5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric identities invlving sums r differences f tw variables Since 75 30 + 5, yu might ask whether cs 75 can be fund by calculating cs (30 + cs (5 The answer is n; cs 75 < 1, whereas 3 3 + cs (30 + cs (5 + > 1 The crrect frmula fr the csine f the sum f tw angles is given by the fllwing therem Therem 51: Csine f a Sum cs ( α + β csα cs β sin α sin β T see why the frmula in Therem 51 is crrect, cnsider the unit circles in Figure 51 (a and (b In Figure 51(a, AOB is α + β In Figure 51 (b, COD is btained by rtating AOB thrugh the angle ( α In Figure 51 (a, we have A ( cs ( α + β, sin ( α + β and B ( 1, 0, s by the distance frmula, AB [ cs( α + β 1] + [ sin( α + β 0] cs ( α + β cs( α + β + 1+ sin ( α + β [cs ( α + β + sin ( α + β ] cs( α + β + 1 cs( α + β + 1 cs( α + β 31
In Figure 51 ( b, C ( cs β, sin β and D ( cs ( α, sin ( α ( α, sin α s, by the distance frmula again, cs, CD ( cs β csα + [sin β ( sinα ] ( cs β csα + ( sin β + sinα cs β cs β cs α + cs α + sin β + sin β sin α + sin α ( cs β + sin β + ( cs α + sin α cs β csα + sin β sinα 1 + 1 ( cs β cs α sin β sinα ( csα cs β sin α sin β Because AOB is cngruent t COD, it fllws that AB CD Hence, cs ( α + β ( csα cs β sin α sin β cs ( α + β ( csα cs β sin α sin β cs ( α + β csα cs β sin α sin β, which is the frmula given in Therem 51 Example 51 ---------------------------- ------------------------------------------------------------ Find the exact value f cs 75 By Therem 51, cs 75 cs( 30 + 5 cs 30 cs 5 sin 30 sin 5 3 1 ( 3-1 6 - -, r Therem 5: Csine f a Difference In Therem 51, replace β by β t btain cs ( α β cs [ α + ( β ] cs α cs (-β sin α sin (-β cs α cs β sin α ( - sin β cs α cs β + sinα sin β Example 5 ---------------------------- ------------------------------------------------------------ Find the exact value f cs 15 30 cs By Therem 51, cs 15 cs( 5 3 1 + cs ( α β csα cs β + sin α sin β ( 3 + 1 5 cs 30 + sin 5 sin 30, r 6 + 3
Therem 53: Cfunctin Therem If θ is a real number r an angle measured in radians, then ( i cs θ sinθ ( ii sin θ csθ ( iii ct θ tanθ ( iv tan θ ctθ ( v csc θ secθ ( vi sec θ cscθ We prve ( i, ( ii, and ( iii here and leave the remaining prfs as exercises: ( i By Therem 5, cs θ cs csθ + sin sin θ 0 csθ + 1 sin θ sinθ ( ii In part ( i, we replace θ by π cs θ Thus, csθ and therefre ( ii hlds ( iii By parts ( i and ( ii, π cs( θ π ct ( θ π sin( θ θ sin θ sin θ, sinθ tanθ csθ t btain Naturally, if we measure an angle α in degrees instead f in radians, the relatinships expressed in Therem 53 are written as cs ( 90 α sin α, sin ( 90 α csα, and s frth Nw by cmbining Therem 5 and Therem 53, we can prve the fllwing imprtant therem Therem 5: Sine f a Sum sin ( α + β sin α cs β + cs α sin β By parts ( i and ( ii f Therem 53, 33
