- Trigonometry is a branch of Mathematics that studies triangles and the relationship between their sides and the angles.
- The Word “ TRIGONOMETRY ” is derived from the Greek words , “ TRI ” (Meaning – Three), “GON ”(Meaning – Sides), “METRON” (Meaning – Measure).
TRIGONOMETRY = TRI + GON + METRON
INTRODUCTION TO TRIGONOMETRY
- Let us consider a right triangle . Here angle A is an acute angle and the position of the side BC with respect to angle A. We will call the side Opposite to angle A. AC is the Hypotenuse of the right triangle and the side AB is the part of angle A. So, we will call it as the side Adjacent to angle A
- The Hypotenuse is the longest side, and is always opposite the right angle.
- The Opposite side is the one opposite the angle we are interested in.
- The Adjacent side is the one next to the angle we are interested in.
- This section presents the 3 basic Trigonometric Ratios Sine, Cosine and Tangent. The concept of similar Triangles and the Pythagorean Theorem can be used to develop the Trigonometry of right Triangles.
- The Trigonometric ratios of an acute angle in a right triangle ABC are defined as follows.
- Sin A = Side Opposite to angle A/Hypotenuse = BC/AC
- Cos A = Side Adjacent to angle A/ Hypotenuse = AB/AC
- Tan A = Side Opposite to angle A/Side Adjacent to angle A = BC/AB
- Since hypotenuse is the longest side in a right triangle, therefore, the value of sin A or cos A is always less than or equal to 1.
Relationship between Trigonometrical Ratios
- Cosec A = 1/Sin A = Hypotenuse/Side Opposite to angle A = AC/BC
- Sec A = 1/Cos A = Hypotenuse/Side Adjacent to angle A = AC/AB
- Cot A = 1/Tan A = Side Adjacent to angle A/Side Opposite to angle A = AB/BC
- Tan A = Sin A/Cos A = BC/AB
An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle.
For any acute angle A, we have
- Sin2A + cos2 A = 1
- 1 + tan2A = sec2A
- 1 + cot2A = cosec2A
NOTE: Trigonometry is the study of triangles. Using trigonometry we can solve many problems precisely where we may have only been able to approximate before. Trigonometry is used in everything from Engineering and Architecture to G.P.S
Question. In triangle ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Find the value of sin A and cos A.
By using the Pythagoras theorem, we have
AC2 = AB2 + BC2
AC2 = (24)2 + (7)2
AC2 = 576 + 49
AC = √625 = 25 cm.
. sin A = BC/AC = 7/25 ,
cos A = AB/AC = 24/25