Write a recursive sequence that represents the sequence defined by the following explicit formula:
Answers
Answer:
0,-2,-4,-6,-8, ...
Step-by-step explanation:
In order to find the sequence represented by the given explicit formula we have to plug in the values of n
Given formula is:
[tex]a_n=2-2n\\a_1=2-2(1)\\=2-2=0\\a_2=2-2(2)\\=2-4\\=-2\\a_3=2-2(3)\\=2-6\\=-4\\a_4=2-2(4)\\=2-8\\=-6\\a_5=2-2(5)\\=2-10\\=-8[/tex]
So the explicit sequence is:
0,-2,-4,-6,-8,..........
Answer:
kmkmlklm
Step-by-step explanation:
Related Questions
Ammo 67 match each definition on the left with the correct term.
Answers
The amount of a person's paycheck p varies directly with the number of hours worked t. For 22 hours of work, the paycheck is $126.50. Write an equation for the relationship between hours of work and pay.
Answers
Answer:
5.75t=p
Step-by-step explanation:
22p=126.50/22
p=5.75 per hour
5.75t=p
The $5.75 is earned for every hour worked.
Graph the following system of linear inequalities. Identify at least two points in the solution: y < 5 - 2x | x + 5y > -7
Answers
Answer:
(1,2) and (2,-1)
Step-by-step explanation:
we have
[tex]y< 5-2x[/tex] ----> inequality A
The solution of the inequality A is the shaded area below the dashed line [tex]y=5-2x[/tex]
[tex]x+5y>-7[/tex] ----> inequality B
The solution of the inequality B is the shaded area above the dashed line [tex]x+5y=-7[/tex]
The solution of the system of inequalities is the shaded area between the two dashed lines
If a ordered pair is a solution of the system of inequalities, then the ordered pair must lie on the shaded area
Two points in the solution are
(1,2) and (2,-1)
see the attached figure
Find the solution(s) to the system of equations. Select all that apply
y=x^2-1
y=2x-2
Answers
Step-by-step explanation:
y = x^2 - 1
y = 2x - 2
x^2 - 1 = 2x - 2
x^2 - 2x + 1 = 0
(x - 1)^2 = 0
x = 1
Because x = 1 the answer must be D (1, 0).
Answer:
(1,0)
Step-by-step explanation:
Replace x values and y values with the answer choices.
[tex]y=(1)^2- 1 = 0[/tex]
[tex]y= 2(1)-2=0[/tex]
Since the coordinate pair [tex](1,0)[/tex] it is safe to assume that D will be the correct answer.
Use the graph to confirm that the solution is (1,0)
The blue line intersects at (1,0), so the solution for the system of equations is (1,0)
5?10 5 5, dot, 10, start superscript, 5, end superscript is how many times as large as 1\cdot10^51?10 5 1, dot, 10, start superscript, 5, end superscript?
Answers
Answer:
5 times as large
Step-by-step explanation:
The ratio of the two numbers tells you how many times as large one is as the other:
[tex]\dfrac{5\cdot 10^5}{1\cdot 10^5}=\dfrac{5}{1}\cdot\dfrac{10^5}{10^5}=5[/tex]
Answer: 5 is the correct answer!
Step-by-step explanation:
This question is that I need help with is "Finding angle measures between intersecting lines"
Answers
Answer:
Step-by-step explanation:
Alright. Looking at the graph, angle CGE is equal to angle FGD.
FGD = 90-73
FGD = 17
Therefore, as angle CGE is equal to angle FGD, angle CGE is equal to 17.
Answer: 17 degrees.
By applying angle relationships and recognizing the equality of angles, we determined that angle CGE measures 17 degrees, as it is equal to angle FGD in the context of the graph.
In geometry, analyzing angles within shapes and figures is fundamental. Here, we have a graph depicting intersecting lines and angles. Let's break down the steps and reasoning for finding the measurement of angle CGE:
1. Angle CGE and Angle FGD:
First, we observe the graph and notice that angle CGE is mentioned to be equal to angle FGD. This equivalence allows us to focus on finding the measurement of angle FGD, which, in turn, will be the measurement of angle CGE.
