MATH Final Exam prep

Find the length to three significant digits of the arc intercepted by a central angle theta in a circle of radius r r = 17.9 cm, theta = 7pi/8 radians

Do the formula s = r theta so 17.9(7pi/8) 49.2

Write the complex number in rectangular form. 14(cos240 + i sin240)

First convert cos240 and sin 240 into its fractional unit circle jawn 14(-1/2 -sqrt(3)/2i) Next multiply 14 to both jawns - 7 -7sqrt(3)i

Use a double-angle identity to find the exact value of the expression 2sin120 cos120

First just find out what sin120 and cos 120 is on the unit circle so it will be... 2(-1/2) and sqrt(3)/2 Then when you combine it will become -1*sqrt(3)/2 = -sqrt(3)/2

TRIANGLE Determine the remaining sides and angles of the triangle ABC a = 12 meters B = 52 degrees C = 66 degrees

First to find A, you gotta subtract 180 from 52 + 66 so... 180-118 = 62 A = 62 degrees To find b you gotta do 12sin(52)/sin(62) to get... b = 10.71 (rounded) to find c you gotta do 12sin(66)/sin(62) to get... c = 12.42 (rounded)

Find the exact value of the expression tan(arccos(1/2))

First you find that cos(1/2) = 30 degrees Then you do tan(30) and find the answer which is... sqrt(3)

A tree casts a shadow 32 feet long. At the same time, the shadow cast by a vertical 5-ft post is 8 ft long. Find the height of the tree.

First you gotta do this formula set up stuff x/5 = 32/8 Then cross multiply so... x * 8 = 5 * 32 8x = 160 then divide x = 20

Find the unknown angles in triangle ABC, if the triangle exists. A = 37.4 degrees a = 3.1 c = 17.5

First you solve for sinC 17.5sin(37.4degrees)/3.1 equals 3.34 which it cant be greater than 1 so no solution NO SOLUTION

Solve the equation for solutions over the interval [0,2pi) by first solving for the trigonometric function. 6sinx+15=12

First you subtract the 15 over to the 12 6sinx = -3 Then you divide 6 by the -3 sinx = -1/2 Then find the values in the interval [0,2pi) that have sin -1/2 (The whole damn unit circle) so 11pi/6, 7pi/6

Use an identity to write the expression as a single trigonometric function sqrt(1-cos26/1 + cos26)

It will be 26 / 2 which equals 13 and then it will be a tan jawn tan(13)

tan77 * tan 43 / 1 - tan77 * tan43

Just add tan (77 + 43) = 120 Then find out what tan 120 is which is... -sqrt(3)

Use the trigonometric function values of the quadrantal angles to evaluate 3sin180 + 9sec0 + 4(cos0)^2

Just convert all them jawns to their jawn so 3sin180 = 0 9sec0 = 9 4cos0^2 = 4 then add them up = 13

Find the exact value of s in the given interval that has the given circular function value. Do not use a calculator. [pi/2, pi]; cos s = -sqrt(3)/2

Just find the jawn on the unit circle and get the radians of it 5pi/6

For u = (4,-5) and v = (-4,3), find u * v

Just multiply the x's and y's together and add it 4*-4 + -5*3 = -16+-15 = -31

csc pi/6

Multiply this jawn by 180/pi csc(30) = 2

For the plane curve. (a) graph the curve, and (b) find a rectangular equation for the curve. x = t + 4 y = t^2, for t in [-1,1]

Pick the graph that goes to the right 4 then use x = t + 4 move the 4 over x-4 = t y=t^2 =(x-4)^2 subtract 1 to x and add 1 to y y = (x-4)^2 for x in the interval [3,5]

Find the following product, and write the product in rectangular form [2(cos60 + isin60)][7(cos120 + isin120)]

Put the jawn into this jawn, (2 * 7)[cos60 + 120] + isin(60 + 120)] 14[cos(180) + isin(180)] (Convert the jawn into unit circle jawn) 14[0 + i(-1)] = -14

Find the exact value of the real number y if it exists. y = cos^-1 (sqrt(3)/2)

Think about the unit circle, it is the jawn in the parenthesis sooo.... pi/6

Find the exact value of the remaining trigonometric functions of theta. Rationalize denominators when applicable. sin theta = sqrt(2)/3 given that theta is in quadrant 1