π sin( α + β cs ( α + β cs α β cs α β cs α cs β + sin α sin β sin α cs β + csα sin β Example 53 ---------------------------- ------------------------------------------------------------ 7π Find the exact value f sin 1 By Therem 5 and the fact that π π 7π +, we have 3 1 sin 7 π sin π π π π π π 1 + sin cs + cs sin 3 3 3 3 + 1 ( 3 + 1, r 6 + The fllwing therem fllws directly frm Therem 5 and the even-dd identities Therem 55: Sine f a Difference sin ( α β sin α cs β csα sin β Replacing β by β in Therem 5, we have sin ( α β sin [ α + ( β ] sin α cs( β + cs α sin( β sin α csβ csα sin β The fur identities cs ( α + β csα cs β sin α sin β sin ( α + β sin α cs β + cs α sin β cs ( α β csα cs β + sin α sin β sin ( α β sin α cs β csα sin β are used s ften that they shuld be memrized We recmmend tht yu memrize the first identity in the frm: The csine f a sum f tw angles is equal t the csine f the first times the csine f the secnd minus the sine f the first times the sine f the secnd The remaining three identities can be stated in a similar manner Example 5 ---------------------------- ------------------------------------------------------------ Simplify each expressin: (a sin( 90 + θ ( b cs + x ( a sin( 90 + θ sin 90 csθ +cs90 sinθ 1 cs θ + 0 sin θ csθ ( b cs π + x cs π cs x sin sin x 0 cs x 1 sin x sin x 3
Example 55 ---------------------------- ------------------------------------------------------------ Suppse that α and β are angles in standard psitin; α is in quadrant I, sinα 5 3, β is in quadrant II, and cs β 13 5 Find: (a csα (b sin β (c sin(α + β (d cs(α + β (e the quadrant cntaining angle α + β in standard psitin (a Since α is in quadrant I, csα is psitive, and csα sin α 3 5 16 5 5 (b Since β is in quadrant I, sin β is psitive, and sin β cs β 5 1 13 (c sin(α + β sinα cs β + csα sin β (d cs(α + β csα cs β sinα sin β 1 1 169 13 3 5 5 1 + 13 5 13 5 3 1 5 13 5 13 33 65 56 65 (e Since sin(α + β is psitive and cs(α + β is negative, it fllws that (α + β is a quadrant II angle Example 56 ---------------------------- ------------------------------------------------------------ Simplify each expressin withut the use f a calculatr r tables: (a cs 5 cs 0 sin 5 sin 0 (b sin π 3π π 3 cs + cs sin π 5 5 5 5 (a Reading the frmula fr the csine f a sum frm right t left, we have cs 5 cs 0 sin 5 sin 0 cs (5 + 0 cs( 5 (b Reading the frmula fr the sine f a sum frm right t left, we have sin π 3π π 3π π 3π cs + cs sin sin + sin π 0 5 5 5 5 5 5 The graphs f the sine and the csine functins suggest that sin( x + π sin x and cs( x + π cs x 35
Example 57 ---------------------------- ------------------------------------------------------------ Prve the identities: (a sin( x + π sin x (b cs( x + π cs x (c cs( x + y cs ( x y cs x sin y (a sin( x + π sin x cs π + cs x sin π sin x ( 1 + cs x 0 sin x (b cs( x + π cs x cs π sin x sin π cs x ( 1 sin x ( 0 cs x (c Beginning with the left side and applying the frmulas fr the csine f a sum and fr the csine f a difference, we have cs( x + y cs ( x y ( cs x cs y sin x sin y( cs x cs y + sin x sin y ( cs x cs y ( sin x sin y cs x cs y sin cs x (1 sin cs x cs x sin cs x sin y x sin y y (1 cs y sin x sin y + cs y x sin Using the frmulas fr the sine and csine f a sum, we can derive a useful frmula fr the tangent f a sum Therem 56: Tangent f a Difference y If csα 0, cs β 0, and cs ( α + β 0 then + tan β tan ( α + β tan α tan β tan( α + β sin( α + β cs( α + β sin α cs β + csα sin β csα cs β sin α sin β Dividing the numeratr and denminatr f the fractin n the right side f the equatin abve by csα cs β, we get sin α csβ csα sinβ sin α sinβ + + csα csβ csα csβ csα csβ + tanβ tan( α + β csα csβ sin α sinβ sin α sin β tanβ 1 csα csβ csα csβ csα cs β 36
