Chapter 8. Analytic Trigonometry. 8.1 Trigonometric Identities

Transcription

1 Chapter 8. Analytic Trigonometry 8.1 Trigonometric Identities Fundamental Identities Reciprocal Identities: 1 csc = sin sec = 1 cos cot = 1 tan tan = 1 cot tan = sin cos cot = cos sin Pythagorean Identities: cos 2 + sin 2 = tan 2 = sec cot 2 = csc 2 Even-Odd Identities: cos( ) = cos, sin( ) = sin, csc( ) = csc, sec( ) = sec(), tan( ) = tan(), cot( ) = cot(). Cofunction Identities: π sin u = cos 2 u π tan u = cot 2 u π sec u = csc u 2 π cos u = sin 2 u π cot u = tan 2 u π csc u = sec u 2

2 Proving Trigonometric Identities Eample: [#46] sin 4 cos 4 = sin 2 cos 2 sec Eample: [#61] = sec (sec + tan ) sec tan Eample: [#53] tan 2 sin 2 = tan 2 sin 2 Eample: [#64] sin cot 1 cos = csc Eample: [#79] (tan + cot ) 2 = sec 2 + csc 2

3 8.2 Addition and Subtraction Formulas Addition / Subtraction formulas sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β tanα + tan β tan( α + β ) = 1 tanα tan β tanα tanβ tan( α β ) = 1+ tanα tanβ Eample: Evaluate sin θ, cos θ, and tan θ, for θ = 5π / 12. Eample: Evaluate sin(80º) cos(20º) cos(80º) sin(20º)

4 8.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-angle formulas sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2sin 2 tan 2 2tan 1 tan = 2 Half-angle formulas (simple versions, a.k.a. Power Reducing Formulas) sin 2 = (1 cos 2 ) / 2 cos 2 = (1 + cos 2 ) / 2 1 cos 2 tan 2 = 1+ cos 2 Half-angle formulas u sin 2 = ± 1 cos u 2 u 1+ cos u cos = ± 2 2 (The sign is determined by the quadrant in which u / 2 lies.) u 1 cos u tan = 2 sin u sin u = 1+ cos u

5 Product-to-Sum Formulas sin(α) sin(β) = [cos(α β) cos(α + β)] / 2 sin(α) cos(β) = [sin(α β) + sin(α + β)] / 2 cos(α) cos(β) = [cos(α β) + cos(α + β)] / 2 Sum-to-Product Formulas α + β α β sinα + sinβ = 2sin cos 2 2 α + β α β sinα sinβ = 2 cos sin 2 2 α + β α β cosα + cosβ = 2 cos cos 2 2 α + β α β cosα cosβ = 2sin sin 2 2

6 8.4 Inverse Trigonometric Functions While trigonometric functions are not one-to-one in their respective nature domain (by definition, no periodic function, in general, is one-to-one), consequently they don t have inverse functions for their native domains. However, if we restrict their domains (to very small intervals) then it is possible to define inverse trigonometric functions. The Inverse Sine Function The inverse sine function, sin 1, has domain [ 1, 1] and range [ π / 2, π / 2]. It is defined by sin 1 = y <=> sin y = The inverse sine function is also called arcsine, denoted by arcsin. sin(sin 1 ) = for 1 1 sin 1 (sin ) = for π / 2 π / 2 Graph of y = arcsin

7 The Inverse Cosine Function The inverse cosine function, cos 1, has domain [ 1, 1] and range [0, π]. It is defined by cos 1 = y <=> cos y = The inverse cosine function is also called arccosine, denoted by arccos. cos(cos 1 ) = for 1 1 cos 1 (cos ) = for 0 π Graph of y = arccos Eample: cos(cos 1 1) = 1, but cos 1 (cos 5π / 4) = 3π / 4 (Why?)

8 The Inverse Tangent Function The inverse tangent function, tan 1, has domain (, ) and range ( π / 2, π / 2). It is defined by tan 1 = y <=> tan y = The inverse tangent function is also called arctangent, denoted by arctan. tan(tan 1 ) = for all real numbers tan 1 (tan ) = for π / 2 < < π / 2 Graph of y = arctan Note the twin horizontal asymptotes y = ± π / 2. It is the first eample we have seen of a function with 2 horizontal asymptotes.

9 The Inverse Secant Function The inverse secant function, sec 1, has domain 1 and range [0, π / 2) and (π / 2, π]. It is defined by sec 1 = y <=> sec y = The inverse secant function is also called arcsecant, denoted by arcsec. sec(sec 1 ) = for all, 1 sce 1 (sec ) = for 0 < π / 2, or π / 2 < π Note: The choice of intervals for the range of sec 1, and as well for csc 1, is not universally agreed upon. Your tetbook, for eample, use [0, π / 2) and [π, 3π / 2) as the range of sec 1. Inverse cotangent and cosecant functions are defined similarly, although they are not used nearly as commonly as the others. Graph of y = arcsec There is a horizontal asymptote y = π / 2.

10 Eample: Evaluate (a) cos 1 ( 1) = π (b) tan 1 (1) = π / 4 (c) sin(cos 1 (1 / 2)) = 3 / 2 (d) cos(tan 1 (2 / 5)) = (e) sec(sin 1 (1 / 3)) = (f) cos(cot 1 ( 4 / 7)) = (g) csc(tan 1 ( 2 / 3)) =

11 8.5 Trigonometric Equations Eample: 2sin Eample: 4cos 2 = 3 Eample: 4sin(3) cos(3) = 1 Eample: 2cos 2 + 7cos 4 = 0 Eample: 2cos 3 3 2cos = 3sin 2

12 Eample: [#56] tan 4 13tan = 0 Eample: [#22] 3tan 3 = tan Eample: [#34] sec = 2 cos 2 Find all solution of the given equation on [0, 2π). Eample: [#61] cos cos 3 sin sin 3 = 0 Eample: [#65] sin 2 + cos = 0 Eample: [#54] 2sin 2 cos = 0