Answers
The trigonometric measures in the context of this problem are given as follows:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{3-\sqrt{3}}{6}}[/tex][tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{3+\sqrt{3}}{6}}[/tex][tex]\tan{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{3 + \sqrt{3}}}[/tex]How to obtain the trigonometric measures?We are given the measure of the sine for the angle and we need to find the three measures, sine, cosine and tangent for half the angle.
To find the measure for half the angle, the cosine is needed to apply the identities, and the cosine is calculated by the following identity:
sin²(x) + cos²(x) = 1
[tex]\left(\frac{2}{3}\right)^2 + \cos^2{x} = 1[/tex]
[tex]\cos^2{x} = 1 - \frac{4}{9}[/tex]
[tex]\cos{x} = \pm \sqrt{\frac{5}{9}}[/tex]
The angle is in the first quadrant, in which the cosine is positive, hence:
[tex]\cos{x} = \sqrt{\frac{5}{9}}[/tex]
[tex]\cos{x} = \frac{\sqrt{3}}{3}[/tex]
Then, applying an identity, the sine for half the angle is given as follows:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \cos{x}}{2}}[/tex]
Considering the calculated value for the cosine:
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 - \frac{\sqrt{3}}{3}}{2}}[/tex]
[tex]\sin{\left(\frac{x}{2}\right)} = \sqrt{\frac{3-\sqrt{3}}{6}}[/tex]
The identity for the cosine is:
[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{1 + \cos{x}}{2}}[/tex]
Hence:
[tex]\cos{\left(\frac{x}{2}\right)} = \sqrt{\frac{3+\sqrt{3}}{6}}[/tex]
The tangent is obtained by the division of the sine and the cosine. The entire division can be placed into a single square root, and the common denominators of six are simplified, hence:
[tex]\tan{\left(\frac{x}{2}\right)} = \sqrt{\frac{3 - \sqrt{3}}{3 + \sqrt{3}}}[/tex]
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Related Questions
A manufacturer knows that their items have a normally distributed length, with a mean of 13.6 inches, and standard deviation of 2.4 inches. If one item is chosen at random, what is the probability that it is less than 13.1 inches long?
Answers
If one item is chosen at random, the probability that it is less than 13.1 inches long is equal to 0.4176.
How to determine the probability?First of all, we would determine the z-score of this data set. Mathematically, the z-score of a given sample score in a normal distribution can be calculated by using this formula:
Z-score = (x - μ)/σ
Where:
x represents the sample score.σ represents the standard deviation.μ represents the mean score.Substituting the given parameters into the formula, we have;
Z-score, z = (13.1 - 13.6)/(2.4)
Z-score, z = -0.5/2.4
Z-score, z = -0.208
If one item is chosen randomly, the probability that it would be less than 13.1 inches long is given by this mathematical expression:
P(x < 13.1) = P(z < -0.208)
P(z < -0.2083) = 0.4176.
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Use the function, to match each key feature listed below.
Answers
The parabola have a vertex of (-1, -64), it's x-intercept is (3, 0), (-5, 0), it's y-intercept is (0, -60) and it's axis of symmetry is x = -1
ParabolaThis is a curve formed by the intersection of a cone with a plane parallel to a straight line in its surface. It can also be defined as a curve formed by a point moving so that its distance from a fixed point is equal to its distance from a fixed line. In mathematics, it is a conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone.
f(x) = 4x² + 8x - 60
The vertex = (-1, -64)
The x-intercept = (3, 0), (-5, 0)
The y-intercept = (0, -60)
The axis of symmetry = x = -1
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Jonathan, Lucas, and Kiley are in a book
club. This past week, Lucas read six less
pages than Kiley read. Jonathan read two
times the amount of pages that Lucas.
read., Altogether, they read 122 pages.
how many pages did each person read? PLS HELP ME
Answers
Answer:
Lucas 29
Kiley 35
Jonathan 58
Step-by-step explanation:
Kiley = Lucas + 6
Jonathan = 2 x Lucas
Kiley + Lucas + Jonathan = 122
1. Replace Kiley and Jonathan with information we have
K + L + J = 122
(L+6) + L + (2L) = 122
4L + 6 = 122
4L = 122 - 6
L = 116/4 = 29
2. Replace Lucas with 29, you have
Kiley = 29 + 6 = 35
Jonathan = 2 × 29 = 58
The 23 members of the westview journalism club are trying to raise at least 2,100 to buy new publishing desgin software. The members have already raised 1180. How much should each student still raise, on average, to meet the goal?