You gotta put it as cos^2 theta = 1 - sqrt(2)/3^2 1-2/9 = 7/9 costheta = sqrt(7/9) = sqrt(7)/3 Just do the rest of the plugging in stuff csc = 3sqrt(2)/2 sec = 3sqrt(7)/7 tan = sqrt(14)/7 cot = sqrt(14)/2

A 7.3 m fire truck ladder is leaning against a wall. Find the distance d the ladder goes up the wall (above the fire truck) if the ladder makes an angle of 38 degrees 25 minutes with the horizontal TRUCK JAWN

d = 7.3 * sin 38 degrees 25 minutes = 7.3(0.6214) answer is 4.54

tanxcosx =

sinx

Write the vector (2,20) in the form ai + bj

so naturally i = (1,0) and j = (0,1) so just put the jawn as 2i + 20j

secxcosx =

1

TRIANGLE A plane flies 2.2 hours at 160 mph on a bearing of 35 degrees. It then turns and flies 3.1 hours at the same speed on a bearing of 125 degrees. How far is the plane from its starting point

180-125 = 55 b = (160)(2.2) = 353 miles a = (160)(3.1) = 496 miles a^2 + b^2 = c^2 496^2 + 352^2 = c^2 answer = 608

Use an appropriate sum or difference identity to find the exact value of the expression 13pi/12

1st Split jawn up to be pi/4 and 5pi/6 Then do sin(pi/4) * cos(5pi/6) + cos(pi/4) * sin(5pi/6) They become sqrt(2)/2 * -sqrt(3)/2 + sqrt(2)/2 * 1/2 Then combine them to make... sqrt(2) - sqrt(6) / 4

(Double angle formula) cos2x =

2cos^2x-1

(Double angle formula) sin2x =

2sinxcosx

(Double angle formula) tan2x =

2tanx/1-tan^2x

Find the cube roots of 64i. Graph each cube root as a vector in the complex plane

4(cos30 + isin30), 4(cos150 + isin150), 4(cos270 + isin270) (Just pick the lowest number and going up) Then for the graph just find use the degrees to find the lines jawns

Find the measure (in degrees)of the central angle. The area of the shaded sector is 960pi sq units CIRCLE 48

Do A = 1/2 r^2 theta 960pi = 1/2 (48)^2 theta multiply these jawns by 2 1920pi = (48)^2 theta 1920pi / 48^2 5pi/6 5pi/6 (180/pi) = 150

Evaluate the expression 5tan^2 120 + 7 sin^2 150 - cos^2 270

Do 180 - 120 to get 60 do -tan60 = -sqrt(3) Do 180 - 150 = 30 sin30 = 1/2 cos270 = 0 maine Now plug in those results into the original equation 5(sqrt(3)^2 + 7(1/2)^2 - 0^2 5(3) + 7(1/4) - 0 = 67/4

For the rectangle coordinates (sqrt(3)/2 , 3/2), (a) plot the point and (b) give two pairs of polar coordinates for the point, where 0 <= theta < 360

Find the graph that has the jawn to the right and up For the polar coordinates do r = sqrt((sqrt(3)/2)^2 + (3/2)^2) this converts to sqrt(3/4+9/4)

Find all values of theta in the interval [0, 360) that have the given function value sin theta = - sqrt(3)/2

Find what this jawn is in degree form from the unit circle sooo.... for some reason the jawn is like the cos value its weird instead of 30 it is 60 YOu feel me. So not (sqrt(3)/2 , 1/2) 30 degrees it is (1/2, sqrt(3)/2) 60 degrees Then do 180 + 60 240 360 - 60 300

The measures of two angles of a triangle are given. Find the measure of the third angle. 41 degrees 38 minutes, 111 degrees 27 minutes

First add the two jawns together so 41 38 + 111 27 = 152 degrees 65' then subtract 180 by it butttt since it is like that jawn you gotta make it 178 degrees 120 minutes in order for you to subtract the minutes correctly 178 degrees 120 minutes - 152 degrees 65 minutes = 26 degrees 55 minutes

Find the quotient and write it in rectangular form using exact values 12(cos100 + isin100) / 2(cos145 + isin145)

First divide 12 by 2 and subtract 100 by 145 so then it is 6(cos -45 + i sin-45) cos-45 = sqrt(2)/2 sin-45 = -sqrt(2)/2 Next you do 6(sqrt(2)/2 + i(sqrt(2)/2) = 3sqrt(2) -3sqrt(2)i

