Introduction to Trigonometry PDF

Introduction to Trigonometry PREPARED BY: ENGR. CRISTEN KATE T. CELESTIAL, REE Introduction Trigonometry is the study of triangles by applying the relations between the sides and the angles. The term "trigonometry" comes from the Greek words trigonon which means "triangle" and metria m...

Introduction to Trigonometry PREPARED BY: ENGR. CRISTEN KATE T. CELESTIAL, REE Introduction Trigonometry is the study of triangles by applying the relations between the sides and the angles. The term "trigonometry" comes from the Greek words trigonon which means "triangle" and metria meaning "measurements. Trigonometry is divided into two branches, namely: 1. Plane Trigonometry deals with triangles in the two dimensions of the plane. 2. Spherical Trigonometry concerns with triangles extracted from the surface of a sphere. Conversion Using the basic relationship 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180° Degrees to Radians 𝜋 𝑟𝑎𝑑 ____° × 180° Radians to Degrees 180° _____𝑟𝑎𝑑 × 𝜋 𝑟𝑎𝑑 Classifications of Triangles According to their Sides: Equilateral triangle Isosceles triangle Scalene triangle – a triangle with all sides – a triangle with two – a triangle no two are equal, and all sides equal and sides are equal. angles are equal. An corresponding two equilateral triangle is angles are also equal. also equiangular. Each interior angle is 60°. Classifications of Triangles According to their Angles 1. Right triangle - a triangle with one angle is a right angle. 2. Acute triangle - a triangle with each interior angle is less than a right angle. 3. Obtuse triangle - a triangle with one angle is greater than right angle. Note: A triangle which is not a right triangle is known as oblique triangle. Both acute triangle and obtuse triangle are oblique triangles. Other Types of Triangles 1. Egyptian triangle - a triangle with sides 3, 4, 5 units. 2 Pedal triangle - a triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from some point inside a given triangle. 3. Golden triangle - an isosceles triangle with sides is to its base in the golden ratio; its angles are 72°, 72°, 36°. 4. 45°- 45° Right Triangle 5. 30°- 60° Right Triangle Example Problems: 1. The longer side of a 30° 60° 90° triangle is 4√3 units. Find the length of its hypotenuse and the other side. 2. If the hypotenuse of a 45° 45° 90° triangle is 3√2 units, what is the length of its other two legs. 3. Find the long leg and the hypotenuse of a 30° 60° 90° triangle if its short leg is equal to 5√3 units.  The Pythagorean Theorem Pythagorean theorem is the most renowned of all mathematical theorems. It is considered as the most proved theorem in mathematics. This was formulated by Pythagoras about 500 B.C. Pythagorean theorem states that, "In a right triangle, the sum of the squares of the sides is equal to the square of the hypotenuse“. Example Problems: 1. If the sides of a right triangles are a, b, and c, such that a = 13 cm, b = 14 cm and c is the hypotenuse. Find the value of c. 2. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs, if one leg is 14 cm longer than the other. Find the value of each of the six trigonometric functions of 𝜃 𝑠𝑖𝑛𝜃= 𝑐𝑜𝑠𝜃= 𝑡𝑎𝑛𝜃= 𝑐𝑠𝑐𝜃= 𝑠𝑒𝑐𝜃= 𝑐𝑜𝑡𝜃= Fundamental Identities Pythagorean Identities Reciprocal Identities 1 1 sin2 𝜃 + cos 2 𝜃 = 1 sin 𝜃 = csc 𝜃 = csc 𝜃 sin 𝜃 1 1 1 + tan2 𝜃 = sec 2 𝜃 cos 𝜃 = sec 𝜃 = sec 𝜃 cos 𝜃 1 1 1 + cot 2 𝜃 = csc 2 𝜃 tan 𝜃 = cot 𝜃 = cot 𝜃 tan 𝜃 Even-odd Identities Quotient Identities sin −𝑥 = − sin 𝑥 csc −𝑥 = − csc 𝑥 sin 𝜃 cos 𝜃 tan 𝜃 = c𝑜𝑡 𝜃 = cos −𝑥 = cos 𝑥 sec −𝑥 = sec 𝑥 c𝑜𝑠 𝜃 sin 𝜃 tan −𝑥 = − tan(𝑥) cot(−𝑥) = − cot 𝑥  VERIFYING TRIGONOMETRIC IDENTITIES 1. sec 𝑥 cot 𝑥 = csc 𝑥 2. sin 𝑥 tan 𝑥 + cos 𝑥 = sec 𝑥 3. cos 𝑥 − cos 𝑥 sin2 𝑥 = cos 3 𝑥 1+sin 𝜃 4. = sec 𝜃 + tan 𝜃 cos 𝜃  Coterminal Angles Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles. Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results in a coterminal angle. Thus, an angle of 𝜃° is coterminal with angles of 𝜃° ± 360°𝑘, where k is an integer. Increasing or decreasing the radian measure of an angle by an integer multiple of 2𝜋 results in a coterminal angle. Thus, an angle of u radians is coterminal with angles of 𝜃 ± 2𝜋𝑘, where k is an integer. Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following: 1. 420° 2. −120° Find a positive angle less than 2𝜋 that is coterminal with each of the following: 17 𝜋 1. 𝜋 2. − 6 12