Cop yri ht 2006, Barr Mabillard.

Transcription

1 Trignmetry II Cpyright Trignmetry II Standards 006, Test Barry ANSWERS Mabillard. 0

2 . If csα, where sinα > 0, and 5 cs α + β value f sin β, where tan β > 0, determine the exact 9 First determine the quadrant α is in: Then determine the quadrant β is in: csα < 0 sinα > 0 sin β > 0 tan β > 0 Nw slve the triangles fr bth α and β; use Pythagras t find the unknwn sides. a + b c + b 5 5 b + b b a + b c + b b + b 65 b 65 At this pint, state all required trignmetric ratis: 65 csα cs β 5 9 sinα sin β 5 9 Finally, evaluate cs ( α+ β ) ( + ) cs α β csαcs β sinαsin β 65 cs( α + β) cs( α + β) 5 5 cs ( α β) Trignmetry II Standards Test ANSWERS

3 . Express sin ( α + β ) using nly csine.. If Start with the identity: sin θ + cs θ Rearrange t get: cs θ sin θ cs α + β sin α + β It fllws that: The answer is cs ( α + β ) csθ, where θ is an acute angle, determine the exact value f θ. π π If yu evaluate c sx, yu get x,. 6 6 Yu d nt want t use any ther angles since nly π 6 is acute (<90 ). Divide by t get the angle θ: π π π *Alternatively, yu culd graph each side f c sθ in yur calculatr using degree mde, then cnvert the answer t a radian fractin.. Determine the exact value f tan ( 5 ) Rewrite as tan and use the frmula 0 0 tan 5 tan 0 tan ( 5 0 ) + tan5 tan0 + () tan ( α β) Get a cmmn denminatr fr tp & bttm 0 Divide fractins by multiplying the reciprcal tanα tan β + tanα tanβ Evaluate the required tan ratis befre ding the calculatin n the left. tan 0 sin 0 cs 0 sin 5 tan 5 cs 5 Trignmetry II Standards Test ANSWERS

4 ct 5. Prve the identity: + ct x sin x cs x x - ct x + ct x - ct x csc x -ct x sin x cs x - sin x sin x sin x-cs x 6. Slve fr x, where 0 x π : csc x csc x Rewrite as csc x - cscx - 0 cscx - cscx + 0 Then factr: Nw equate each set f brackets t zer and slve fr x. cscx - 0 cscx sinx π 5 π x, 6 6 cscx + 0 cscx - sinx - π x The answer is π 5π π x,, Slve fr x: x x cs + cs 0 Factr t btain: ( csx - )( csx + ) 0 Nw equate each set f brackets t zer and slve fr x. csx - 0 csx csx π 5 π x, csx + 0 csx - x π The answer is π 5 π x, π, Trignmetry II Standards Test ANSWERS

5 sin x ct x+ cs x 8. Simplify: sin x csx sinx + csx sinx sinx csx + csx sinx csx sinx ctx π 9. Slve ( csθ + ) ( tanθ ) 0 fr θ in the interval x π Equate each set f brackets t zer and slve fr θ. csθ +0 csθ - csθ - π π θ, tanθ -0 tanθ π 5 π θ, π 5π π π θ,, (Omit since it lies utside the specified dmain.) 0. Determine the exact value f π a) cs π First cnvert t degrees. This will make the remainder f the calculatin easier than π 80 wrking in radian fractins. 65 π Write as cs ( ), then apply the csine sum frmula cs cs0 cs5 - sin0 cs π b) sec Trignmetry II Standards Test ANSWERS sec65 cs65 This is simply the reciprcal f the answer yu fund in part a) The answer is - - 6

6 ctθsec θ. Prve the identity tanθ ct θ + ctθsec θ ct θ + ctθsec θ csc θ ctθ cs θ sin θ csθ sin θ sinθ cs θ sinθ csθ tanθ. If tanθ and π < θ < π, state the exact value f cs θ Draw the triangle and find the hyptenuse using Pythagras: a + b c + 9 c c c + c Frm the diagram, cs θ Nw use the frmula: cs θ cs θ - csθ - csθ - 8 csθ - 5 csθ - Trignmetry II Standards Test ANSWERS 5

7 sin θ T slve this equatin algebraically, yu need t determine the slutins within rtatins f the unit circle.. Slve fr θ ver [ 0, π ]: Slve the equatin sinx t get the angles π and π, bth f which are within the first rtatin. Add the perid t each ne t get the c-terminal angles in the secnd rtatin. π π 6π 7π + π + π π 6π 8π + π + Finally, take all yur slutins π, π, 7π, 8π and divide by. π π 7π 8π The answer is,,, Slve the fllwing equatin fr x R : csθ + 0 Get csθ by itself befre trying t determine the angles: csθ - - csθ T slve this equatin algebraically, yu need t determine the slutins within rtatins f the unit circle. csx - t get the angles bth f which are within the first rtatin. Slve the equatin π π and, Add the perid t each ne t get the c-terminal angles in the secnd rtatin. π π 6π 8π + π + π π 6π 0π + π + π π π 8π 0π Finally, take all yur slutins,,, and divide by. The answer is π, π, 8π, 0π Trignmetry II Standards Test ANSWERS 6

8 5. If α and β are secnd quadrant angles, and determine the exact value f sin ( α β ) csα and sin β, Slve the triangles fr bth α and β; use Pythagras t find the unknwn sides. a + b c + b 9 + b b 5 b 5 + a b c + b + 6 b b 5 b 5 At this pint, state all required trignmetric ratis: csα 5 cs β 5 sinα sin β Use a negative since the b-value lies n the negative x-axis. Finally, evaluate sin ( α - β ) ( ) sin α β sinαcs β csαsin β 5 5 sin ( α β) 75 sin ( α β) + sin ( α β) Slve fr x ver the interval [ 0, π ] fr sin x sin x sinx 0 x0, π,π Set the equatin t zer: si n x - sinx 0 si nx sinx - 0 Then factr: Nw set each factr equal t zer and slve fr the angles sinx - 0 sinx π x The slutin is π x0,, π,π Trignmetry II Standards Test ANSWERS 7

