ 1

Let sin–1  x = y.

Then x = sin y = cos [π/2 – y ]  

Therefore cos–1 x = π /2 − y = π /2 – sin–1 x

Hence sin–1 x + cos–1 x = π/2

Similarly, we can prove the other parts.

5. (i) tan–1 x + tan–1 y = tan–1 {( x + y)/ 1 – xy} , xy < 1

(ii) ttan–1 x – tan–1 y = tan–1{ x – y/ 1 + xy} , xy > – 1

Let tan–1 x = θ and tan–1 y = φ.

Then x = tan θ, y = tan φ

Now tan( θ+φ) = (tanθ + tanφ) /( 1- tanθ tanφ) = (x + y)/(1 − xy)

This gives θ + φ = tan–1 [ (x+y) / ( 1-xy)]

Hence tan–1 x + tan–1 y = tan–1 [ (x+y) / ( 1-xy)]

In the above result, if we replace y by – y, we get the second result and by replacing y by x, we get the third result as given below.

6. (i) 2tan–1x = sin–1 2x/ (1 + x2 ), | x | ≤ 1

(ii) 2tan–1 x = cos–1 (1 – x2 )/ (1 + x2 ) , x ≥ 0

(iii) 2tan–1x = tan–1 2x/ (1 – x2 ) , – 1 < x < 1

Let tan–1 x = y, then x = tan y.

Now  sin–1 2x/ (1 + x2 ) =  sin–1   2 tan y / (1 + tan2 y)  = sin–1 (sin 2y) = 2y = 2tan–1 x

Also cos–1 (1 – x2 )/ (1 + x2 ) = cos–1  (1 – tan2 y)/(1 + tan2 y) =  cos–1  (cos 2y) = 2y = 2tan–1 x

(iii) Can be worked out similarly.

Karnataka Class 12 Commerce Maths Inverse Trigonometric Functions

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