2. Calculation of Angle FGD:
We know that angle FGD is related to a right angle, as indicated by the angle symbol "90" next to it. To calculate the measurement of angle FGD, we use the fact that the sum of angles around a point is 360 degrees, and the angle opposite to a right angle is complementary, totaling 90 degrees.
So, we calculate angle FGD as follows:
Angle FGD = 90 degrees (right angle) - 73 degrees (given angle)
Angle FGD = 17 degrees
3. Angle CGE:
Since angle CGE is stated to be equal to angle FGD, we determine that angle CGE is also 17 degrees.
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Peter has saved $10. He doubles the amount he saves each week. Does this represent a exponential function? If so, please write down the function.
Answers
Answer:
[tex]y=10(2)^x[/tex]
Step-by-step explanation:
This is an exponential function. Peter is not simply adding $10 a week to his savings, he's doubling each value each week. The first case of adding $10 a week is linear.
The standard form of an exponential equation is
[tex]y=a(b)^x[/tex]
where a is the initial amount and b is the growth or decay factor. We know a = 10 because we are told he started with $10. After the first week he would have $20 then because $10 doubled is $20. That coordinate is (1, 20). We will use that in place of x and y and solve for b:
[tex]20=10(b)^1[/tex]
b to the first is just b, so what we have essentially is 10b = 20, so b = 2. The equation then is:
[tex]y=10(2)^x[/tex]
Sam recorded the number of points she scored on each weekly test out of the number of points possible for the test. In which two subjects did Sam score the lowest? Choose exactly two answers that are correct. Language Arts History Science Math Sam's Test Scores Math 2225 Science 1920 Language Arts 84100 History 910
Answers
Makes no sense.
He scored the lowest in History and Science. Compare the numbers to help you prepare the obvious chart.
Sam scored the lowest in Language Arts with 84% and Math with 88%.
To determine this, We need to calculate Sam's percentage scores for each subject:
Math: [tex]\frac{25}{25}*100[/tex] = 88%Science: [tex]\frac{19}{20}*100[/tex] = 95%Language Arts: [tex]\frac{84}{100} *100[/tex] = 84%History:[tex]\frac{9}{10}*100[/tex] = 90%Based on these calculations, the two subjects where Sam scored the lowest are Language Arts (84%) and Math (88%).
complete question:
Sam recorded the number of points she scored on each weekly test out of the number of points possible for the test.In which two subjects did Sam score the lowest?Choose exactly two answers that are correct. Sam's Test Scores in Math 22/25 ,in Science 19/20 ,in Language Arts 84/100 ,in History 9/10.
The sum of two consecutive integers is 9. Find the numbers.
Answers
[tex]n,n+1[/tex] - two consecutive integers
[tex]n+n+1=9\\2n=8\\n=4\\n+1=5[/tex]
4 and 5
Given the following linear function identify the slope in the Y intercept of the function
Answers
Answer:
Choice #2
Step-by-step explanation:
Your linear function is in the slope-intercept form of a line, y = mx + b, where m is the value of the slope, and b is the value of the y-intercept. The number in the m position in your equation is 1/6, and thee number in the b position in your equation is 7. So the second choice is the one you want.
the coordinates of a triangle are given as A(3,2), B(-4,1),C(-3,-2). what are the coordinates of the image after the triangle is reflected in the line y=x?
Answers
A reflection is a transformation where the mirror image of a figure is shown directly opposite its line of reflection.
To find an image that has been reflected across the line y = x, switch the x- and y-coordinates.
Therefore, the rule for reflecting an image across the line y = x can be described as (x, y) → (y, x).