TRIANGLE Solve the triangle shown to the right Round the lengths of sides to the nearest tenth and angles to the nearest degree. b = 5 a = 7 C = 97 degrees

First do the jawn you know c^2 = a^2 + b^2 - 2ab cosC c^2 = (7)^2 + (5)^2 - 2(7)(5)cos(97degrees) c^2 = 82.53085404 Then do the square root of that jawn c = 9.08 Next do: b/sinB = c/sinC 5/sinB = 9.08/sin(97degrees) (Cross multiply) 9.08*sinB = 5 * sin(97degrees) [5*sin(97)/9.08] = 0.546556251 Then do the sin^-1 of that to get B = 33.13 or 33 degrees To find A you gotta do 180 - 33 -97 to get 50 A = 50 degrees

Use a double angle formula to find the exact value of the given expression 1 - 2sin^2 112.5

First find that the 1-2sin^2 x matches with the double angle formula for cos2x So then you plug the jawn in so it is cos(2 * 112.5) which equals = cos(225) Then find what cos(225) is on unit circle... -sqrt(2)/2

Solve the equation on the interval [0,2pi) sin2x = -sqrt(3)sinx

First move the -sqrt(3)sinx to the left sin2x + sqrt(3)sinx = 0 Then take out the sines sinx(2 cosx + sqrt(3)) = 0 Next set both sides equal to 0 sinx=0 or (2cos x + sqrt(3)) = 0 Then move the jawns over to the right sooo... cosx = -sqrt(3)/2 Then use sinnn***** 0,pi,5pi/6,7pi/6

Find the magnitude and positive direction angle of the vector (-2sqrt(3), -2)

First use formula |v| = sqrt(a^2 + b^2) so... sqrt(-2sqrt(3)^2 + -2^2) which equals 4 [Just add the jawns up soit would be -12 + 16] Next find the directional angle which is find the tan so tan theta = -2/-2sqrt(3) which equals sqrt(3) Then put in calculator tan^-1 (sqrt(3) = 30 degrees and add 180 to that soooo... 210 degrees

Find the following power. Write the answer in rectangular form [5(cos30 + i sin30)]^4

First you do 5^4 which = 625 and then you do 4 * 30 for both sin and cos which equals 120 625(cos120 + isin120) 625(sqrt(3)/2 + i -1/2) = -625/2 + 625sqrt(3)/2i

Find the area of the triangle ABC a = 102.4 m, b = 74.8 m, c = 99.4 m

First you do a + b + c / 2 so... 102.4 + 74.8 + 99.4 / 2 s = 226.9 next plug it into the formula Area = sqrt(226.9(226.9 - 102.4) (226.9 - 74.8) (226.9 - 99.4)) (make sure you subtract what is in the parenthesis and plug that result in, in order to get the right answer) 124.5,152.1,127.5 3502

For the point(1,120) in polar coordinates (a) plot the point, (b) give two other pairs of polar coordinates for the point, and (c) give the rectangular coordinates for the point

For the graph with all them lines and one point find the degree and choose that one. the points that are equivalent to (1,120) are (-1,300) and (1,480) [One you find the opposite way so its -1,300 and then you add 180 to the original degree of 120 to get 1,480 To find the coordinates you gotta do x = rcos theta and y = rsin theta sooo... x = 1cos120 (-1/2) y = 1sin120 (sqrt(3)/2) {-1/2 , sqrt(3)/2}

Solve the equation for exact solutions over the interval [0,2pi) 7sin^2x + 14sinx + 7 = 0

I guess if it looks like this and has a zero it is gonna be... 3pi/2

For the plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve x=4sint, y=4cost, for t in [0,2pi]

It be a circle graph x^2 + y^2 = 16, for x in [-4,4]

Convert the following angle to decimal degree notation 72 degrees 33 minutes

Just divide the 33 minutes by 60 72.55 degrees

Solve the equation for exact values over the interval [0,360) 6sin(theta/2) = -6 cos (theta/2)

Just divide the jawn by 6 sin(theta/2) = cos(theta/2)

Rewrite cot79 in terms of its cofunction cot79

Just do 90 - 79 which equals 11 then turn it into tan = tan11

Find the exact value sin(2pi/3)

Just find the degree of the jawn. ANd then find the sin of that jawn sqrt(3)/2

Find all values of theta, if theta is in the interval [0, 360) and has the given function value cot theta = sqrt(2)/2