9 7. Prve + cscθ + csθ csθ -csθ +csθ + + csθ - csθ - csθ + csθ -csθ +csθ + + csθ - csθ - csθ + csθ - csθ + csθ + - cs θ - cs θ - csθ ++csθ - cs θ - cs θ sin θ cscθ 8. Slve fr x ver the interval [ ] 0, π : + tan x. State the slutins as radians t three decimal places First islate ta nx : +tan x tan x tanx ± Nw slve in yur calculatr by graphing Y tan x Y Y + The answer is x ,.86,.097, 5.8 Trignmetry II Standards Test ANSWERS 8

10 9. a) Prve sin x tan x + csx sinx + csx sinxcsx + cs x - sinxcsx cs x sinx csx tanx sin x b) State a value f x where is undefined + csx The expressin is undefined when + csx 0 + csx 0 csx - First slve csx - ver tw rtatins: The slutin t this is x π, π Nw divide by the t get the slutins t cs x - π π The final answer is x, (Yu culd als slve this by graphing Y csx & Y - in yur calculatr and finding the pints f intersectin) π 0. Find the exact value f cs 8 Use the identity π csα cs - 8 csα cs α - What ges here is duble what ges here. π π 8 We nw have the equatin The answer is: π π cs cs - 8 π cs Trignmetry II Standards Test ANSWERS 9

11 cs x sin x + sin x. Simplify csc x cs x sinx + sinx cscx cs x sinx + sinx sinx sin x+cs x. Find the exact value f: 7π a) cs 7 π 80 First express the angle in terms f degrees: 05 π cs 05 cs cs cs60 cs5 - sin60 sin5-6 7π b) sec This is simply the reciprcal f the answer yu fund in part a) 7π sec - 6 Trignmetry II Standards Test ANSWERS 0

12 . Express tan θ using nly sinθ Start with the identity tan θ sin θ cs θ Rewrite the identity sin θ+cs θ as cs θ -sin θ Finally, sin θ sin θ cs θ - sin θ. Prve: cs x sin x+ + sec x sin x+ cs x csx csx sinx + sinx + + sinx + csx csx sinx + cs x sin x + sinx + + csx sinx + csx sinx + cs x + sin x + sinx + csx sinx + + sinx + csx ( sinx +) +sinx csx sinx + +sinx csx sinx + csx secx 5. Slve fr θ in the interval [ 0, π ]: sin θ sinθ + 0 Factr t btain: ( sinθ -)( sinθ -) 0 sinθ -0 sinθ sinθ π 5 π θ, 6 6 Nw slve fr the angles: sinθ -0 sinθ π θ The answer is π π 5 π θ,, 6 6 Trignmetry II Standards Test ANSWERS

13 6. If csθ and cscθ > 0, determine the exact value f: 5 a) tan θ First determine which quadrant Nw fill in the triangle the angle is fund in: and use Pythagras t determine the unknwn side: a + b c + b b b 9 b tanθ Use the frmula tanθ -tan θ - tanθ Frm the triangle, sinθ 5 csθ - 5 tanθ b) sin θ Use the frmula s inθ sinθcsθ sinθ sinθ - 5 Trignmetry II Standards Test ANSWERS

14 7. Slve the equatin cscθ, where θ R. State yur slutin t three decimal places. Slve by graphing: The questin specifies the dmain as θ R, s a general slutin is required. The (psitive) slutins are.8 and 5.9 Since the slutins repeat themselves every perid, write the general slutin as: θ.8+ k π, k I 5.9+ k π, k I 8. Shw that sin 8x is equivalent t sinxcsx sin x sinxcsx Start with the identity What ges here is duble what ges here. In the equatin si n8x sinxcsx the x here becmes the 8x here 9. Express ctθ secθ as a single trignmetric functin ctθsecθ csθ sinθ csθ sinθ cscθ Trignmetry II Standards Test ANSWERS

15 0. If sinθ and csθ < 0, find the exact value f a) tanθ First determine which quadrant the angle is fund in: Frm the triangle, the trig ratins are: sinθ a + b c + b 69 + b b 65 b 5 5 csθ - tanθ - 5 The answer is tanθ - 5 π b) cs θ π Expand cs θ - using the subtractin law fr csine: π π π cs θ - csθcs + sinθsin Nw use the values fund in part a) frm the triangle, and values frm the unit circle: π π 5 csθcs + sinθsin Trignmetry II Standards Test ANSWERS

16 . Find the exact values f x in the interval [ 0, π ]: cs x + cs x+ sin x cs x + csx + sin x cs x + csx sin x - cs x + csx -cs x cs x + csx 0 csx csx + 0 Nw slve fr x: csx 0 π π x, csx + 0 csx - csx - π π x, The full slutin set is π π π π x,,,. Prve the identity: cs x cs x ct x sin x csx - cs x sin x csx - cs x sin x csxsin x sin x csx sinx ctx Trignmetry II Standards Test ANSWERS 5

17 . Given the pint shwn n the circle, determine the value f tan β Recall that the x-crdinate is cs θ, and the y-crdinate is s inθ. 5 adj csβ hyp pp sinβ hyp Therefre, tanβ 5 tanθ Nw use the identity tanθ -tan θ 5 tanθ - 5 tanθ 5-5 tanθ tanθ tanθ tanθ - 9 Trignmetry II Standards Test ANSWERS 6