PLEASEE ASAAPP
Answers
Based on the number of members in the Westview Journalism Club, the amount that each student should raise on average to meet the goal is $40.
How to find the average amount?The amount that the students would have to raise is the amount that is remaining from the amount already raised. This can be found as:
= Amount that club needs to raise - Amount already raised
= 2, 100 - 1, 180
= $920
The average amount that each student has to raised to meet the target would then be based on the number of students in the club which is 23 members.
The average amount is therefore:
= Amount that still needs to be raised / Number of students in the Westview Journalism Club
= 920 / 23
= $40
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Mae has $215 in her account and withdraws $120 one time. Each month her balance is below $100, she pays a $7 fee. Write an equation to express her balance for the next x months, if she makes no deposits or withdrawals .
Answers
Answer: 930$ and the expression is 1000$ - 70$ = 930$
Step-by-step explanation:
if she has under 100 dollars in her bank account every month for 10 months therefore multiplying it by 10 giving you 1000 but we cant forget about the 7$ monthly fee. 7 times 10 = 70 1000- 70 = 930$
Answer:
Step-by-step explanation:
PLS HELP! 50 points
Answers
2
x
=
24
25
Explanation:
sin
2
x
=
2
sin
x
cos
x
cos
x
=
−
√
1
−
sin
2
x
(cosx <0 in quadrant III)
so
cos
x
=
−
√
1
−
9
25
that's
cos
x
=
−
4
5
and
sin
2
x
=
2
(
−
3
5
)
(
−
4
5
)
that's
24
25
A weightlifter lifts
72
k
g
. Which of the following weights is the same as
72
k
g
?
(
1
k
g
=
1
,
000
g
)
720 g
720,000 g
72,000 g
30,000 g
Answers
Answer:
I think 72,000 because since the weightlifter is lifting 72 and it is kg. It is turned into grams. multiplied by 1000 so 72000
For every dozen bagels you buy, the cost increases $15. Which equation represents the cost, y, and the number of bagels, x?
a. y=1.25 + x
b. y=15x
c. x=15y
d. y=1.25x
Answers
The equation that represents the cost, y, and the number of bagels, x is D. y=1.25x
What is an equation?An equation is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario.
It should be noted that 1 dozen = 12
The equation the represents the information will be:
y = (15/12) × x
y = 1.25 × x
y = 1.25x
where x = number of bagels
The equation is y = 1.2x.
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what is the point slope for the top and slope intercept for the bottom? TIA
Answers
Answer:
It looke like the point given is (1,2) and the slope is 3
Point slope form: y - 2 = 3(x -1)
Slope Intercept form: y = [tex]\frac{-1}{3}[/tex]x + 1
Step-by-step explanation:
Point slope As they name implies, you need a point and a slope. The picture is a little hard to read, but it looks like the point plotted is (1,2) The slope is the change in y over the change in x. It looks like from that point to get back on the line only using horizontal and vertical moves, we would move up three and 1 to the right to get back on the line at point (2,5).
The point slope form of a line is:
y - [tex]y_{1}[/tex] = m( x - [tex]x_{1}[/tex]) Plug in 2 for [tex]y_{1}[/tex] and 1 for [tex]x_{1}[/tex] amd 3 for m
y = 2 = 3( (x-1)
Slope intercept form: As the names implies we need a slope and a yi-intercept.
The slope is [tex]\frac{-1}{3}[/tex]. Select a point on the line. If you can only move veritacl and horizontal. You would move down 1 unit and three units to the right. The y intercept is where the line crosses the y axis. In this case 1
y = [tex]\frac{-1}{3} x + 1[/tex]
graph this problem and post the answer
Answers
A graph of the given set of functions has been plotted and shown in the image attached below.
What is a function?In Mathematics, a function simply refers to a mathematical expression which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair.
This ultimately implies that, a function is typically used in mathematics for uniquely mapping an input variable to an output variable.
What is a graph?In Mathematics, a graph is a type of chart that is typically used for the graphical representation of data points on both the horizontal and vertical lines of a cartesian coordinate, which are the x-coordinate (x-axis) and y-coordinate (y-axis).
