Applied Maths-Trigonometric & Inverse Trigonometric Function

Trigonometric Functions

• Positive and negative angles

• Measuring angles in radians and in degrees and conversion of one into other

• Definition of trigonometric functions with the help of unit circle

• Signs of trigonometric functions

• Domain and range of trigonometric functions and their graphs

• Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application

• Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x

• General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

Inverse Trigonometric Functions

• Definition, range, domain, principal value branch

• Graphs of inverse trigonometric functions

• Elementary properties of inverse trigonometric functions

SUMMARY

Trigonometric Functions

1. If in a circle of radius r, an arc of length l subtends an angle of θ radians, then l = r θ

2. Radian measure = π 180 × Degree measure

3. Degree measure = 180 π × Radian measure

4. cos (2nπ + x) = cos x

5. sin (2nπ + x) = sin x

6. sin (– x) = – sin x

7. cos (– x) = cos x

8. cos (x + y) = cos x cos y – sin x sin y

9. cos (x – y) = cos x cos y + sin x sin y

10. cos ( π/2 − x ) = sin x

11. sin ( π/2 − x ) = cos x

12. sin (x + y) = sin x cos y + cos x sin y

13. sin (x – y) = sin x cos y – cos x sin y

14. cos (π – x) = – cos x                   sin (π – x) = sin x

     cos (π + x) = – cos x                   sin (π + x) = – sin x

     cos (2π – x) = cos x                    sin (2π – x) = – sin x

15. (i) 2cos x cos y = cos ( x + y) + cos ( x – y)       (ii) – 2sin x sin y = cos (x + y) – cos (x – y)     

     (iii) 2sin x cos y = sin (x + y) + sin (x – y)          (iv) 2 cos x sin y = sin (x + y) – sin (x – y).

16. sin x = 0 gives x = nπ, where n ∈ Z.

17. cos x = 0 gives x = (2n + 1) π/2 , where n ∈ Z.

18. cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.

19. tan x = tan y implies x = nπ + y, where n ∈ Z.

Inverse Trigonometric Functions

1. sin–1x should not be confused with (sin x) –1. In fact (sin x) –1 = 1 sin x and similarly for other trigonometric functions.

2. The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.

3. For suitable values of domain, we have

y = sin–1 x ⇒ x = sin y

x = sin y ⇒ y = sin–1 x

sin (sin–1 x) = x

sin–1 (sin x) = x

sin–1 1/x = cosec–1 x

cos–1 (–x) = π – cos–1 x

cos–1 1/x = sec–1x

cot–1 (–x) = π – cot–1 x

tan–1 1/x = cot–1 x

sec–1 (–x) = π – sec–1 x

sin–1 (–x) = – sin–1 x

tan–1 (–x) = – tan–1 x

tan–1 x + cot–1 x = π/2

cosec–1 (–x) = – cosec–1 x

sin–1 x + cos–1 x =  π/2

cosec–1 x + sec–1 x = π/2

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

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

4. sin–1 x should not be confused with (sin x) –1. In fact (sin x) –1 = 1/sin x and similarly for other trigonometric functions.

5. Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function.

6. The value of an inverse trigonometric functions which lies in the range of principal branch is called the principal value of that inverse trigonometric functions.