Now, apply rule to coordinates ABC:
A': (2, 3)
B': (1, -4)
C': (-2, -3)
Answer:
got it right on odyssey
A'(2,3)
B'(1,-4)
C'(-2,-3)
Step-by-step explanation:
Please help me with this?
Answers
Answer:
A) The population in the year that she was born.
Step-by-step explanation:
The multiplier of an exponential function is the value that function has when the exponent is zero -- the initial value. The initial value of population in this context is the population in the year Adriana was born.
Vlad spent 20 minutes on his history homework and then completely solved x math problems that each took 2 minutes to complete. What is the equation that can be used to find the value of y
Answers
Answer:
y = 2x + 20
Step-by-step explanation:
2 * x is the number of minutes from the math problems
20 is the number of minutes for the history homework
Answer:
y= 2x=20
Step-by-step explanation:
A beam of gamma rays is to be used to treat a tumor known to be 1.8cm beneath the patients skin to avoid damaging a vital organ the radiologist moves the source over 5.3 cm at what angle to the patient's skin must the radiologist aim the gamma ray source to hit the tumor?
Answers
Answer:
about 18.8°
Step-by-step explanation:
The depth of the tumor (1.8 cm) is the leg of the right triangle that is opposite the angle of interest. The offset distance (5.3 cm) is the adjacent leg of the triangle.
We know that ...
tan(angle) = opposite/adjacent = (1.8 cm)/(5.3 cm) = 18/53
Then the angle is found from ...
angle = arctan(18/53) ≈ 18.76°
_____
The arctangent, or inverse of the tangent function, is also written tan⁻¹. It may be a "second function" of your calculator's tan key.
The angle to aim the gamma ray source to hit the tumor 1.8cm beneath the skin without damaging the vital organ, while moving the source over 5.3cm, can be determined by calculating the inverse tangent (arctan) of the ratio 1.8cm / 5.3cm.
Explanation:In physics, this problem involves the concept of triangles and trigonometry. Since a right triangle can be formed in this configuration with the distance beneath the patient's skin being one side (1.8cm), the distance over which the radiologist moves the gamma ray source being the second side (5.3cm), the angle required would be an inverse tangent (arctan) of the ratio between these two lengths.
The calculation is thus as follows:
Firstly, set up the ratio of the opposite over the adjacent side of the triangle. This equates to 1.8cm / 5.3cm.
Next, compute the inverse tangent (arctan) of this ratio. You can compute this using a scientific calculator. The answer will be in degrees.
Thus, the required angle to aim the gamma ray source to hit the tumor without damaging the vital organ will be the result of the above calculation.
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A car, originally valued at 70,000 in 2006 depreciates exponentially at a rate of 4% each year. Round the expected value of the car in 2018 to the nearest dollar. Round the expeated value of the car in 2018 to the nearest dollar
Answers
Answer:
$42,890
Step-by-step explanation:
The standard form for an exponential equation is
[tex]y=a(b)^x[/tex]
where a is the initial amount value and b is the growth rate or decay rate and t is the time in years. Since we are dealing with money amounts AND this is a decay problem, we can rewrite accordingly:
[tex]A(t)=a(1-r)^t[/tex]
where A(t) is the amount after the depreciation occurs, r is the interest rate in decimal form, and t is the time in years. We know the initial amount (70,000) and the interest rate (.04), but we need to figure out what t is. If the car was bought in 2006 and we want its value in 2018, a total o 12 years has gone by. Therefore, our equation becomes:
[tex]A(t)=70,000(1-.04)^{12}[/tex] or, after some simplification:
[tex]A(t)=70,000(.96)^{12}[/tex]
First rais .96 to the 12th power to get
A(t) = 70,000(.6127097573)
and then multiply.
A(t) = $42,890
For the functions f(x) = x^2+ 8x + 2 and g(x) = -5+9, find (f•g)(x)and (f.g)(-1).
