The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ’metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.
The earliest known work on trigonometry was recorded in Egypt and Babylon. Early astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrically concepts.
We will learn some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle. We will restrict our discussion to acute angles only.
However, these ratios can be extended to other angles also. We will also define the trigonometric ratios for angles of measure 0° and 90°. We will calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.
Here, ∠ CAB (or, in brief, angle A) is an acute angle. Note the position of the side BC with respect to angle A. It faces ∠ A. We call it the side opposite to angle A. AC is the hypotenuse of the right triangle and the side AB is a part of ∠ A. So, we call it the side adjacent to angle A.
The trigonometric ratios of the angle A in right triangle ABC are defined as follows :
Sine of ∠ A = side opposite to angle A/hypotenuse = BC/AC
Cosine of ∠ A = side adjacent to angle A/hypotenuse = AB/AC
Tangent of ∠ A = side opposite to angle A/side adjacent to angle A = BC/AB
Cosecant of ∠ A = hypotenuse/side opposite to angle A = AC/BC
Secant of ∠ A = hypotenuse/side adjacent to angle A = AC/AB
Cotangent of ∠ A = hypotenuse/side adjacent to angle A = AB/BC
The ratios defined above are abbreviated as sin A, cos A, tan A, cosec A, sec A and cot A respectively. Note that the ratios cosec A, sec A and cot A are respectively, the reciprocals of the ratios sin A, cos A and tan A.
So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.
The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.
We all are already familiar with the construction of angles of 30°, 45°, 60° and 90°. Here we will find the values of the trigonometric ratios for these angles and, of course, for 0°.
In the trigonometric ratios of angle A, if it is made smaller and smaller in the right triangle ABC, till it becomes zero. As ∠ A gets smaller and smaller, the length of the side BC decreases.The point C gets closer to point B, and finally when ∠ A becomes very close to 0°, AC becomes almost the same as AB.
When ∠ A is very close to 0°, BC gets very close to 0 and so the value of sin A = BC/AC is very close to 0. Also, when ∠ A is very close to 0°, AC is nearly the same as AB and so the value of cos A = AB/AC is very close to 1.
|
∠ A |
0° |
30° |
45° |
60° |
90° |
|
sin A |
0 |
1/2 |
1/√2 |
√3/2 |
1 |
|
cos A |
1 |
√3/2 |
1/√2 |
1/2 |
0 |
|
tan A |
0 |
1/√3 |
1 |
√3 |
Not defined |
|
cosec A |
Not defined |
2 |
√2 |
2/√3 |
1 |
|
sec A |
1 |
2/√3 |
√2 |
2 |
Not defined |
|
cot A |
Not defined |
√3 |
1 |
1/√3 |
0 |
We learnt that the two angles are said to be complementary if their sum equals 90°.
So, sin (90° − A) = cos A,
cos (90° − A) = sin A, tan (90° − A) = cot A,
cot (90° − A) = tan A,
sec (90° − A) = cosec A,
cosec (90° − A) = sec A,
For all values of angle A lying between 0° and 90°. Check whether this holds for A = 0° or A = 90°.
Note : tan 0° = 0 = cot 90°, sec 0° = 1 = cosec 90° and sec 90°, cosec 0°, tan 90° and cot 0° are not defined.
An equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved.
(cos A)2 + (sin A)2 = 1
i.e., Cos2 A + sin2 A = 1
1 + Tan2 A = sec2 A
Cot2 A + 1 = cosec2 A
Sec2 A − tan2 A = 1 for 0° ≤ A < 90°, cosec2 A = 1 + cot2 A for 0° ∠ A ≤ 90°.