math final

use (A = A0 e ^kt) the population 6 years ago was 1,600, today only 1100 are alive, once the population drops below 100, the situation will be irreversible, when will this happen

*

a point on the terminal side of angle theta is given, find the exact value of the indicated trig function of theta: (-1/2, 1/5) find cosO

- 5 sp rt 29/29

polar coordinates of a point are given, find the rectangular coordinates of the point: (-9, 3 pi/4)

- 9 sq rt 2/2, 9 sq rt 2/2

use trig identities to find the exact value: tan85 - tan (-35)/1 + tan 85 tan (-35)

- sq rt 3

evaluate the expression without using a calculator: log6 1/6

-1

use the reference angles to find the exact value of the expression, do not use a calculator: sin 3 pi/2

-1

evaluate or simplify the expression without using a calculator: ln 1

0

solve the equation on the interval [0, 2 pi): cos x + 2 cos x sin x = 0

0, pi/2, 7 pi/6, 3 pi/2

solve the exponential equation, use a calculator to obtain a decimal approximation, correct to two decimal places: e^5x = 7

0.39 (take natural log)

a point of the terminal side of angle O is given, dind the exact value of the indicated trigonometric function of O: (-10, 24) find sinO

12/13

find a positive angle less than 360 degrees or 2 pi that is cotermimal with the given angle: 534 degrees

174 degrees

determine the period of y = sin 3x

2 pi/3

use the unit circle to find the value of the trig function: csc pi/6

2 sp rt 3/3

find sinO, give an exact answer witha rationalized denominator: H-10, bottom-9

9/10

write the partial fraction decomposition of the rational expression: 4x² - x - 9/x(x+1)(x-1)

9/x + -2/x+1 + -3/x - 1

use the given vectors to find u + v: u = 10i + 4j and v = 6i + 8j

92

find the angle between the given vectors, round to the nearest tenth of a degree: u = 4j, v = 9i + 6j

56.3 degrees

use properties of logarithms to condense logarithmic expression as much as possible, write the expression as a single logarithm whose coefficient is 1, where possible evaluate logarithmic expressions: 3 ln (x-5) - 10 ln x

ln (x-5)³/x¹⁰

use properties of logarithms to condense logarithmic expression as much as possible, write the expression as a single logarithm whose coefficient is 1, where possible evaluate logarithmic expressions: log3 (x+8) - log3 (x+6)

log3 (x+8/x-6)

write the equation in its equivalent logarithmic form: 5⁻³ = 1/125

log5 1/125=-3

use properties of logarithms to expand logarithmic expression as much as possible, where possible evaluate logarithmic expressions without using a calculator: log6 (7x11/13)

log6 7 + log6 11 - log6 13

use properties of logarithms to expand logarithmic expression as much as possible, where possible evaluate logarithmic expressions without using a calculator: log5 (7/5)

log6 7 - log6 5

write the equation in its equivalent logarthmic form: 15^x = 225

long15 225 = x

use the dot product to determine whether the vectors are parallel, orthogonal, or neither: v = 3i + 2j, w = 2i - 3j

orthogonal

find the exact value of the expression: cos ^-1(-1)

pi

find a positive angle less than 360 degrees or 2 pi that is coterminal with the given angle: 21 pi/10

pi/10

solve the equation by expressing each side as a power of the same base and then equatiing exponents: e ^x+10 = 1/e⁷

{-17}

solve the logarithmic equation, be sure to reject any value that is not in the domain of the original logarithmic expressions: 4 ln (4x) = 24

{e⁶/4}

let a = [-5 2] and b = [1 0] find 3a + 4b

[-11 6]

the given angle is in standard position. determine the quadrant in which the angle lies: 346 degrees

quadrant IV

use the reference angles to find the exact value of the expression, do not use a calculator: sin 4 pi/3

- sp rt 3/2

complete the identity: (sin x + cos x)²/ 1 + 2sinxcosx

1

determine the period of y = -2 cos 1/3 x

6 pi

use a calculator to find the approximate value of the expression, round to two decimal places: sin87 degrees

1.00

approximate the number using a calculator; round your answer to three decimal places: 4¹.²

5.278

if emery has $1200 to invest at 8% per year compounded monthly, how long will it be before he has $1900? if the compounding is continous, how long will it be

5.763 yrs, 5.744 yrs

one rope pulls a barge directly east with a force of 59 newtons, and another rope pulls the barge directly north with a force of 65 newtons, find the magnitude of the resultant force acting on the barge, round to the nearest whole number

88 newtons

solve the exponential equation, express the solution set in terms of natural logarithms: e^x+8 = 3

{ln 3-8}

evaluate the expression without using a calculator: log7 sq rt 7

1/2

determine the period of y = 3 sin x

2 pi

find the product ab if possible: a = [1, 3, -2. 3, 0, 4] (2x3) b = [3, 0, -2, 1, 0, 1] (3x2)

[-3, -5, 9, 16] (2x2)

let A = [6, -9, -5, -3, -9, 7] (3x2) and B = [8, -8, -5, 2, -3, -9] (3x2) find A + B

[14, -17, -10, 0, -12, -2] (3x2)

identify a and b in the following expression which is the right side of the formula for cos (a-b): cos (7 pi/18)cos (2 pi/9) + sin (7pi/18) sin (2pi/9)

a = 7 pi/18, b = 2pi/9

find the domain of the logarithmic function: f(x)=log5 (x+1)