Answers
Answer: (f·g)(x) = -5x³ - 31x² + 62x + 18
(f·g)(-1) = -70
(fog)(x) = 25x² - 130x + 155
(fog)(-1) = 310
Step-by-step explanation:
f(x) = x² + 8x + 2 g(x) = -5x + 9
(f·g)(x) = (x² + 8x + 2)(-5x + 9)
= -5x³ + 9x²
- 40x² + 72x
- 10x + 18
= -5x³ - 31x² + 62x + 18
(f·g)(-1)= -5(-1)³ - 31(-1)² + 62(-1) + 18
= -5(-1) - 31(1) - 62 + 18
= 5 - 31 - 62 + 18
= -70
****************************************************************************************
(fog)(x) = (-5x + 9)² + 8(-5x + 9) + 2
= 25x² - 90x + 81
- 40x + 72
+ 2
= 25x² - 130x + 155
(fog)(-1) = 25(-1)² - 130(-1) + 155
= 25 + 130 + 155
= 310
It wasn't clear if you wanted multiplication or composition so I solved both.
The composition (f⋅g)(x) is 50, and the product of functions (f.g)(-1) is -20.
To find the composition of the functions f(x) and g(x), denoted as (f⋅g)(x), we first need to determine g(x). Given g(x) = -5 + 9, we simply add these numbers together to get g(x) = 4. With g(x) found, we can now substitute g(x) into f(x) to find the composition. To do this, wherever there is an x in f(x), we replace it with 4.
So, (f⋅g)(x) = f(g(x)) = f(4) = 4^2 + 8(4) + 2 = 16 + 32 + 2 = 50.
Next, to find (f.g)(-1), which represents the product of the two functions evaluated at x = -1, we evaluate both f(-1) and g(-1). We already know that g(x) is constant at 4, so g(-1) = 4. Now we find f(-1):
f(-1) = (-1)^2 + 8(-1) + 2 = 1 - 8 + 2 = -5.
Thus, the product at x = -1 is:
(f.g)(-1) = f(-1) ⋅ g(-1) = (-5) ⋅ 4 = -20.
The measure of an angle's supplement is 24 less than twice the measure of the angle. Find the measure of the angle and its supplement.
a. 38, 52
b. 52, 38
c. 68, 112
d. 112, 68
Answers
Answer:
c. 68, 112
Step-by-step explanation:
The angle is our unknown, so we will call it x. If the angle is x, then its supplement is 180 - x (supplementary angles add up to equal 180). The word "is" means "equals", so putting the equation together looks like this:
180 - x = 2x - 24. Add x to both sides and at the same time add 24 to both sides (combining like terms, in other words):
204 = 3x so
x = 68
x is the angle measure, so the angle is 68 and its supplement is 180 - 68 = 112
The angle will be 68 and its supplement will be 112.
It is given that supplement of an angle is 24 less than twice the measure of the angle.
We have to find out the measure of the angle and its supplement.
What are the supplementary angles ?
The supplementary angles are the angles whose sum is equal to 180° i.e., sum of angles 150° and 30° equal to 180°.
The angle is unknown. Let's assume angle be x.
We know that , supplementary angles add up to equal 180.
If the angle is x, then its supplement will be :
180 - x ----------- Equation 1
Also ;
angle's supplement is equal to ;
2 × x - 24 ----------- Equation 2
Keeping both the equations equal ;
180 - x = 2x - 24.
180 + 24 = 3x
204 = 3x
x = 68
Supplement will be ; 180-x = 180 - 68 = 112
Thus , x is the angle measure, so the angle is 68 and its supplement is 112.
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If $34,500 is invested at 6.9% for 30 years, find the future value if the interest is compounded:
A-annually
E- daily
Answers
Answer:
A) 255,358.46
E) 273,353.92
Step-by-step explanation:
The formula for future value of principal P at interest rate r per year compounded n times per year for t years is ...
FV = P(1 +r/n)^(nt)
Filling in the numbers and doing the arithmetic, we have ...
