Name: --=K_e..-4-y ___ _ I

Date: __ _ Period : --- PRECALCULUS (May 2016) 2"d Semester Exam Review

This test will consist of 25 multiple choice questions ... you MUST have a pencil or pen the day of your test! You will be allowed to use your notecards during the test.

I recommend that you study old tests and make sure that you can work all the following problems.

A. Trigonometry (Unit Circle) 1. Determine in which quadrant each angle is located:

a. -3: I[[ b. s; ][ c. 640° JS[. d. -300° r

2. Classify and determine a positive and negative coterminal angle in bot h degrees and radians: -~,, ~ ~ 60°/{Jo"/-3o0° b. 22s 0

1 !>RS:-13~0" ~ -3~

B. Graphing Trig Functions: Sine, Cosine, Tangent, Cotangent, Cosecant, Secant 3. Complete the following table about the characteristics (domain, range, amplitude, period, vertical

asymptotes) of all six trigonometric functions/graphs.

y = sinx y = cosx y = tanx y = cscx y = secx y = cotx

Domain Range Period

;)..?(

a'ir 1r ~rr

1T Tr

Amplitude

I I

Vertical Asymptotes

4. Find the 6 trig functions that satisfy the following information: 01'\ nol,...book f<ft.r a. Angle formed through the point (-2, 5) b. Angle formed through the point (5, 4)

5. Find the requested information for the following functions. Sketch a graph of each function.

NOTE: a· sin(b(x - c)) + d; a= vertical stretch/shrink (amplitude); b = horizontal stretch/shrink (period);

c = horizontal shift left/right; d = vertical shift up/down parent function period

b = NEW period

a. y = 3 sin 4(x - rr) + 1 Period= ~ : %

rr c. y = 3 tan 4-(x - - ) + 1

Period = Jt,;, 2

e. y=4csc3x-1 ~ Period= ">:!

Amplitude = .3 Domain = (~--04-1- eo .... )--Range = [-2, I.(] Transfor1J1ations = t/. r+re.ftk I,? J, hor.~I .s4n~l k._ 'I. 6_4f V,

'<( I , • x

b. y = -2 cos-Period =

2 a~,. =- l/ 'fY

Amplitude= -l Domain = (--06-,,-00-,c-)- --Range = [- J. , ~J Transformations= t~-+I~ o~ X-AJc,'.r, v . .rl-rc:1-d. 1, ,. }.. • k. :sl ,-,.1,J, &,)' YA I •

VA= )( = .. 0 1 T, .-l~ .... Domain= X ~- .• o, fl; l.'ir' Range= (-06, oo) Transformations= t/. ~hit '-t 1.,

~ · Ju. '.J. ~ J '<1,6.~f ~/A I uf f x

d. y = -2sec 2 Period= --=- ---:---VA = X ::: .. • 4 fl;"' 3~ . .. Domain= Xi, .. -7r, fr. 3Tr Range= (-t,,o-QiJ[j,"°) Transformations= 1.,.1}~ i>Vv x-4,)(. ,'r~

\J , .sl-rc/.&, '°l ~i h. .,-b_tc.4 '7h

VA= )( .:: ... 0 1 fr1 ~'Ir Domain= {-OIO - ,Jo[J, (}la ">: Range = ><.1'..P. ,r; ~11; .. . Transformatio~s ; ~. ~k.4 7 'C h. r/,.r.~ , ... 3, ~ ...... I

I

C. Inverse Trig Functions J,....!2. 6. Graph each function: A.ff..,.

a. y = sin-1 2x b. y = 3 arccos(x - 1) c. y = 4 arctan(x + 1)

7. Evaluate each expression. (Hint: Use the unit circle when possible.) !)~ ,iokl,cot r .. ,~ ,-a. cos ( tan-1 D b. sin ( tan-1

~) c. sec ( cos-1 D d. (. -1 ffi_) csc sm -

5-

8. Solve each equation. Find all values of x such that O ~ x ~ 2:rr.

a. sin x = ~ b. cos 2x = 0 c. tan x = -1 d. 2 sin x = ~ 2

e. (2cosx-1)(cosx+1)=0 f. 2cos2 x-2cosx=O

D. Trig Identities (the pink sheet) 9. Use fundamental trig identities to simplify the following. Your final answer should not involve a factional answer.

a. cos2 (} (1 + tan2 8) b. cos(} csc (} +sin(} sec(} c. cos2 x sec2 x - cos 2 x

10. Use sum and difference formulas to find the sine and cosine of the given angle. a. 195° = 225° - 30° b. 285° = 225° + 60°

