Trigonometry-PPT.pptx

Plane Trigonometry Trigonometry Came from the Greek word “trigonon” which means triangle and “metria” which means measurement. It is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. Hipparchus is rec...

Plane Trigonometry Trigonometry Came from the Greek word “trigonon” which means triangle and “metria” which means measurement. It is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. Hipparchus is recognized as the Father of Trigonometry. ANGLE Angle is the space between two intersecting lines and the point of intersection of this two lines is called vertex. Units: 1 rev = 360 degrees = 2π radians = 400 grads = 400 gons = 6400 mils Name given to angles and its equivalent Names Angle Equivalent Oblique Angles – angles Zero (Null) θ = 0° which are non-right and Acute 0 < θ < 90° non-straights. Right θ = 90° Obtuse 90° < θ < 180° Vertical Angles – opposite Straight θ = 180° angles formed between Reflex 180° < θ c b + c > a c + a > b A + B + C = 180° Right Triangle Pythagorean Theorem Trigonometric Functions Trigonometric Identities Reciprocal Identities Pythagorean Identities Trigonometric Identities Sum & Differences Double Angle Identities Oblique Triangles Cosine Law Sine Law Area of Triangle Given base and height Triangle Circumscribing a c Circle a r b Given 2 sides and included angle Triangle Inscribed in a 𝜃 Circle Given 3 sides, Heron’s Formula Triangle with Escribed Circle c a r , semi-perimeter b Theorems on Circle Arc and central angle: Angles subtended by the same arc: Inscribed angle and central angle: Intersecting tangent and chord: Theorems on Circle Intersecting chords: Intersecting tangent and secant: Intersecting secants: Quadrantal Trigonometric Functions ACT S+ Other Trigonometric Formulas co Spherical Trigonometry Spherical Trigonometry It is the branch of mathematics which focuses on the measurement of triangles on the spheres. It is principally used in navigation and astronomy Comparison between plane triangle and spherical triangle Sides (a, b, c) Sides (a, b, c) Linear measurements Angular measurements Sum of interior angle Sum of interior angle A + B + C = 180° 180 < A + B + C < 540° Basic Formula Area of Spherical Triangle, Volume of Spherical Triangle, Where: Spherical Excess, Spherical Defect, Spherical Triangles Right Spherical Triangle Oblique Spherical Triangle Triangle on the sphere Triangle on the sphere having atleast one interior which has no right angle angle equal to 90 (90 degrees) on its degrees. included angles. Solutions to Right Spherical Triangle Rule 1: The sine of any middle part is equal to the product of the tangents of the adjacent parts. Sin-Tan-Ad Rule 2: The sine of any middle part is equal to the product of the cosine of the opposite parts. Sin-Cos-Op Solutions for Oblique Spherical Triangle Sine Law: Cosine Law for Sides: S.P.A.N. Sides Positive Angles Negative Cosine Law for Angles: Terrestrial Sphere In spherical trigonometry, earth is assumed to be a perfect sphere. Units: 1 minute = 1 nautical mile 1 nautical mile = 6080 feet 1 statute mile = 5280 feet 1 knot = 1 nautical mile rearth = 6400 km = 3959 miles per hour 1° = 60 nautical miles Terrestrial Sphere Latitudes are small circles parallel to the equator. These will serve as angular elevation above and below the equator. Longitudes or meridians are semicircles that run from the north and south poles and used to locate how far east or west from Greenwich, England. Prime Meridian is the semicircle running from the north to south pole through Greenwich, England. Opposite to it is the International Date Line (IDL) Terrestrial Sphere Bearings Azimuths Measurements from the Clockwise angles usually north or south, clockwise measured from a meridian or counterclockwise. It is line thus azimuths used quadrantal, therefore it either north or south as does not exceed 90°. their reference.