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 + 45, yu might ask whether cs 75 can be fund by calculating cs (30 + cs (45 The answer is n; cs 75 < 1, whereas 3 3 + cs (30 + cs (45 + > 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, ΔCOD is btained by rtating ΔAOB 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 1 cs( α + β + 1 cs( α + β Figure51 ( a A y Figure 51 ( b y C unit circle O x α + β B x unit circle O β α D 3
In Figure 51 ( b, C ( cs β, sin β and D ( cs ( α, sin ( α ( cs α, sin α, s, by the distance frmula again, CD ( cs β - csα + [sin β - ( -sinα ] ( cs β - sinα + ( 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 cs α + sin β sinα 1 + 1 ( cs β cs α sin β sinα ( cs α cs β sin α sin β Because Δ AOB is cngruent t Δ COD, it fllws that AB CD Hence, cs ( α + β ( cs α cs β sin α sin β cs ( α + β ( cs α cs β sin α sin β cs ( α + β csα cs β sinα 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 + 45 cs 30 cs 45 sin 30 sin 45 3 1 ( 3 1 6, r 4 4 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 By Therem 51, cs 15 cs( 45 cs ( α β csα cs β + sin α sin β 3 30 cs 1 + 45 cs 30 + sin 45 sin 30 ( 3 + 1 6 +, r 4 4 33
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 sinθ 0 csθ + 1 sin θ sinθ ( ii In part ( i, we replace θ by ( θ t btain cs - ( -θ sin ( θ Thus, csθ sin ( θ, and therefre ( ii hlds ( iii By parts ( i and ( ii, cs( θ sinθ ct ( θ sin( θ tanθ csθ 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 54: Sine f a Sum sin ( α + β sin α cs β + cs α sin β By parts ( i and ( ii f Therem 53, sin( α + β cs ( α + β cs α β cs α β 34
cs α cs β + sin α sin β sin α cs β + csα sin β Example 53 ---------------------------- ------------------------------------------------------------ 7 Find the exact value f sin 1 By Therem 54 and the fact that + 7, we have 3 4 1 sin 7 sin + sin cs + cs sin 1 3 4 3 4 3 4 3 1 + ( 3 + 1, r 6 + 4 4 The fllwing therem fllws directly frm Therem 54 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 54, 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 54 ---------------------------- ------------------------------------------------------------ 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 cs cs x - sin sin x x - sinx + x 0 cs x 1 sin 35
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 1 sin α 3 1 16 4 5 5 5 (b Since β is in quadrant I, sin β is psitive, and sin β 1 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 144 169 13 3 5 4 1 + 5 13 5 13 65 4 5 3 1 56 5 13 5 13 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 3 3 cs + cs sin 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( 45 (b Reading the frmula fr the sine f a sum frm right t left, we have sin 3 3 3 cs + cs sin sin + sin 0 5 5 5 5 5 5 The graphs f the sine and the csine functins suggest that sin( x + sin x and cs( x + cs x Example 57 ---------------------------- ------------------------------------------------------------ Prve the identities: (a sin( x + sin x (b cs( x + cs x (c cs( x + y cs ( x y cs x sin y 36
(a sin( x + sin x cs + cs x sin sin x ( -1 + cs x 0 sin x (b cs( x + cs x cs sin x sin 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 x sin y cs x (1 - sin y (1 - cs x sin y cs x - cs x sin y sin y + cs cs x sin 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 sin( α + β sinα csβ + csα sinβ tan( α + β cs( α + β csα csβ sinα sinβ 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 37
Therem 57: Tangent f a Difference If csα 0, cs β 0, and cs ( α β 0 then tan β tan ( α β 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 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 + 45 cs 105 3 sin 195 Hint: 195 150 + 45 4 tan 195 17 5 cs 1 Hint: 17 5 17 + 6 ct 1 4 6 1 7 sin 11 1 Hint: 11 7 1 6 4 11 8 sec 1 13 13 3 9 tan Hint: + 1 1 4 3 10 ct 13 1 19 11 tan Hint: 1 1 19 11 6 4 19 1 csc 1 In prblems 13 t 34, simplify each expressin withut the use f a calculatr 13 sin ( 180 θ 14 cs ( + t 180 15 cs ( α 16 sin ( x 70 17 sin ( 360 s 18 cs ( γ 19 tan ( 180 t 0 sec ( + u 38
1 csc ( + β 3 tan ( θ 70 + 3 sin 17 cs 13 + sin 13 cs 17 4 cs 43 cs 17 sin 43 cs17 17 5 sin cs 36 11 36 11 17 sin cs 36 36 5 5 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 4 + tan18 31 tan 4 tan18 0 tan 50 tan 0 3 tan 50 tan 0 33 tan 8x tan 7x tan 8x tan 7x 34 tan +θ 35 Suppse that α and β are angles in standard psitin; α is in quadrant I, 4 1 sinα, β is in quadrant II, and sin β Find: 5 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 < s <, ( a sin ( s t ( b cs ( t < t <, sin s 5, and tan t 13 4 Find: s ( 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 3 56 5 37 If α and β are psitive acute angles, sin(α + β, and sin β, find sinα 65 13 Hint: sin α sin (( α + β β 38 Derive a frmula fr tan (α + β fr the case in which csα 0 and sin β 0 In prblems 39 t 48, shw that each equatin is an identity 39 sin( α + β sin α cs β 1 + ct α tan β 40 tan (θ + 39
41 sin t + cs t sin t 4 tan x tan x 3 6 43 cs ( x+y cs ( x y 1 sin x sin y 44 tan β 45 ct (x+y 47 tan θ 4 tan θ 4 ct x ct y 1 ct x + ct y 46 sinθ cs θ 48 tan t + sin( s + t sin( s t sec sin( α β csα cs β tan s + tan t tan s tan t t sin t csc t 1 49 Prve parts (iv, (v, and (vi f Therem 53 tan y tan 50 If x + y 4, shw that tan y x 40