A) FV = $34,500(1 + 0.069)^30 ≈ $255,358.46
__
E) FV = $34,500(1 + 0.069/365)^(365·30) ≈ $273,352.92
PLEASE HELP ME FAST
The table shows values for functions f(x) and g(x). What are the two solutions to f(x) = g(x)? (Hint: What X values have the same y values?)
x = -3
x = -2
x = -1
x = 0
x = 1
x = 2
x = 3
Answers
Answer:
The Answer is: -1 and 1
Step-by-step explanation:
Harry asked a sample of 4 people how many siblings they had. Here are their responses:
0, 0, 1, 3
The mean is x-bar = 1 sibling.
What formula gives the standard deviation?
Answers
Answer:
Formula of standard deviation = √Variance
Standard deviation = 1
Step-by-step explanation:
X-bar is the variance
Therefore, the answer would be
√X-bar
√1 = 1
!!
Answer:
C
Step-by-step explanation:
Priscilla graphed function g, a transformation of the quadratic parent function f(x)=1/2f(x+2)-1 which statement is correct?
A.)Priscilla made a mistake when applying the horizontal shift.
B.)Priscilla made a mistake when applying the vertical shift.
C.)Priscilla made a mistake when applying the vertical compression.
D.)Priscilla correctly graphed the transformed function .
Answers
Answer:
Step-by-step explanation:
A.)Priscilla made a mistake when applying the horizontal shift. That "x+2" indicates a horiz. shift to the LEFT, not to the right.
Answer:
Option A.
Step-by-step explanation:
Priscilla graphed function g(x), a transformation of the quadratic parent function f (x) = [tex]\frac{1}{2}[/tex] f ( x+2 ) - 1
(1) She graphed the vertical compression correctly
(2) Priscilla graphed the vertical shift of (-1) means 1 unit downwards on y-axis correctly.
(3) In the last she made a mistake because for f(x+2) the parent function should have been shifted left on x-axis by two units f (x) ⇒ f[ x- (-2)]
Therefore Option A will be the answer.
please help!
The total number of fungal spores can be found using an infinite geometric series where a1 = 11 and the common ratio is 2. Find the sum of this infinite series that will be the upper limit of the fungal spores.
465
280
The series is divergent
125
Answers
Answer:
The series is divergent
Step-by-step explanation:
An infinite geometric series with a common ratio greater than 1 does not converge to a sum. The series is divergent.
the number of bacteria growing in an incubator culture increases with time according to b(x)=8500(3)^x where x is the time in days. After how many days will the number of bacteria in culture be 2,065,500?
Answers
Answer:
5 days
Step-by-step explanation:
Put the given number in the formula and solve for x.
2065500 = 8500·3^x
243 = 3^x . . . . . . . . . . . divide by 8500
log(243) = x·log(3) . . . take logarithms
log(243)/log(3) = x = 5 . . . . divide by the coefficient of x
After 5 days, the number of bacteria in culture will be 2,065,500.
Woodland Mound Park sells annual visitor passes for $12.50. Last year the park raised $53,500 in annual visitor pass sales. How many annual visitor passes were sold?
Answers
Answer:
4280 passes were sold
Step-by-step explanation:
The unknown we are looking for the number of passes that were sold. We don't know how MANY were sold, but we do know that the cost of ONE is 12.50. We represent the number of passes at that price per pass as
12.50x
We also know that money earned from the sale of this unknown number of passes came to 53,500. Our equation then becomes
12.50x = 53,500
Solving for x, we divide both sides by 12.50 to get that
x = 4280
Final answer:
To find the number of annual visitor passes sold at Woodland Mound Park, divide the total sales ($53,500) by the price per pass ($12.50), resulting in 4,280 passes sold.