By critically observing the graph of this set of functions (see attachment), we can reasonably infer and logically deduce that they both represent parallel lines, with function f(x) being transformed to produce function h(x).
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In ΔNOP, \overline{NP} NP is extended through point P to point Q, m∠OPQ = (6x-15)∘, m∠PNO=(2x+18)∘ , and m∠NOP=(2x−13)∘. What is the value of x?
Answers
Simplify
10x - 10 = 180
+10
10x = 190
/10
x = 19
Answer = 19
suppose that $36,000 is deposited in an account and the balance increases to $39,786.15 after 2.5 years. how long will it take for the account to grow to $51,806.67? Assume continous compounding
It will take about _ years for $36,000 to grow to $51,806.67
Answers
It will take about 9.1 years for $36000 to grow with interest to $51806.67 after continuous compounding .
What is continuous compounding?
The process of continuously compounding involves calculating interest and reinvested it into the balance of an account over an infinite number of periods. When they specifically state that the amount is "compounded continuously" in a problem, they should use the continuous compounding formula. The mathematical constant "e" is used in this formula, and its value is roughly equal to 2.7182818. Here is the formula for continuous compounding.
Here the Principal P = $36000 , Total amount A= $39786.15 and
Time t=2.5 yrs.
Now using continuous compounding formula,
=> A = P[tex]e^(rt)[/tex]
=> 39786.15 = 36000*[tex]e^(r*5/2)[/tex]
=> r = 0.04 = 4%
Now A=51806.67 and Again using continuous compounding formula ,
=> A = P[tex]e^(rt)[/tex]
=> 51806.67 = 36000*[tex]e^(0.04*t)[/tex]
=> t =9.1 years.
Therefore it will take about 9.1 years for $36000 to grow to $51806.67 .
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It takes 1 3/4 of a pint of paint
to paint one wall of a room
and 1 4/5 of a pint of paint for
the second wall. How
many pints of paint are
used on the two walls?
Answers
3 11/20
Step-by-step explanation:
1 3/4+1 4/5=
7/4+9/5=
7/4x5/5=35/20
9/5x4/4=36/20
35/20+36/20=71/20=3 11/20
Help please i need your help
Answers
There are 28 groups of 1/5 are in 5.
And, The solution of the expression 5 ÷ 4/5 will be 25/4.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
There is a number line shown in figure.
Now,
In a number line,
The groups of 1/5 in 5 = 28 - 0
= 28
Thus, There are 28 groups of 1/5 are in 5.
And, The expression is,
⇒ 5 ÷ 4/5
Solve aa;
⇒ 5 × 5/4
⇒ 25 / 4
Thus, There are 28 groups of 1/5 are in 5.
And, The solution of the expression 5 ÷ 4/5 will be 25/4.
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The following angles are supplementary to each other.
m∠A = (4x + 30)° and m∠B = (2x − 30)°
Determine x.
15
20
30
60
Answers
According to the sum of the given supplementary angles, we find out that the value of x is 30.
It is given that ∠A and ∠B are supplementary where,
∠A = (4x + 30)° and,
∠B = (2x - 30)°
We have to find out the value of x.
We know that two supplementary angles add up to 180°.
Since ∠A and ∠B are supplementary angles, therefore
∠A + ∠B = 180°
=> (4x + 30) + (2x - 30) = 180°
=> 6x = 180°
=> x = 30
Thus, for the given supplementary angles, the value of x is 30.
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**According to the sum of the given supplementary angles, we find out that the value of x is 30.
It is given that ∠A and ∠B are supplementary where,
∠A = (4x + 30)° and,
∠B = (2x - 30)°
We have to find out the value of x.
We know that two supplementary angles add up to 180°.
Since ∠A and ∠B are supplementary angles, therefore
∠A + ∠B = 180°
=> (4x + 30) + (2x - 30) = 180°
=> 6x = 180°
=> x = 30 **
Use the annuity formula to determine the accumulated amount in the annuity if $80 is invested for 40 years at 6.0% compounded semiannually .
Answers
The accumulated amount in the annuity if $80 is invested for 40 years at 6% is $25,709.04.
What is the accumulated amount?An annuity is when a series of payment is made over a specified period of time. An annuity can either be an ordinary annuity or a beginning annuity. An ordinary annuity is when the series of payment is made at the end of the period. A beginning annuity is when the annuity payment is made at the beginning of the period.