E. Law of Sines and Cosines Law of Sines Law of Cosines Area Formulas

b2+c2-a2 1 1 1 a b c a2 = b2 + c2 - 2bc cos A cosA = --=--=-- 2bc Area= 2bc sin A= 2ac sinB = 2ab sin C sin A sinB sinC

b2 = a 2 + c2 - 2accosB cosB =

a2+c2-b2 Area= .Js(s - a)(s - b)(s - c), sin A sinB sinC 2ac a+b+c --=--=-- s=--

a b a2+b2-c2 2 c

c2 = a 2 + b2 - 2abcosC cosC = 2ab

11. Solve each triangle. (Find the measure of all three angles and the lengths of all three sides.) i. ii. iii. A

A A

8

c

12. Calculate the area of each triangle above. (Make sure you know how to use each formula)

s

c

13. Equilateral means o./1 .r,'Ar ~ ; isosceles means h.;i;i_ +..,,, slSJ.<r ~ ~,1-t. ~k-.l ~

B

F. Vectors 14. Find the following:

i) component form of the vector connecting the given points (first point = initial point; second point = terminal point)

iii) magnitude of each vector a. (3, -8) and (-5, 1) b. (9, 3) and (1, 11) c. (-1, 4) and (2, 2)

15. Performthefollowingoperationsifu = (-3,-5), v = (-4,6), w = (-7,-7). a. u + w b. 2v c. w + 3u

G. Sequences: an = a1 + (n - 1)d an= a1rn-1

16. For an arithmetic sequence a 3 = 13 and a 9 = 25. Determine the first term and the explicit formula for the nth term. Then find the 20th term.

17. For a geometric sequence a2 = 25 and a3 = 5. Determine the explicit formula for the nth term. 18. Determine the explicit formula for each sequence.

a. -6,-1,4,9,14, ... b. -1, 2, 5, 8, 11, ... 1 1 c. 9, 3, 3, 9, ...

H. Series

Sum of a Finite Arithmetic Series:

Sum of a Finite Geometric Series:

f 1c-1 (1-rn) Sn = L a1 r = a1 1 - r

k=1

Sum of an Infinite Geometric Series:

Sequence Formulas:

an = a1 + (n - l)d

19. Evaluate the given sum: 150

a. Ic6k-k2)

k=l

500

b.Icrnk-4) k=l

20. Find the 1ooth partial sum of: 00

Summation Formulas:

n L k = n(n2+ 1)

k=1

n L k 2 = n(n + 1)6(2n + 1)

k=1

n

I n 2 (n + 1)2

k3 = ----4

k=1

n ,

4 _ n(n + 1)(2n + 1)(3n2 + 3n - 1)

Lk - 30 k=l

n ,

5 _ n2 (n + 1)2 (2n2 + 2n - 1)

Lk - 12 k=l

6

d. I soo c1.s)k-l k=l

Ic4k) k=l

I. Conics Circle Ellipse

(x - h) 2 + (y - k) 2 = r 2 (x - h)2 (y - k)2

a2 + b2 = 1

(y - k) 2 (x - h)2

a2 + b2 = 1

Hyperbola (x - h)2 (y - k)2

az - b2 = 1

(y-k) 2 (x-h) 2

az - b2 = 1

Parabola

X - h = _.!._ (y- k)2 4p

1 y-k =-(x-h)2

4p

21. Find the center of a circle that has a diameter with end points (3, -1) and (7, -1).

22. Find the radius of the circle in the question above.

23. Find the equation of an ellipse with center (-1, 2), vertex (-1, 7), and focus (-1, 5).

24. Describe the conic represented by each equation below:

(x-4)2 - (y-2)2 = 1 a. 49 36

b. (x - 4)2 + (y - 2) 2 = 16

(x-4)2 (y-2)2 c. --+--=1

49 36

d. (x - 4) + (y - 2) 2 = 16

6.

a.

n/2

-1 -0.5

-n/2

Y = 81 csrn(x) f(x)=at csin(2x)

c.

-10 -8 -6

y = a, ctan(x)

y = 4arctan(x+ 1)

0.5

2n

-2rt

3n

y = 3arccos(x-1) Sn/2

2n

· - · 311/2

y = a1 ccos(x)

b. -1 D 2

4 6 8 10

I

b)

)(:: ~ ~.I 0%/ ;z~

c..) ./-"'11. x .:: - / x ~ j-{A~ _, C-1) .:: ~/ 7.%;

e..) {?.cosx..- t) (cosx +-1) = D ~ c «:M" X.. - I = 0 c...o..r)( +I ~ 0 J c.e,s X =- I cos)( rt:.-/ ~SIC = Y.t ~ )(.: Cot-'(-0 )if, )( a Cos • 1 (-J,) = 11.11 L'.!_)

f} ~ Co.s-~ X' - ;\ cos->< ::. D l Cos X ( C\:).r X - I) :::. 0

2 Co.r)( = 0 Ccu-Jt - I =- 0 Cosx :::. /

X ::: Co!' - I I

)t := ()I ).1r

q~~cos-lG (! f-/-0,1-/8) = e.o~;GJ ( se.c ~e)

