Exam Review Trigonometry

Transcription

1 Exam Review Trignmetry (Tyler, Chris, Hafsa, Nasim, Paniz,Tng) Similar Triangles Prving Similarity (AA, SSS, SAS) ~ Tyler Garfinkle 3 Types f Similarities: 1. Side Side Side Similarity (SSS) If three pairs f rrespnding sides are prprtinal AB XY BC AC This is knwn as SSS~. Y Z XZ 2. Side Angle Side (SAS) If tw pairs f rrespnding sides are prprtinal and the ntained angles are equal. This is knwn as SAS~. 3. Angle Angle (AA) If tw pairs f rrespnding angles are equal. Fr example: ㄥ B ㄥ Y, ㄥ C ㄥ Z (see first diagram). This is knwn as AA~. Triple Rati Statements: ex: AB BC XY Y Z AC XZ When slving fr similar triangles yu must inlude the fllwing steps: 1. State the similar angles 2. State ABC~ XYZ beause f (state similarity i.e. SSS, SAS, AA) 3. Write triple rati statements 4. Fill in knwn values 5. Slve fr the unknwn sides/angles Pratie Questin: Prve similarity and slve fr x and y.

2 Slutin: 1. ㄥA ㄥX, ㄥC ㄥZ 2. Therefre, ABC~ XYZ beause f AA~ 3. AB BC AC XY Y Z XZ 5 12 x 7 y y x y x11 5 Trig Ratis When slving fr a side length When Slving fr a side Equatin Sh Cah Ta 1) S in θ O H (This is the equatin when given 1 internal angle and a side length)θ 2) S in 32 O 6 (We fill in all the infrmatin given) 3) 6 Sin 32 O (We then islate fr O by mving the 6 ver t the ther side f the sign by mpleting the ppsite funtin) 4) 3.18

3 When slving fr an angle Equatin Sh Cah Ta 1) Tan θ O H (Tan is the trig rati that is used beause the 2 piees f infrmatin are frm the ppsite and adjaent side length) 2) θ T an 1 O H (We islate fr the angle beause that is what we want t slve. T d this, we must bring Tan ver by mpleting its ppsite funtin. This is alled inverse Tan) 3) θ T an (Sub number values in fr the Oppsite and Adjaent side lengths) 4) θ Sine law Sine Law is used t find the value f a side r an angle, when yu have a mplete prprtin with an inmplete prprtin. Yu uld have an angle, with its side length, alng with anther unknwn side, r an angle with its side length alng with anther unknwn angle. Example 1 : Determine the indiated value () t tw deimal plaes: a sina 32 sin sin sin112 32sin46 sin112 b sinb b sinb sin46 sin46 sinc sin46 Therefre, the length f is units.

4 Example 2 : Tw supprting wires, as shwn in the sketh, are 17.0 m, and 10.0 m in length. They are attahed t the tp f a TV twer frm the tw pints M and N. The angle f elevatin fr the lnger wire is Hw far apart are the pints M and N? Step 1 In rder t slve fr, we need O, and in rder t get O, we need N. m sinm n sinn sino Step 2 Write ut frmula ( ), plug in given values, slve fr N. 10 sin sin sinn 17 sino Step 4 Slve fr using Sine Law sinn sinm sinn N sin 1 ( 10 ) 10 sin sin28.1 m n 17 sin N sin sin98.7 Step 3 Slve fr O sin (SATT) 21 sino sin98.7 sin98.7 Therefre, the distane frm pint M t N is 21 m. Pythagrean therem ~ Tng Li 2 a 2 +b 2.. When C is the hyptenuse f the right triangle Rules 2 side must be identified in rder fr Pythagrean therem t be able t wrk. Example Questin #1 Find X Using Pythagrean therem x 2

5 Csine law ~ Nasim

6 Ex.2 A triangle has sides f length 12m, 17m, and 9.5m. Determine the measure f the largest angle in the triangle? (the largest angleppsite f lngest side) 1) determine whether r nt yu shuld use sine law.(hw many variables d yu have? what are they, angles r sides?) 2) arrange yur sine law and fill in yur infrmatin 3) alulate step by step(put brakets arund the: 2abCsC) 4) square rt yur answer if trying t find the missing side. slutin: ACs 1 (b a 2 )/2b ACs 1 ( )/2(9.5)(12) ACs 1 ( 54.75)/(228) A degrees Therefre the largest angle is 104 degrees if runded t the nearest degree.

7 Try it: DON T FORGET YOUR DIAGRAM! 1. The pst f a hkey gal are 2.0m apart. a player tries t sre a gal by shting the puk alng the ie frm a pint 4.8m frm ne pst and 6.7m frm the ther. Within what angle must the player sht the puk? Inlude a diagram. 2. find the perimeter f triangle SAT, if angle S 56 degrees, a14m, and t 25m. *NOTE: remember yu dn t always have t use the sine r sine law. Yu an use SATT, PLT Z, r divide yur triangle int tw right triangles and use the pythagrean therem. Use the sine and sine law when yu dn t have enugh infrmatin t mplete the prblem. * 1

8 Elevatin/Depressin ~ Paniz Ad The angle f depressin is the angle between the hrizntal and the line f sight frm the frm the bserver's eye t a pint belw eye level. The angle f elevatin is the angle between the hrizntal and the line f sight frm the bserver s eye t a pint abve eye level 1) draw a diagram using the given infrmatin 2) determine the primary trig rati that yu must use as yur equatin 3) slve it. Ex) A flagple asts a shadw f 75m when the angle f elevatin f the sun is 33. Determine the height f the flagple. The flagple is x (ppsite side) and the 75 metres is the adjaent side therefre we use tan /a Slutin: Tan33 x/75 (multiply bth sides by 75) 75tan33 x 48.7 x Slving trig prblems invlving navigatin (bearings) Steps! 1) Draw the mpass at ne f the pints 2) use the mpass diretin given (ex. N48 W) t draw the send pint 3) Draw anther mpass at send pint and use infrmatin given t draw the third pint

9 4) use ur therems (ex. PLT Z, SATT, CA) t find the missing angles inside the triangle 5) Use sine law a sina a sin42 b sinb b sin118 60sin42 a sin20 sinc 60 sin20 a b 60sin118 sin20 b 154.9