Explanation:
The question asks how many annual visitor passes were sold at Woodland Mound Park last year if each pass costs $12.50 and the total amount raised from pass sales was $53,500. To find the number of passes sold, you'll need to divide the total sales by the price per pass. Here is the step-by-step calculation:
Identify the total revenue from sales: $53,500.Identify the price of one annual visitor pass: $12.50.Divide the total revenue by the price per pass to get the quantity sold: $53,500 / $12.50 = 4,280.Isaac downloaded 7 ringtones. Each polyphonic ringtone costs $3.25, and each standard ringtone costs $1.50. If he spends a total of $21 on ringtones, find the number of polyphonic and standard ringtones he downloaded.
Answers
Answer:
[tex]6\ polyphonic\ ringtones[/tex] and [tex]1\ standard\ ringtone[/tex]
Step-by-step explanation:
Let
x -----> the number of polyphonic ringtones
y ----> the number of standard ringtones
we know that
[tex]x+y=7[/tex]
[tex]x=7-y[/tex] -----> equation A
[tex]3.25x+1.50y=21[/tex] -----> equation B
Solve the system of equations by substitution
Substitute equation A in equation B and solve for y
[tex]3.25(7-y)+1.50y=21[/tex]
[tex]22.75-3.25y+1.50y=21[/tex]
[tex]3.25y-1.50y=22.75-21[/tex]
[tex]1.75y=1.75[/tex]
[tex]y=1\ standard\ ringtones[/tex]
Find the value of x
[tex]x=7-1=6\ polyphonic\ ringtones[/tex]
Answer:
6 polyphonic, 1 standard
Step-by-step explanation:
Julie has 5 cherry lollipops,1 lime lollipops, and 2 grape lollipops in a bag. She is going to select one lollipop, replace the lollipop in the bag, and then select a second one. What is the probability that Julie will select a cherry lollipop and then a lollipop other than grape?
a.)6/8
b.)11/16
c.)15/32
d.)10/64
Answers
Answer: [tex]\dfrac{15}{32}[/tex]
Step-by-step explanation:
Given : The number of cherry lollipop = 5
The total number of lollipop = 8
the number of lollipops other than grape =6
The probability of selecting a cherry lollipop is given by :_
[tex]\text{P(Cherry)}=\dfrac{5}{8}[/tex]
The probability of selecting a lollipop other than grape is given by :_
[tex]\text{P(Other than grape)}=\dfrac{6}{8}[/tex]
Since, there is replacement , then the events are independent of each other.
Now, the probability that Julie will select a cherry lollipop and then a lollipop other than grape is given by :-
[tex]\text{P(Cherry and other than grape)}=\dfrac{5}{8}\times\dfrac{6}{8}=\dfrac{15}{32}[/tex]
Hence, the required probability =[tex]\dfrac{15}{32}[/tex]
find f(g(x)) for the functions f(x) = (x+1)^3 -5 and g(x) = ^3sqrt(x) -1
are these functions inverses?
Answers
Answer:
f(g(x)) = x-5the functions are NOT inversesStep-by-step explanation:
Substitute g(x) for x in f(x) and evaluate:
f(g(x)) = f(x^(1/3) -1)
= ((x^(1/3) -1) +1)^3 -5
= (x^(1/3))^3 -5
= x^(3/3) -5
f(g(x)) = x -5
This is confirmed by a graphing calculator. (See attached.)
If the functions were inverses, the value of f(g(x)) would be x. It is not, so the functions are not inverses.
