Name: Date: Block: Advanced(Math(Mid-Term(Review((short)( ( Evaluatethefollowingexpressionsusingthetrianglegivenbelow.Expressanswers insimplifiedfractionalform.(nosquarerootsinthedenominator) 1. ( ( Solvefortheindicatedvariable(s)onthefollowingtriangles.Showallworkand roundanswerstothreedecimalplaces. 2. x= y=
3.Find w, x, y, and z. 4. Solve for w, x, y, and z. 5. Solve for x and y.
6.Find the missing sides and/or angles using special right triangles a. b. 7.Aladderofunknownlengthisleaningagainstawall.Iftheladderreaches10feet upthewallandtheangleformedbetweenthegroundandtheladderis57degrees, thenhowlongistheladder? Sketch: Answer:
8.Sketchtheangleandthendeterminethereferenceangle a.θ = 510 b.θ = 14π 5 9.DetermineifthefollowingtwoanglesarecoRterminal a. 23 and 743 b. 5π 6 and π 6 10.Convertthefollowinganglesfromdegreestoradiansorradianstodegrees 5π a. 200 b. 12 c. 2.35 radians
11. Given the terminal side of θ intersects ( 4, 2) (hint:makeadrawing!) a.findtheangleθ b.findthereferenceangleforθ c.evaluateθforallsixtrigfunctions sin(θ)= cos(θ)= tan(θ)= csc(θ)= sec(θ)= cot(θ)= 12.If tan(θ) = 2 and θ is in Q3 5 a.findtheangleθ b.findthereferenceangleforθ c.evaluateθforallsixtrigfunctions sin(θ)= cos(θ)= tan(θ)= csc(θ)= sec(θ)= cot(θ)=
13.If csc(θ) = 7 and θ is in Q4 3 a.findtheangleθ b.findthereferenceangleforθ c.evaluateθforallsixtrigfunctions sin(θ)= cos(θ)= tan(θ)= csc(θ)= sec(θ)= cot(θ)= 14.Name the quadrant that satisfies each piece of information. a. sec(a) is negative, sin(a) is positive b. sec(b) is positive, tan(b) is negative 15.Evaluatethefollowingexpressionsexactly;leaveanswersinsimplifiedradical form. a. tan( 60 ) b. csc 405 ( ) c. cos 17π 6
Identifythetransformationsforthefollowingtrigonometricfunctions: Graphthefollowingequation(indegrees)Be(sure(to(label(the(x(and(y(axis( 16. f (x) = sin 2 x 45 ( ( )) + 3 Amplitude: Period: Interval: Frequency(b): Phase/HorizontalShift: VerticalShift: Graphthefollowingequation(indegrees)Be(sure(to(label(the(x(and(y(axis( 17. f (x) = 2cos 1 ( 2 x +180) 3 Amplitude: Period: Interval: Frequency(b): Phase/HorizontalShift: VerticalShift:
Graphthefollowingequation(inradians)Be(sure(to(label(the(x(and(y(axis( 18. f (x) = 2sin 1 ( 2 x π ) +1 Amplitude: Period: Interval: Frequency(b): Phase/HorizontalShift: VerticalShift: Graphthefollowingequation(inradians)Be(sure(to(label(the(x(and(y(axis( 19. f (x) = 4cos 2 x π 4 Amplitude: Period: Interval: Frequency(b): Phase/HorizontalShift: VerticalShift:
Directions:Writethesin(x)andcos(x)equationsofthegraphsdepictedinthe questionsbelow;showallworktofindallcoefficients: 20. Amplitude (a) = Frequency (b) = Period (2pi / b) = Vertical Shift (d) = HorizontalshiftforSine= cforsine= HorizontalshiftforCosine= cforcosine= sin(x): cos(x):
Note:thex axisscaleforthisgraphis90degrees!(doworkindegreesthistime) 21. Amplitude (a) = Frequency (b) = Period (360 / b) = Vertical Shift (d) = HorizontalshiftforSine= cforsine= HorizontalshiftforCosine= cforcosine= sin(x): cos(x):
22.Thedepthofwaterattheendofapiervariessinusoidallywiththetides throughouttheday.todaythefirstlowtideoccursat2amwithadepthof6feet. Thefirsthightideoccursat10AMwithadepthof16feet. Let12AMrepresenttime=0 Sketchagraphandwritetheequationthatmodelsthesituation(picksinorcos)!!!! Amp: Period: Interval: phase: vert: A= B= C(forsin)= C(forcos)= D= intermsofsin(x): orcos(x): Atwhattime(s)willtheheightofthewaterbe12feet?
