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highschool/Grade 10/Math/MPM2DZ/Trig Quiz 1.md
2019-11-13 20:39:34 +00:00

2.6 KiB

Question 1

B=B(Corresonding Line theorem)`\because \angle B^\prime = \angle B \quad (\text{Corresonding Line theorem})`

C=C(Corresponding Line theorem)`\because \angle C^\prime = \angle C \quad (\text{Corresponding Line theorem})`

ABCABC( AA )`\therefore \triangle AB^\prime C^\prime \sim \triangle ABC \quad (\text{ AA } \sim) `

ABBC=ABBC`\therefore \dfrac{AB^\prime}{B^\prime C^\prime} = \dfrac{AB}{BC} `

3014=30+x22`\therefore \dfrac{30}{14} = \dfrac{30+x}{22}`

14(30+x)=22(30)`14(30+x) = 22(30) `

x=22(30)1430`x = \dfrac{22(30)}{14} - 30 `

x=17.142857117.14`x = 17.1428571 \approx 17.14 `

ACBC=ACBC`\dfrac{AC^\prime}{B^\prime C^\prime} = \dfrac{AC}{BC} `

y14=y+1522`\dfrac{y}{14} = \dfrac{y+15}{22} `

22y=14y+14(15)`22y = 14y + 14(15) `

8y=14(15)`8y = 14(15) `

y=26.25`y = 26.25`

Question 2

h=bsinA`h = b \sin A`

h=11.3sin32`h = 11.3 \sin 32`

h=5.99`h = 5.99`

h<6.8<11.3`\because h \lt 6.8 \lt 11.3`

2s exist`\therefore 2 \triangle 's \text{ exist}`

 Lets call point T is the height that is perpendicular on side AB and connects to point C. and B be the other possible point of B.`\text{ Lets call point } T \text{ is the height that is perpendicular on side } AB \text{ and connects to point } C. \text { and } B^\prime \text{ be the other possible point of } B.`

 Case 1:`\text{ Case } 1:`
CBT=sin1(5.996.8)`\angle CB^\prime T = \sin^{-1} \Bigl(\dfrac{5.99}{6.8} \Bigr)`
CBT=61.75o`\angle CB^\prime T = 61.75^o`
ABC=18061.75=118.25o( Complentary Angle Theorem)`\angle AB^\prime C = 180 - 61.75 = 118.25^o (\text{ Complentary Angle Theorem})`
ACB=180118.2532=29.75o(ASTT)`\angle ACB^\prime = 180 - 118.25 - 32 = 29.75^o (\text{ASTT})`
ABsinACB=CBsinA`\dfrac{AB}{\sin \angle ACB^\prime} = \dfrac{CB^\prime}{\sin A}`
ABsin29.75=6.8sin32`\dfrac{AB}{\sin29.75} = \dfrac{6.8}{\sin 32}`
AB=sin29.75×6.8sin32`AB = \dfrac{\sin 29.75 \times 6.8}{\sin32}`
AB=6.37`AB = 6.37`

 Case 2:`\text{ Case } 2: `

ABC=61.75`\angle ABC = 61.75`

ACB=1803261.75=86.25o( ASTT)`\angle ACB = 180 - 32 - 61.75 = 86.25^o (\text{ ASTT})`

ABsinC=CBsinA`\dfrac{AB}{\sin C} = \dfrac{CB}{\sin A}`

ABsin86.25=6.8sin32`\dfrac{AB}{\sin 86.25} = \dfrac{6.8}{\sin 32}`

AB=sin86.25×6.8sin32`AB = \dfrac{\sin 86.25 \times 6.8}{\sin 32}`

AB=12.8`AB = 12.8`

Question 3

let the square be ABCD and the inner triangle AEF`\text{let the square be } \square ABCD \text{ and the inner triangle } \triangle AEF `

sin(β)=EFAE=EF1=EF`\sin (\beta) = \dfrac{EF}{AE} = \dfrac{EF}{1} = EF`

sin(α)sin(β)=EFAE×ECEF=ECAE=EC1=EC`\sin (\alpha) \sin(\beta) = \dfrac{EF}{AE} \times \dfrac{EC}{EF} = \dfrac{EC}{AE} = \dfrac{EC}{1} = EC`

cos(α)sin(β)=CFEF×EFAE=CFAE=CF1=CF`\cos(\alpha) \sin(\beta) = \dfrac{CF}{EF} \times \dfrac{EF}{AE} = \dfrac{CF}{AE} = \dfrac{CF}{1} = CF`

Draw a parallel line to CD that connects point E to AD.sin(α+β)=CDAE=CD1=CD`\text{Draw a parallel line to } CD \text{ that connects point } E \text{ to } AD. \sin(\alpha + \beta) = \dfrac{CD}{AE} = \dfrac{CD}{1} = CD`

cos(α)cos(β)=ADAF×AFAE=AD1=AD`\cos(\alpha) \cos(\beta) = \dfrac{AD}{AF} \times \dfrac{AF}{AE} = \dfrac{AD}{1} = AD`

sin(α)cos(β)=FDAF×AFAE=FD1=FD`\sin(\alpha) \cos(\beta) = \dfrac{FD}{AF} \times \dfrac{AF}{AE} = \dfrac{FD}{1} = FD`

cos(β)=AFAE=AF1=AF`\cos(\beta) = \dfrac{AF}{AE} = \dfrac{AF}{1} = AF`

Draw a parallel line to CD that connects point E to AD.cos(α+β)=BEAE=BE1=BE`\text{Draw a parallel line to } CD \text{ that connects point } E \text{ to } AD. \cos(\alpha + \beta) = \dfrac{BE}{AE} = \dfrac{BE}{1} = BE`