Page 1 INTRODUCTION TO TRIGONOMETRY Trigonometry: Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Trigonometric Ratios: The ratios of the sides of a right triangle are called trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are the three common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) are the reciprocal of the ratios sin, cos and tan respectively. These are defined for acute angle A in right angled triangle ABC below: Sine of ?A = Cosine of ?A = Tangent of ?A= Cosecant of ?A= Secant of ?A = Cotangent of ?A = Side adjacent to angle A Side adjacent to angle A C B A Side opposite to angle A Side adjacent to angle A Hypotenuse Hypotenuse Side opposite to angle A BC AC = Hypotenuse Side adjacent to angle A AB AC = Side opposite to angle A BC AB = Hypotenuse Side opposite to angle A AC BC = Hypotenuse Side adjacent to angle A AC AB = Side opposite to angle A AB BC = Page 2 INTRODUCTION TO TRIGONOMETRY Trigonometry: Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Trigonometric Ratios: The ratios of the sides of a right triangle are called trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are the three common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) are the reciprocal of the ratios sin, cos and tan respectively. These are defined for acute angle A in right angled triangle ABC below: Sine of ?A = Cosine of ?A = Tangent of ?A= Cosecant of ?A= Secant of ?A = Cotangent of ?A = Side adjacent to angle A Side adjacent to angle A C B A Side opposite to angle A Side adjacent to angle A Hypotenuse Hypotenuse Side opposite to angle A BC AC = Hypotenuse Side adjacent to angle A AB AC = Side opposite to angle A BC AB = Hypotenuse Side opposite to angle A AC BC = Hypotenuse Side adjacent to angle A AC AB = Side opposite to angle A AB BC = INTRODUCTION TO TRIGONOMETRY Also, observe that tan A = and cot A = Trigonometric ratios of some specific angles: In this we will be calculating the trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and we will also develop the relationship between them. Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then the other angle is also 45°. ? ?A = ?C = 45° also AB = BC The sides in this triangle are in the ratio 1:1: v2 Using the definition of trigonometric ratios, we have, Sin 45° = BC AC = a v2a = 1 v2 Cosec 45° = 1 Sin 45° = v2 Cos 45° = AB AC = a v2a = 1 v2 Sec 45° = 1 Cos 45° = v2 Tan 45° = BC AB = a a = 1 Cot 45° = 1 Tan 45° = 1 Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 30° then the other angle is 60°. Side opposite to 30° is always half of the hypotenuse; side opposite to 60° is always v3 times its adjacent side. ?The sides in this triangle are in the ratio 1:2: v3 Using the definition of trigonometric ratios, we have v2a a a Cos A Sin A Sin A Cos A 45° 45° C B A a v3a a 2a 30° 60° C B A Page 3 INTRODUCTION TO TRIGONOMETRY Trigonometry: Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Trigonometric Ratios: The ratios of the sides of a right triangle are called trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are the three common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) are the reciprocal of the ratios sin, cos and tan respectively. These are defined for acute angle A in right angled triangle ABC below: Sine of ?A = Cosine of ?A = Tangent of ?A= Cosecant of ?A= Secant of ?A = Cotangent of ?A = Side adjacent to angle A Side adjacent to angle A C B A Side opposite to angle A Side adjacent to angle A Hypotenuse Hypotenuse Side opposite to angle A BC AC = Hypotenuse Side adjacent to angle A AB AC = Side opposite to angle A BC AB = Hypotenuse Side opposite to angle A AC BC = Hypotenuse Side adjacent to angle A AC AB = Side opposite to angle A AB BC = INTRODUCTION TO TRIGONOMETRY Also, observe that tan A = and cot A = Trigonometric ratios of some specific angles: In this we will be calculating the trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and we will also develop the relationship between them. Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then the other angle is also 45°. ? ?A = ?C = 45° also AB = BC The sides in this triangle are in the ratio 1:1: v2 Using the definition of trigonometric ratios, we have, Sin 45° = BC AC = a v2a = 1 v2 Cosec 45° = 1 Sin 45° = v2 Cos 45° = AB AC = a v2a = 1 v2 Sec 45° = 1 Cos 45° = v2 Tan 45° = BC AB = a a = 1 Cot 45° = 1 Tan 45° = 1 Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 30° then the other angle is 60°. Side opposite to 30° is always half of the hypotenuse; side opposite to 60° is always v3 times its adjacent side. ?The sides in this triangle are in the ratio 1:2: v3 Using the definition of trigonometric ratios, we have v2a a a Cos A Sin A Sin A Cos A 45° 45° C B A a v3a a 2a 30° 60° C B A INTRODUCTION TO TRIGONOMETRY Sin 30° = BC AC = a 2a = 1 2 Cosec 30° = 1 Sin 30° = 2 Cos 30° = AB AC = v3a 2a = v3 2 Sec 30° = 1 Cos 30° = 2 v3 Tan 30° = BC AB = a v3a = 1 v3 Cot 30° = 1 Tan 30° = v3 Similarly, Sin 60° = AB AC = v3a 2a = v3 2 Cosec 60° = 1 Sin 60° = 2 v3 Cos 60° = BC AC = a 2a = 1 2 Sec 60° = 1 Cos 60° = 2 Tan 60° = AB BC = v3a a = v3 Cot 60° = 1 Tan 60° = 1 v3 