Trigonometrical Ratios
1. Trigonometry is the study of relationship between the sides and the angles of the triangle.
2. The word trigonometry is derived from the Greek words ‘tri’ meaning three, ‘gon’ meaning sides and
‘metron’ meaning measure.
3. Angle measured in anticlockwise direction is taken as positive angle.
4. Angle measured in clockwise direction is taken as negative angle.
5. Ratio of the sides of the right triangle with respect to the acute angles is called trigonometric ratios
of the angle.
6. Trigonometric ratios of acute angle A in right triangle ABC:
side opposite to A p
i. sin A
hypotenuse h
side adjacent to A b
ii. cos A
hypotenuse h
side opposite to A p
iii. tan A
side adjacent to A b
hypotenuse h
iv. cos ecA
sideopposite to A p
hypotenuse h
v. s ecA
side adjacent to A b
side adjacent to A b
vi. co t A
side opposite to A p
7. Each trigonometric ratio is a real number. It has no unit.
8. Only symbols cosine, sine, tangent, cotangent, sec and cosec have no meaning.
sin
is generally written as sinn , n being a positive integer. Similarly, other trigonometric ratios
n
9.
can also be written.
10. The values of the trigonometric ratios of an angle do not vary with the length of the sides of the
triangle, if the angles remain the same.
11. Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square
of the other two sides.
12. When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras
theorem.
13. Relation between trigonometric ratios:
sin
i. tan
cos
1
ii. cos ec
sin
1
iii. s ec
co s
1 co s
iv. cot
tan sin
14. Values of Trigonometric ratios of some specific angles:
A 0 30 45 60 90
sin A 0 1 1 3 1
2 2 2
cos A 1 3 1 1 0
2 2 2
tan A 0 1 1 3 Not defined
3
cosec A Not defined 2 2 2 1
3
sec A 1 2 2 2 Not defined
3
cot A Not defined 3 1 1 0
3
15. The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater
than 1 or equal to 1.
16. The value of sin increases from 0 to 1 when increases from 00 to 900.
17. The value of cos decreases from 1 to 0 when increases from 00 to 900.
18. Trigonometric ratios of complementary angles:
i. sin (90o – A) = cos A
ii. cos (90o – A) = sin A
iii. tan (90o – A) = cot A
iv. cot (90o – A) = tan A
v. sec (90o – A) = cosec A
vi. cosec (90o – A) = sec A
19. An equation involving trigonometric ratios of an angle, say , is termed as a trigonometric identity if
it is satisfied by all values of .
20. Basic trigonometric identities:
i. sin2 co s2 1
ii. 1 tan2 sec 2
iii. 1 cot 2 sec 2