Just find the degree of this jawn = 45 and find it as the same jawn in another spot 45, 315

Which of the graphs shown below is the graph of the polar equation r = 4cos2theta

Makes a jawn that looks like a cross

Convert the angle to degrees, minutes and seconds form. 51.27 degrees

Multiply .27 by 60 to get 16.2 then multiply the .2 by 60 to get 12 = 51 degrees 16 minutes 12 seconds

Convert the following degree measure to radian measure 225 degrees

Multiply 225 by pi/180 which equals 5pi/4

Find the exact value of the real number y if it exists. y = csc^-1 (-2/sqrt(3))

So first find the inverse jawn to see that it is... -sqrt(3)/2 and then find it on the unit circle (also this is sin) so sin(-sqrt(3)/2) = - pi /3

Given the vectors u and , find (a) 6u (b) 6u + 5v (c) v - 5u u = {4,2}, v = {2,0}

So for these you gotta do 6 * 4 and 6 * 2 to get the solution set {24,12} for b you gotta find 5v which is 5 * 2, 5 * 0 so {10,0} then combine the x and y from each jawn sooo 24+10, 12+0 to get {34,12} to find c you gotta find 5u which is 5 * 4 and 5 * 2 so {20,10} then subtract it from v {2,0} {2-20 , 0 - 10} {-18,-10}

Perform the calculation 74 degrees 13 minutes - 23 degrees 49 minutes

To find the answer you gotta subtract one from the degree and add 60 to the minutes 73 degrees 73 minutes - 23 degrees 49 minutes = 50 degrees 24 minutes

Find the compliment and the supplement of the given angle 20 degrees

To find the complement you gotta subtract it 20 by 90 complement = 70 To find the supplement you gotta subtract it by 180 supplement = 160

Find the exact value of the remaining trigonometric functions of theta. Rationalize denominators when applicable. tan theta = -4/3, given that theta is in quadrant IV

To find the missing jawn you gotta do r = sqrt(x^2 + y^2) This is where you would find the value to fit into the sin cos and theta equations r = sqrt(-4^2 + -3^2) sqrt(16+9) sqrt(25) r= 5 Put the jawns in the jawn sin = -4/5 cos = 3/5 csc = -5/4 sec = 5/3 cot = -3/4

Find the distance in kilometers between the following pair of​ cities, assuming they lie on the same​ north-south line. The radius of the Earth is approximately 6400 km. City A, 55 N and City B, 35

When you add the jawns together you get 90 degrees. Then you do 90 * pi/180 = pi/2 s = r theta s = 6400(pi/2) 10,053

sin(pi/2 - x)

cosx

Find the ordered pair that corresponds to the given pair of parametric equations and value of t x=t +4, y = t^2 - 7, t = 5

first you add 4 with 5 to get 9, x = 9 then you do y y = 5^2 - 7 y = 18 (9,18)

TRIANGLE Find h as indicated in the figure 442, 11.7 degrees, 55.5 degrees

h = (x + 389)tan 17.5 xtan47.3 = xtan17.5 + 389tan 17.5 xtan47.3 - tan17.5 = 389 tan 17.5 x = 159.62 h = 173 (IDK)

The equation, with a restriction on x, is the terminal side of an angle theta in standard position. 3x + y = 0, x>= 0

let x = 1 and y = -3 then do the formula r = sqrt(x^2 + y^2) which will be sqrt(1^2 + -3^2) = sqrt(10) then you find the sin cos and tan of that sin = -3sqrt(10)/10 cos = sqrt(10)/10 tan = -3 csc = -sqrt(10)/3 sec = sqrt(10) cot = -1/3

Find the exact value of the expression below cos(105)

sqrt(2) - sqrt(6)/ 4

Find the exact value of the six trigonometric functions of the angle 2130

subtract 360 until you get the jawn between (0 - 360) youll get 330 sin330 = -1/2 cos330 = sqrt(3)/2 tan330 = -sqrt(3)/3 csc330 = -2 sec330 = 2sqrt(3)/3 cot330 = -sqrt(3)

TRIANGLE Find the exact length of the unknown side of the triangle shown to the right. 6, 60 degrees, and 16

use the jawn a^2 = b^2 + c^2 - 2bc cosA a^2 = (6)^2 + (16)^2 - 2(6)(16) cos (60degrees) a^2 = 196 Then you take the square root of that jawn a = 14