23. A weight attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the ground varies sinusoidally with time. You start a stopwatch. When the stopwatch reads 1.1 seconds the weight reaches a minimum point of 20 cm above the floor. This minimum point is followed by a maximum point of 80 cm above the floor at a time of 1.9 seconds. Amp: Period: Interval: phase: vert: A= B= C(forsin)= C(forcos)= D= intermsofsin(x): orcos(x): Atwhattime(s)willtheweightbe30cmabovethefloor?
24. Consider the equation: where y is height in cm and x is time in seconds. a. How far off the ground is the weight when the stopwatch starts? b. How far off the ground is the weight after 5 seconds? c. How many seconds have passed when the weight is first 70 cm above the ground 25.Evaluatethefollowingexpressionswithoutacalculator;expressanswersin simplifiedfractionalform: a. cos 1 0 c. tan 1 3 ( ) b. sin 1 1 2 ( ) d. tan 1 1 ( )
Solvethetrianglebelowforallmissingvalues;roundanswerstothreedecimal places;expressanglesindegreeform: 26. 27.Makeadrawingtorepresenteachproblembelowandthensolve;round answerstothreedecimalplacesandexpressanglesindegrees a.you are standing 50 feet from the base of a flag on a golf course. The flag is 8 feet tall. Find the angle of elevation to the top of the flag to the nearest tenth of a degree. b.you are standing on a roof of a factory and you are looking down at the base of a silo that is 70 feet away. The factory is 25 feet tall. Find the angle of depression.
28.Solvethefollowingequationsandlistallsolutions:(theyshouldbeunitcircle values!) a. sin x = 1 2 b. 2cos x = 2 c.3cot x = 3 d. e. 4cos 2 x +1 = 4 f. tan 2 x 2 = 1
29.Solvethefollowingequationsforallsolutionsbetween0and360;round answersto3decimalplaces:(feelfreetousetheblankunitcircletohelp!) a. 3cos x 2 = 0 b. sec x = 6.2 c. tan x = 11 9
30.Sketchandthensolvethefollowingtrianglesforallmissingsidesandangles; showallwork: a. In ABC, a = 114, <B=61º, and <C = 47º. b. In ABC, a = 4, <A=53.13º, and b = 5. c. In ABC, <B=20º, <C=50º, and c = 20.
d. In ABC, a=8, b=15, and c = 20. e. In ABC, a=12, b=10, and <C=78º. 31. In, e = 24, f = 8, E = 100, F = 20. What is the area of the triangle to the nearest tenth?
32.Thefollowingsidesandanglescanbedrawnas2/possible/triangles;drawboth trianglesandsolveforallmissingsidesandangles: AngleB=35 sidea=11 sideb=7 33.ProvethefollowingstatementsbyCLEARLYshowingeverystep a. b. c. d.
e. f. 34.Graph each polar coordinate and complete the equivalent coordinate. A (5, 165º) = ( 5, ) B ( 3, 270º) = ( 3, ) C (4, 60º) = (, 120º) D ( 2, 210º) = (, 330º)
35. Change from polar coordinates to rectangular coordinates. E ( 7, 270º) F ( 5, 235º) 36. Change from rectangular coordinates to polar coordinates. G ( 4 3, 4) H (9, -2)