What is the equation of a line, in general form, that passes through point (1, -2) and has a slope of 1/3
3x - y - 7 = 0
x - 3y + 7 = 0
x - 3y - 7 = 0
Answers
Answer:
x-3y-7=0
Step-by-step explanation:
Given
m=1/3
The standard form of point slop form is:
y=mx+b
To find the value of b, we will put the point in the standard form
So,
[tex]-2=\frac{1}{3}(1)+b[/tex]
Solving for b[tex]-2=\frac{1}{3}+b\\-2-\frac{1}{3}=b\\\frac{-6-1}{3}=b\\b=\frac{-7}{3}[/tex]
Putting the values of b and m in standard form:
[tex]y=\frac{1}{3}x+\frac{-7}{3}\\y=\frac{1}{3}x-\frac{7}{3}Multiplying\ both\ sides\ by\ 3\\3y=x-7\\-x+3y+7=0\\Can\ also\ be\ written\ as\\x-3y-7=0[/tex]
What is the solution of √1-3x=x+3
Answers
Answer:
x = -1
Step-by-step explanation:
[tex]\sqrt{1-3x}=x+3\\\\1-3x=x^2+6x+9 \qquad\text{square both sides}\\\\x^2+9x+8=0 \qquad\text{put in standard form}\\\\(x+8)(x+1)=0 \qquad\text{factor}\\\\\left \{ {{x=-8} \atop {x=-1}} \right. \qquad\text{values that make the factors zero}[/tex]
Only the solution x = -1 will work for this equation. The other solution is extraneous.
If sinθ = 2/3 and θ is located in Quadrant II, then tan2θ = _____.
Answers
Answer:
[tex] -4 \sqrt{5} [/tex]
Step-by-step explanation:
Quadrant 2 means cosine is negative.
So [tex] \sin(\theta)=\frac{2}{3} =\frac{\text{ opp }}{\text{ hyp }} [/tex]
So the adjacent side is [tex] \sqrt{3^2-2^2}=\sqrt{9-4}=\sqrt{5} [/tex]
So [tex] \cos(\theta)=-\frac{\sqrt{5}}{3} [/tex]
Now to find [tex] \tan(2 \theta) [/tex]
[tex] \tan(2 \theta) =\frac{2\tan(\theta)}{1-\tan^2(\theta)}[/tex]
We will need [tex] \tan(\theta) [/tex] before proceeding.
[tex] \tan(\theta) =\frac{\sin(\theta)}{\cos(\theta)}=\frac{\frac{2}{3}}{\frac{-\sqrt{5}}{3}}=\frac{-2}{\sqrt{5} } [/tex]
Now plug it in and the rest is algebra.
[tex] \tan(2 \theta) =\frac{2\tan(\theta)}{1-\tan^2(\theta)} =\frac{2 (\frac{-2}{\sqrt{5}}}{1-\frac{4}{5}} [/tex]
Now the algebra, the simplifying.... We need to get rid of the compound fraction. We will multiply top and bottom by [tex] 5 \sqrt{5} [/tex]
This will give us
[tex] \frac{-4(5)}{5 \sqrt{5}-4 \sqrt{5}} [/tex]
[tex] \frac{-20}{\sqrt{5}} [/tex]
Multiply top and bottom by [tex] \sqrt{5} [/tex]
[tex] \frac{-20 \sqrt{5}}{5} [/tex]
The answer reduces to
[tex] -4 \sqrt{5} [/tex]
Final answer:
To find tan2θ when sinθ = 2/3 and θ is in Quadrant II, we use the Pythagorean identity to find cosθ and the double angle identity for tangent. After calculation, tan2θ = -4√5.
Explanation:
If sinθ = 2/3 and θ is located in Quadrant II, we first have to find cosθ and then use the double angle identity for tangent to find tan2θ. Since sinθ is positive in Quadrant II and cosθ must be negative (as the x-values are negative in Quadrant II), we can use the Pythagorean identity sin2θ + cos2θ = 1 to find cosθ. Thus, cosθ = -√(1 - sin2θ) = -√(1 - (2/3)2) = -√(1 - 4/9) = -√(5/9) = -√5/3.
The double angle identity for tangent is tan2θ = 2tanθ / (1 - tan2θ). But first, we find tanθ = sinθ/cosθ = (2/3) / (-√5/3) = -2/√5. Then, tan2θ = 2(-2/√5) / (1 - (-2/√5)2) = -4/√5 / (1 - 4/5) = -4/√5 / (1/5) = -20/√5. Simplifying further, we multiply by √5/√5 to rationalize the denominator, which gives us tan2θ = -20√5/5 = -4√5.