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 Chapter 1 Trigonometric Functions

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 x abscissa Vertical y ordinate Quadrant I (+,+) Quadrant II (-,+) Quadrant III (-,-) Quadrant IV (+,-)

Pythagorean Theorem a 2 + b 2 = c 2 hypotenuse c leg a leg b C B A

Distance Formula a = (x 2 – x 1 ) b = (y 2 – y 1 ) c = √ a 2 + b 2 or distance = √ (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2

Midpoint Formula The Midpoint Formula: The midpoint of a segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) 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 one- dimensional 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 - 1/360 th 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 - ‘, 1/60 th of a degree second – “, 1/60 th of a minute, 1/3600 th 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

Geometric Properties (cont 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 & /

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 30à45à60à90à sin1/2ñ2/2ñ3/21 cosñ3/2ñ2/21/20 tanñ3/31ñ3Und csc2ñ22ñ3 3 1 sec2ñ3 3 ñ22Und cotñ31ñ3/30

Trigonometric Functions Values for Quadrant Angles 0à0à90à180à270à sin010 cos100 tan0 Undefined 0 csc Undefined 1 sec1 Undefined Undefined cot Undefined 0 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 £ = csc £ = cos £ = sec £ = tan £ = cot £ = 1 csc £ 1 sec £ 1 cot £ 1 sin £ 1 cos £ 1 tan £

Signs and Ranges of function values £ in Quadrant sin £cos £tan £ cot £ sec £ csc £ I II III IV-+--+-

All Students Take Calculus Quadrant I (+,+) Quadrant II (-,+) Quadrant III (-,-) Quadrant IV (+,-) All functions are positive Sin & Csc are positive Tan & Cot are positive Cos & Sec are positive x>0 y>0 r>0 x<0 y>0 r>0 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: < sin £ < 1 and -1 < cos £ < 1; 2. tan £ and cot £ may be equal to any real number; 3. sec £ 1 and csc £ 1 (Notice that sec £ and csc £ are never between -1 and 1.)

The Pythagorean Identities Remember in a right triangle a 2 + b 2 = c 2 or using x, y, and r x 2 + y 2 = r 2 Dividing by r 2 x y r

x 2 + y 2 = r 2 or cos 2 θ + sin 2 θ = 1 or sin 2 θ + cos 2 θ = 1 r2r2 r2r2 r2r2 x y r θθ This is our first trigonometric identity

cos 2 θ + sin 2 θ 1 or 1 + tan 2 θ = sec 2 θ or tan 2 θ + 1 = sec 2 θ x y r θθ Basic trigonometric identities cos 2 θ =

cos 2 θ + sin 2 θ 1 or cot 2 θ + 1 = csc 2 θ or 1 + cot 2 θ = csc 2 θ x y r θθ Basic trigonometric identities sin 2 θ =

The quotient Identities tan £ = = cot £ = = sin £ cos £ sin £ yxyx xyxy