~ 1 cos~~ ( c~a~) :: OJ P) Cos$ C.Sc <a -1-..r,~G s~c.e

~

:: eosG}{r~) +s,~G(c~e) ~ ;:: '- ~:

=Jeo-1-~ + ./--. ... e }

c..) aosi)(sec....1 >< - c..os~ X = Cos~X (s~'°x - 1)

: Mx.( ~n~x) ~ eos~x (:j~) = [.s,'n~x }

eo.s(r1!f) : c().s(~~1""-~o) = CA!' .;i..t,"a:u--30 "'° JI~ JAS-..r,~ Jo

•• §.. a + --H.J. _ -a _ !i. ,.l-«-r. j A"- A.l\.._. 'f '( 'f

b) ;,·~ (2~ -c: s,~(~S'+be) ::. §1~ ~~!>° c~,o +- Cos~·u"'..r,~ ,o fi J- +- -,r,; E -R a t-G -n; ~ -T. A T . 5: ~ 'I -T ~ '{

('O,S'c~~ -cos~-lf'-1-bO) =- ('c,.f AJ.~co.r6o - .SI~ ;i;~ J,}.,o

" _ fi . J. __ ,r; .,{i ~ 4 .,. a . r-~] ~ ~ A A ,,,, L/ '-{

5.5S

/f .:~. Cl = )O

h :: 't c..= /)..

iv'. SAA

-'( 'f~ 1-/:,..C-10') ... 0 A = c.os ~c,))~> - /'it/ , I{ o :. eos--' l.20~ ,.a .. -,. = I':>. ;l. 9 0

J:> v ~(Ao)(a.) o c = =- ac. ~,

~ = '{6 /. 3:,. A = I{ 9 ° ::t I 1-o - ( -1-2 + ,r;) 1, = ~Jt{ g = b</. c.. .: s-~,. 'I c_ ~ =t-..i O

°'- - g;-,A '{ --,- ..... , .s-,111/1 .S ' "' n

i~ ,: . s A s 0... :: 8 I J "f A ~ ~'I O

b = ~ B = 3+, 5'/0

c. ~ 1- c..: 5"t-.'190 CA.~::. 5"~-+- 7- ~ - ~ ( ~)( 1-) cor(i'i) 5,~ B ~

r

/).. ~. Ar~ .. =f¥(11--ao)(~ -1)(!!J-- 1i)

f .> 1 . 6 ,;- ,.,tc,'/..r- szt....&. l ~ ~~ ~~"" = t(s-~~)(5"t?/.'-t).s,~ (f1)

:/ I I 4 / 16 ;;. . tr Urt, 4 -z,. .• AJ 1 ' 1 '

(AA~.

1,. ct= (-3 -s-> v~ ~ <-'{ b > tS = <-~ -~> , ,I ~ . / ;'

o..) tJ-,. (j' : <-3 +- -?- ) -~ +-1> a[ <-Jo., -,~

I:,) ;;I_ if :: <-'J(.1) I b(.1) > -" t<-} / /;2 >] c...) ,;:J f-3 it = < -7 ,L 3 f3) I - f' + 3(-r;)) {<-/6., -.l.2,>)

~ 0= ~ ~ ;t(~o-1) :a 1 1- 3~

l O.;io =: qr]

17._ a._ == i 4f' a 3 = 5'

~ ;-. ;l 6 = o.., ( f )a .. ,. ;­I l5' = o..,

I~. °') _,/-I/'-(/ 't., I~ . - . v vv '-' ., ,-s /.~ .,.,,

~" :. -b f- 0(1t.-1) :: -{, r5'11. - ~

J, \ .. J). ~ J; ~ I/ •. , ~ -vvv"'~

'f-J "J .. , .,.,

a,. =-/-1-3(,r-1) ~ -/ +- 'Jlf -3

c) 11 1/3 .. :(, . .. r-= ~, :. j/3

/a,, ~ 'f(!Y-'l

JJ.

~3 . " . 'F'.

c. •

(x +-8 +­,

-- ~1u- ~ ( .. fl .Pt) lA::: 6 b :: r.---5.'1-_ ,-....,l I :' [ti;'::_ l/ c.. ::: .3

:J.'I. a) JI 1pt,Lol 4.

b) c:r-cle

c) E.Jl ;r.tt.

!) ) P~,°' bol ..