Therem 57: Tangent f a Difference If csα 0, cs β 0, and cs ( α β 0 then tan β tan ( α β 1+ tan α tan β In Therem 56, we replace β by β t btain + tan (-β tanβ tan ( α β tan[ α + ( β ] tan α tan (-β 1+ tan α tanβ Sectin 5 Prblems--------------- ------- ----------------------------------------------------------- In prblems 1 t 1, find the exact value f the indicated trignmetric functin D nt use a calculatr t slve 1 sin 105 Hint: 105 60 + 5 cs 105 3 sin 195 Hint: 195 150 + 5 tan 195 17π 5 cs 1 Hint: 17π 5π π 17π + 6 ct 1 6 1 7 sin 11π 1 Hint: 11π 7 π 1 6 11π π 8 sec 1 13π 13π 3π π 9 tan Hint: + 1 1 3 10 ct 13π 1 19π 11 tan Hint: 1 1 19π 11 π 6 19π π 1 csc 1 In prblems 13 t 3, simplify each expressin withut the use f a calculatr 13 sin ( 180 θ 1 cs ( + t 180 15 cs ( π α 16 sin ( x 70 17 sin ( 360 s 18 cs ( π γ 19 tan ( 180 t 0 sec ( π + u 37
3 1 csc + β tan ( 70 + θ 3 sin 17 cs 13 + sin 13 cs 17 cs 3 cs 17 sin 3 cs17 17π 5 sin cs 36 11π 36 11π 17π sin cs 36 36 5π π π 5π 6 cs sin + cs sin 7 7 7 7 7 cs x cs x + sin x sin x 8 sin ( α β cs β + cs ( α β sin β 9 cs ( α + β cs β + sin ( α + β sin β 30 cs ( x + y cs ( x y tan + tan18 31 tan tan18 0 tan 50 tan 0 3 1+ tan 50 tan 0 33 tan 8x tan 7x 1+ tan 8x tan 7x 3 tan +θ 35 Suppse that α and β are angles in standard psitin; α is in quadrant I, 3 1 sinα, β is in quadrant II, and sin β Find: 13 ( a cs α ( b cs β ( c sin (α + β ( d cs (α + β ( e sin ( α β ( f tan (α + β ( g The quadrant cntaining the angle (α + β in standard psitin 36 Suppse that π < s < π, π < t < π, sin s 5, and tan t 3 13 Find: ( a sin ( s t ( b cs ( s t ( c sin ( s + t ( d sec ( s + t ( e The quadrant cntaining the angle θ in standard psitin if the radian measure f θ is s + t 56 5 37 If α and β are psitive acute angles, sin(α + β, and sin β, 65 13 then find sinα Hint: sin α sin ( ( α + β β 38 Derive a frmula fr tan ( α + β fr the case in which cs α 0 and sin β 0 38
In prblems 39 t 8, shw that each equatin is an identity 39 sin( α + β sin α cs β 1 + ct α tan β 0 tan (θ + π 1 sin t + cs t sin t tan x + tan x 3 6 3 cs ( x+y cs ( x y 1 sin x sin y tan β sin( α β csα cs β 5 ct ( x + y ct x ct y 1 ct x + ct y 6 sin( s + t sin( s t tan s + tant tan s tant 7 π tan θ π 1+ tan θ sinθ cs θ 8 tan π t sec t sin t π csc t 1 9 Prve parts (iv, (v, and (vi f Therem 53 π tan y 50 If x + y, shw that tan y 1+ tan x 51 ( a Use Therem 51 t derive the duble angle frmula fr csine: cs θ cs θ sin θ ( b Use yur result frm (a and the Pythagrean identity invlving sine and csine t prve: (i cs (θ cs θ 1 and (ii cs (θ 1 sin θ 5 Use Therem 5 t prve the duble angle frmula fr sine: sin (θ sinθ csθ In prblems 53 and 5, find all values f θ [ 0, π ] that satisfy the given equatin 53 sin (θ + sinθ 0 5 3 sin θ + cs (θ 1 39