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.