(-1, infinity)

write the equation in its equivalent exponential form: log2x=3

2³=x

use a calculator to find the approximate value of the expression: csc 16 degrees

3.63

complete the identity: cos²x - sin²x/1-tan²x

cos²x

complete the identity: sin 2x / 1 - cos 2x

cot x

solve the equation on the interval [0, 2 pi): sin²2x = 1

pi/4, 3 pi/4, 5 pi/4, 7 pi/4

find a cofunction with the same value as the given expression: cos81 degrees

sin 9 degrees

use A=Pe^rt, find the accumulated value of an investment of $5,000 at 9% compounded continuously for 4 years

$7,1655.65 (5000e^(0.09)(4)

use A=P(1+r/n)^nt, find the accumulated value of an investment of $5000 at 5% compounded monthly for 8 years

$7452.93 (5000(1+0.05/12)^8x12)

the rectangular coordinates of a point are given, find the polar coordinates of the point, express theta in radians: (4, -4 sq rt 3)

(8, 5 pi/3)

find the length of the arc on a circle of radius r intercepted by a central angle theta: r = 50 in, theta=55 degrees

*

find the exact value of cos theta: tanO = -2/3, theta in quadrant 2

- 3 sq rt 13/13

use periodic properties of the trig functions to find the exact value of the expression: sin 10 pi/3

- sp rt 3/2

find the exact value of the expression: sin 250 cos 10 - cos 250 sin 10

- sq rt 3/2

use the calculator to find the value of the expression in radians rounded to two decimal places: sin ^-1 (-1/4)

-0.25

use even and odd properties of the trig functions to find the exact value of the expression: cos (- pi/3)

-1/2

use periodic properties of the trig functions to find the exact value of the expression: cos 20 pi/3

-1/2

find the exact value of the expression: cos(255 degrees - 15 degrees)

-1/2 (cos255cos15+sin255sin15)

use the given information to find the exact value of the expression: sin a = 24/25, a lies in quadrant 2, and cos b = 2/5, b lies in quadrant 1, find cos (a-b)

-14 + 24 sq rt 21/125

theta is an acute angle and sinO and cosO are given, use identites to find the indicated value: sinO = - sq rt 5/3, cosO = 2/3, find cotO

-2 sp rt5/5

use the given vectors to find (4u) + v: u = 6i - 7j, v = -9i + 5j

-356

convert the angle in radians to degrees: -pi/5

-36 degrees (pi r/5 x 180/pi r)

solve the exponential equation, use a calculator to obtain a decimal approximation, correct to two decimal places: 10^x = 3.95

0.60 (take log of both sides x= 3.95(log))

evaluate the expression without using a calculator: log10 10

1

use an identity to find the value of the expression, do not use a calculator: sin50 degrees x csc 50 degrees

1

use an identity to find the value of the expression, do not use a calculator: sin50 degress x csc 50 degrees

1

use properities of logarithms to expand the logarithmic expression as much as possible, where possible evaluate logarithmic expressions without using a calculator: log7(7x)

1+log7 x

evaluate the expression without using a calculator: log10 1000

10

find the measure of the side of the right triangle whose length is designated by a lower case letter: degree= 32, long= 16, bottom=a

10 (a = 16xtan32)

evaluate the expression without using a calculator: ln e¹³x

13x

use the given information to find the exact value of the expression: cos a = - 7/25, a lies in quadrant 3 and sin b = sq rt 21/5, b lies in quadrant 2, find cos (a + b)

14 + 24 sq rt 21/125

use properties of logarithms to expand logarithmic expression as much as possible, where possible evaluate logarithmic expressions without using a calculator: ln (e²/9)

2-ln 9

a child throws a ball with a speed of 3 feet per second at an angle of 67 degrees with the horizontal, express the vector described in terms of i and j, round each component to 3 decimals

2.762 i + 1.172 j

find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s: r=1/2 feet, s= 12 ft

24 radians

find the reference angle for the given angle: -38.4 degrees

38.4 degrees

use properties of logarithms to expand logarithmic expression as much as possible, where possible evaluate logarithmic expressions without using a calculator: log7 x³

3log7 x

convert the angle in degrees to radians, express the answer as a multiple of pi: 270 degrees

3pi/2 radians (270/1 x pi r/180)

complete the identity: sec³x+sec²x tan² x - 2 tan⁴x

3sec² x - 2

determine the amplitude of y = -4 sin x

4

use a sketch to find the exact value of the expression: cos(sin^-1(3/5))

4/5

a surveyor is measuring the distance across a small lake, he has set up his transit on one side of the lake, 110 ft from a piling that is directly across from a pier on the other side of the lake, from the transit the angle between the piling and the pier is 60 degrees, what is the distance between the piling and the pier to the nearest foot?

64 ft

solve the equation on the interval [0, 2 pi): cos 2 x = sq rt 3/2

pi/12, 11 pi/12, 13 pi/12, 23 pi/12

solve the equation on the interval [0, 2 pi): sin²x - cos²x=0

pi/4, 3 pi/4, 5 pi/4, 7 pi/4

find the exact value of the expression if possible, do not use a calculator: cos ^-1 [ cos(- pi/6)]

pi/6

find the reference angle for the given angle: 7 pi/8

pi/8

use the pythagorean theorem to find the length of the missing side, then find secO, give an exact answer with a rationalized denominator: 7-long, 5-bottom

sp rt 74/5

find all the solutions of the equation: 2 cos x - 1 = 0

x = pi/3 + 2 n pi or x = 5 pi/3 + 2 n pi

solve the equation by expressing each side as a power of the same base and then equatiing exponents: 4^(5-3x) = 1/256

{3}

solve the logarithmic equation, be sure to reject any value that is not in the domain of the original logarithmic expressions: log6 (x-2) = 1

{8} (6¹=(x-2))