Chapter 5 Trigonometric Functions

Transcription

1 Chapte 5 Tignmetic Functins Sectin 5.2 Tignmetic Functins 5-5. Angles Basic Teminlgy Degee Measue Standad Psitin Cteminal Angles Key Tems: vetex f an angle, initial side, teminal side, psitive angle, negative angle, quadantal angle Basic Teminlgy A cunteclckwise tatin geneates a measue, and a clckwise tatin geneates a measue. Degee Measue EXAMPLE Finding the Cmplement and the Supplement f an Angle F an angle measuing 40, find the measue f (a) its cmplement and (b) its supplement. EXAMPLE 2 Finding Measues f Cmplementay and Supplementay Angles Find the measue f each maked angle in the figue.

2 5-2 Chapte 5 Tignmetic Functins Standad Psitin An angle is in if its vetex is at the igin and its initial side lies n the psitive x-axis. The figue shws anges f angle measues f each quadant when Quadantal Angles Angles in standad psitin whse teminal sides lie n the, such as angles with measues 90, 80, 270,and s n, ae quadantal angles. Cteminal Angles Angles with measues 60 and 420 have the same initial side and the same teminal side, but diffeent amunts f tatin. Such angles ae. Thei measues diffe by a multiple f. EXAMPLE 5 Finding Measues f Cteminal Angles Find the angles f least psitive measue that ae cteminal with each angle. (a) 908 (b) 75 (c) 800 EXAMPLE 6 Analyzing the Revlutins f a CD Playe CD Playes always spin at the same speed. Suppse a playe makes 480 evlutins pe min. Thugh hw many degees will a pint n the edge f a CD mve in 2 sec?

3 Sectin 5.2 Tignmetic Functins Tignmetic Functins Tignmetic Functins Quadantal Angles Recipcal Identities Signs and Ranges f Functin Values Pythagean Identities Qutient Identities Key Tems: sine (sin), csine (cs), tangent (tan), ctangent (ct), secant (sec), csecant (csc) Tignmetic Functins Let ( xybe, ) a pint the than the igin n the teminal side f an angle in standad psitin. The 2 distance fm the pint t the igin is x y 2. The six tignmetic functins f ae defined as fllws. y x sin cs tan y ( x 0) x csc ( y 0) sec ( x 0) ct x ( y 0) y x y Recipcal Identities F all angles f which bth functins ae defined, the fllwing identities hld. sin cs tan csc sec ct csc sec ct sin cs tan EXAMPLES Finding Functin Values f an Angle The teminal side f an angle in standad psitin passes thugh the pint given. Find the values f the six tignmetic functins f angle. Example ) thugh the pint (8, 5). Example 2) thugh the pint (, 4).

4 5-4 Chapte 5 Tignmetic Functins EXAMPLE Finding Functin Values f an Angle Find the six tignmetic functins f the angle in standad psitin, if the teminal side f is defined by x 2y 0, x 0. EXAMPLE 4 Finding Functin Values f Quadantal Angles Find the values f the six tignmetic functins f each angle. (a) an angle f 90 (b) an angle in standad psitin with teminal side thugh (, 0) Cnditins f Undefined Functin Values Identify the teminal side f a quadantal angle. If the teminal side f the quadantal angle lies alng the y-axis, then the tangent and secant functins ae undefined. If the teminal side f the quadantal angle lies alng the x-axis, then the ctangent and csecant functins ae undefined. The values given in this table can be fund with a calculat that has tignmetic functin keys. Make sue the calculat is set in degee mde. One f the mst cmmn es invlving calculats in tignmety ccus when the calculat is set f adian measue, athe than degee measue. Which w in the table gives the tignmetic functin values f an angle measuing 270? Measuing 50?

5 Sectin 5.2 Tignmetic Functins 5-5 EXAMPLE 5 Using the Recipcal Identities Find each functin value. (a) cs, given that 5 sec (b) sin, given that csc 2 2 Signs f Functin Values θ in Quadant sin θ cs θ tan θ ct θ sec θ csc θ I II III IV EXAMPLE 6 Detemining Signs f Functins f Nnquadantal Angles Detemine the signs f the tignmetic functins f an angle in standad psitin with the given measue. (a) 87 (b) 00 (c) 200 EXAMPLE 7 Identifying the Quadant f an Angle Identify the quadant(s) f an angle that satisfies the given cnditins. (a) sin 0, tan 0 (b) cs 0, sec 0

6 5-6 Chapte 5 Tignmetic Functins Tignmetic Functin f Range (Inteval Ntatin) sin, cs [, ] tan, ct (, ) sec, csc (, ] [, ) EXAMPLE 8 Deciding Whethe a Value Is in the Range f a Tignmetic Functin Decide whethe each statement is pssible impssible. (a) sin 2.5 (b) tan 0.47 (c) sec 0.6 EXAMPLE 9 Finding All Functin Values Given One Value and the Quadant 2 Suppse that angle is in quadant II and sin. Find the values f the the five tignmetic functins. Pythagean Identities F all angles f which the functin values ae defined, the fllwing identities hld. 2 2 sin cs 2 2 tan sec 2 2 ct csc Qutient Identities F all angles f which the denminats ae nt ze, the fllwing identities hld. sin cs tan cs sin ct EXAMPLES: Find Functin Values Ex 0) Find sin and tan, given that cs and sin 0. 4 Ex ) Find sin and cs, given that 4 tan and is in quadant III.