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Chapter 1 Trigonometric Functions Section 1.1 Basic Concepts Section 1.2 Angles Section 1.3 Angle Relationships Section 1.4 Definitions of Trig Functions Section 1.5 Using the Definitions Section 1.1 Basic Concepts In this section we will cover: • Labeling Quadrants • Pythagorean Theorem • Distance Formula • Midpoint Formula • Interval Notation • Relations • Functions The Coordinate Plane Horizontal Quadrant II Quadrant I (-,+) (+,+) Quadrant III Quadrant IV (-,-) (+,-) x abscissa Vertical y ordinate Pythagorean Theorem B a leg a2 + b2 = c2 C leg b A Distance Formula a = (x2 – x1) b = (y2 – y1) c = √ a2 + b2 or distance = √ (x2 – x1)2 + (y2 – y1)2 Midpoint Formula The Midpoint Formula: The midpoint of a segment with endpoints (x1 , y1) and (x2 , y2) has coordinates Interval Notation • Set-builder notation {x|x<5} the set of all x such that x is less than 5 • Interval notation (-∞, 5) the set of all x such that x is less than 5 (-∞, 5] the set … x is less than or equal to 5 the first is an open interval the second is a half-opened interval [0, 5] is an example of a closed interval Relations and Functions A relation is a set of points. A dependent variable varies based on an independent variable. For example y = 2x y is the dependent variable x is the independent variable A relation is a function if each value of the independent variable leads to exactly one value of the dependent variable. The values of the dependent variable represent the range. The values of the independent variable represent the domain. A relation is a function if a vertical line intersects its graph in no more than one point. (Vertical Line Test) Section 1.2 Angles In this section we will cover: • Basic terminology • Degree measure • Standard position • Co terminal Angles Basic Terminology • line - an infinitely-extending onedimensional figure that has no curvature • segment - the portion of a line between two points • ray - the portion of a line starting with a single point and continuing without end • angle - figure formed through rotating a ray around its endpoint Basic Terminology (cont) • • • • initial side - ray position before rotation terminal side - ray position after rotation vertex - point of rotation positive rotation - counterclockwise rotation • negative rotation - clockwise rotation • degree - 1/360th of a complete rotation Basic Terminology (cont) • acute angle - angle with a measure between 0° and 90° • right angle - angle with a measure of 90° • obtuse angle - angle with a measure between 90° and 180° • straight angle - angle with a measure of 180° • complementary - sum of 90° • supplementary - sum of 180° Basic Terminology (cont) • minute - ‘ , 1/60th of a degree • second – “ , 1/60th of a minute, 1/3600th of a degree • standard position - an angle with a vertex at the origin and initial side on the positive abscissa • quadrantal angles - angles in standard position whose terminal side lies on an axis • co terminal angles - angles having the same initial and terminal sides but different angle measures Section 1.3 Angle Relationships In this section we will cover: • Geometric Properties – Vertical angles – Parallel lines cut by a transversal • Corresponding angles • Same side interior and exterior angles • Applying triangle properties – Angle sum – Similar triangles Geometric Properties • Vertical angles are formed when two lines intersect. They are congruent which means they have equal measures. • When parallel lines are cut by a third line, called a transversal, the result is to sets of congruent angles. 1 2 3 4 6 5 7 8 Geometric Properties (cont 1 2 3 4 6 5 7 8 So here angles 1, 4, 5, and 8 are congruent and angles 2, 3, 6, and 7 are congruent. Corresponding pairs are / 1 & / 5, / 2 & / 6, / 3 & / 7, and / 4 & / 8. Triangle Properties The sum of the interior angles of a triangle equal 180°. Acute – 3 acute angles Right – 2 acute and one right angle Obtuse – 1 obtuse and two acute angles Equilateral – all sides (and angles) equal Isosceles – two equal sides (and angles) Scalene – no equal sides (or angles) Triangle Properties (cont) Corresponding parts of congruent triangles are congruent. Corresponding angles of similar triangles are congruent. Corresponding sides of similar triangles are in proportion. Section 1.4 Definitions of Trigonometric Functions In this section we will cover: • Trigonometric functions – Sine – Cosine – Tangent • Quadrantal angles –Cosecant –Secant –Cotangent Trigonometric Functions • • • • • • Sine = opposite /hypotenuse = y/r Cosine = adjacent/hypotenuse = x/r Tangent = opposite/adjacent = y/x Cosecant = hypotenuse/opposite = r/y Secant = hypotenuse/adjacent = r/x Cotangent = adjacent/opposite = x/r Special Triangles Special Trig Values sin cos tan csc sec cot 30à 1/2 ñ3/2 ñ3/3 2 45à ñ2/2 ñ2/2 1 ñ2 90à 1 0 Und 1 ñ2 60à ñ3/2 1/2 ñ3 2ñ3 3 2 2ñ3 3 ñ3 1 ñ3/3 0 Und Trigonometric Functions Values for Quadrant Angles sin cos tan csc sec cot 0à 0 1 0 90à 1 0 Undefined 180à 0 -1 0 270à -1 0 Undefined Undefined 1 Undefined -1 1 Undefined -1 Undefined Undefined 0 Undefined 0 Section 1.5 Using the Definitions of Trigonometric Functions In this section we will cover: • The reciprocal identities • Signs and ranges of function values • The Pythagorean identities • The quotient identities The Reciprocal Identities sin £ = 1 csc £ = csc £ 1 sin £ cos £ = 1 sec £= sec £ 1 cos £ tan £ = 1 cot £ =£ cot 1 tan £ Signs and Ranges of function values £ in Quadrant sin £ cos £ tan £ cot £ sec £ csc £ I + + + + + + II + - - - - + III - - + + - - IV - + - - + - All Students Take Calculus x<0 y>0 r>0 x<0 y<0 r>0 Quadrant II Quadrant I (-,+) (+,+) Sin & Csc are positive All functions are positive Tan & Cot are positive Cos & Sec are positive Quadrant III Quadrant IV (-,-) (+,-) x>0 y>0 r>0 x>0 y<0 r>0 Ranges for Trig Functions For any angle £ for which the indicated functions exist: 1. 2. 3. -1 < sin £ < 1 and -1 < cos £ < 1; tan £ and cot £ may be equal to any real number; sec £ < -1 or sec £ > 1 and csc £ < -1 or csc £ > 1 (Notice that sec £ and csc £ are never between -1 and 1.) The Pythagorean Identities Remember in a right triangle a2 + b2 = c2 or using x, y, and r r x2 + y2 = r2 y Dividing by r2 x x2 + y2 = r2 r2 r 2 r 2 or r y cos2θ + sin2θ = 1 or θ x sin2θ + cos2θ = 1 This is our first trigonometric identity Basic trigonometric identities cos2θ + sin2θ 1 cos2θ cos2θ= cos2θ or 1 + tan2θ = sec2θ r or tan2θ +1 = y sec2θ θ x Basic trigonometric identities cos2θ + sin2θ 1 = sin2θ 2 2 sin θ sin θ or cot2θ + 1 = csc2θ r or 1 + cot2θ = y csc2θ θ x The quotient Identities tan £ = sin £ = cos £ y x cot £ = cos £ sin £ x y =