Trigonometric Ratios of 0° & 90° : if angle A is made smaller and smaller angle C becomes larger and larger. When angle C becomes smaller, side BC also decreases & finally when ?A becomes very close to 0°, AC will be almost the same as AB and side BC gets very close to zero. Therefore the value of sin A is very close to 0 and cos A is very close to 1. Thus we have, Sin 0° = 0 Cosec 0° = 1 Sin 0° , which is not defined Cos 0° = 1 Sec 0° = 1 Cos 0° = 1 Tan 0° = Sin 0° Cos 0° = 0 1 = 0 Cot 60° = 1 Tan 0° , which is again not defined. Now, if angle A is made larger and larger angle C becomes smaller and smaller. Therefore the length of side AB goes on decreasing. Point A gets closer to B. Finally when ?A is very close to 90°, ?C becomes very close to 0°. Side AC almost coincides with side BC so, sin A is very close to 1 and cos A is very close to 0. C B A Page 4 INTRODUCTION TO TRIGONOMETRY Trigonometry: Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Trigonometric Ratios: The ratios of the sides of a right triangle are called trigonometric ratios. Sine (sin), cosine (cos), and tangent (tan) are the three common trigonometric ratios, cosecant (cosec), secant (sec) and Cotangent (cot) are the reciprocal of the ratios sin, cos and tan respectively. These are defined for acute angle A in right angled triangle ABC below: Sine of ?A = Cosine of ?A = Tangent of ?A= Cosecant of ?A= Secant of ?A = Cotangent of ?A = Side adjacent to angle A Side adjacent to angle A C B A Side opposite to angle A Side adjacent to angle A Hypotenuse Hypotenuse Side opposite to angle A BC AC = Hypotenuse Side adjacent to angle A AB AC = Side opposite to angle A BC AB = Hypotenuse Side opposite to angle A AC BC = Hypotenuse Side adjacent to angle A AC AB = Side opposite to angle A AB BC = INTRODUCTION TO TRIGONOMETRY Also, observe that tan A = and cot A = Trigonometric ratios of some specific angles: In this we will be calculating the trigonometric ratios value for different angles such as 0°, 30°, 45°, 60° & 90° and we will also develop the relationship between them. Trigonometric Ratios of 45°: In a right angled triangle if one angle is of 45° then the other angle is also 45°. ? ?A = ?C = 45° also AB = BC The sides in this triangle are in the ratio 1:1: v2 Using the definition of trigonometric ratios, we have, Sin 45° = BC AC = a v2a = 1 v2 Cosec 45° = 1 Sin 45° = v2 Cos 45° = AB AC = a v2a = 1 v2 Sec 45° = 1 Cos 45° = v2 Tan 45° = BC AB = a a = 1 Cot 45° = 1 Tan 45° = 1 Trigonometric Ratios of 30° & 60° : In a right angled triangle if one angle is of 30° then the other angle is 60°. Side opposite to 30° is always half of the hypotenuse; side opposite to 60° is always v3 times its adjacent side. ?The sides in this triangle are in the ratio 1:2: v3 Using the definition of trigonometric ratios, we have v2a a a Cos A Sin A Sin A Cos A 45° 45° C B A a v3a a 2a 30° 60° C B A INTRODUCTION TO TRIGONOMETRY Sin 30° = BC AC = a 2a = 1 2 Cosec 30° = 1 Sin 30° = 2 Cos 30° = AB AC = v3a 2a = v3 2 Sec 30° = 1 Cos 30° = 2 v3 Tan 30° = BC AB = a v3a = 1 v3 Cot 30° = 1 Tan 30° = v3 Similarly, Sin 60° = AB AC = v3a 2a = v3 2 Cosec 60° = 1 Sin 60° = 2 v3 Cos 60° = BC AC = a 2a = 1 2 Sec 60° = 1 Cos 60° = 2 Tan 60° = AB BC = v3a a = v3 Cot 60° = 1 Tan 60° = 1 v3 Trigonometric Ratios of 0° & 90° : if angle A is made smaller and smaller angle C becomes larger and larger. When angle C becomes smaller, side BC also decreases & finally when ?A becomes very close to 0°, AC will be almost the same as AB and side BC gets very close to zero. Therefore the value of sin A is very close to 0 and cos A is very close to 1. Thus we have, Sin 0° = 0 Cosec 0° = 1 Sin 0° , which is not defined Cos 0° = 1 Sec 0° = 1 Cos 0° = 1 Tan 0° = Sin 0° Cos 0° = 0 1 = 0 Cot 60° = 1 Tan 0° , which is again not defined. Now, if angle A is made larger and larger angle C becomes smaller and smaller. Therefore the length of side AB goes on decreasing. Point A gets closer to B. Finally when ?A is very close to 90°, ?C becomes very close to 0°. Side AC almost coincides with side BC so, sin A is very close to 1 and cos A is very close to 0. C B A INTRODUCTION TO TRIGONOMETRY So we define sin90° = 1 & cos90° = 0, similarly other trigonometric ratios can be found. Trigonometric Ratios of Complementary angles: If the sum of two angles is one right angle or 90°, then one angle is said to be complementary of the other. Thus, ?° and (90 - ?)° are complementary to each other. sin (90°- A) = cos A ; cos (90°- A) = sin A tan (90°- A) = cot A; cot (90°- A) = tan A sec (90°- A) = csc A; csc (90°- A) = sec A These relations are valid for all the values of A lying between 0° and 90 Trigonometric Identities: An equation is called an identity when it is true for all the value of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all the values of the angle(s) involved. sin 2 ???? + cos 2 ???? = 1 1 + tan 2 ? = sec -1 ???? 1 + cot 2 ? = cosec -1 ???? We obtain these identities by using Pythagoras theorem so these are also known as Pythagorean identities. ?° (90- ?)° C B ARead More

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