In this question, the annuity would last for 40 years and payments would be made twice a year.
The formula that can be used to determine the accumulated amount is:
Accumulated amount = annuity factor x payment
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest rate = annual interest rate / number of compounding = 6% / 2 = 3%n = number of payment period = number of years x number of compounding = 2 x 40 =80 yearsAccumulated amount = $80 x [{(1.03^80) - 1} / 0.03]
Accumulated amount = $80 x 321.36302
Accumulated amount = $25,709.04
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A baseball is hit into the air, and its height h in feet after t seconds is given by h(t)=-16t^2+ 32t+ 1
a What is the height of the baseball when it is hit?
(b) After how many seconds does the baseball reach its maximum height?
(c) Determine the maximum height of the baseball.
Answers
a) The ball was struck from a height of one foot.
b) It took the object one second to reach its highest point.
c) The baseball may fly up to a height of 17 feet.
Determining the maximum height of an object using functions.Given the function that represents the height of an object in air expressed as;
h(t)=-16t^2+ 32t+ 1
a) The ball is hit at the point where the time is zero. Equating the function to zero.
h(0) = -16t^2+ 32t+ 1
h(0) = -16(0) + 32(0) + 1
h(0) = 1 feet
Hence the ball was hit at the height of 1 feet
b) At maximum height, the velocity of the object is zero. Hence;
v(t) = -32t + 32
v(t) = 0 = -32t + 32
32t = 32
t = 32/32
t = 1sec
Hence the object reach its maximum height after 1 seconds.
c) Substitute t = 1 into the function to have:
h(1)=-16(1)^2+ 32(1)+ 1
h(1) = -16 + 33
h(1) = 17 feet
The maximum height reached by the baseball is 17 feet.
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Brett and Julian sign up to run in the Memorial Day race in their town. There are two different events at this race, a 5k and a 10k. Brett signed up for the 5k and Julian signed up for the 10k. Think of when both Brett and Juan would have finished 3/5 of their races. How many kilometers has each person run at this point?
Answers
The number of kilometers that Brett and Julian would have run at the 3 / 5 point are:
Brett - 3kmJulian - 6 km How to find the number of kilometers?Brett is running the 5k race at the Memorial Day race which means that when he runs 3 / 5 of the race, he would have ran a total of :
= Distance on Brett's race x Proportion ran
= 5 x 3 / 5
= 15 / 3
= 3 km
Julian on the other hand, is in the 10 k race which means that when they get to 3 / 5 of the race, they would have ran:
= 10 x 3 / 5
= 30 / 5
= 6 km
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Triangle DEF, with vertices D(3,-5), E(8,-6), and F(4,-2), is drawn inside a rectangle, as shown below.
Answers
The area of triangle DEF, in square units, is given as follows:
4.08 units squared.
What is the area of a triangle?The area of a triangle is given by half the multiplication of it's base by it's height, as follows:
Area = 0.5 x base x height.
We consider the base as the segment DE, hence it's length is given by the distance between these two points, obtained as follows:
[tex]b = \sqrt{(8 - 3)^2 + (-6 - (-5))^2}[/tex]
[tex]b = \sqrt{26}[/tex]
b = 5.1 units.
The midpoint of the base is given as follows:
(5.5, -5.5).
As it is obtained with the mean of the coordinates.
The height is the distance between the midpoint of the base and the other vertex, hence:
[tex]h = \sqrt{(5.5 - 4)^2 + (-5.5 - (-5))^2}[/tex]
h = 1.6 units.
Hence the area of the triangle is obtained as follows:
Area = 0.5 x 5.1 x 1.6 = 4.08 units squared.
Missing InformationThe problem asks for the area of the triangle, as shown in the image given at the end of the answer.
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You are on vacation in New York City, and you need to get around town to different locations. Below are the rates for 2 different cab companies, locally dubbed "The Red Cabs" and "The Green Cabs".
How many miles would you travel for the cost to be the same for both cabs and how much will that cost you?
What is the difference in cost between the two companies if you need to travel 8 miles?
After seeing all your vacation photos on S n a p b o o k, your cousin decided they wanted to visit New York City also. They asked for some general advice on traveling around the city. What advice would you give them about which cab company they should use when they travel around the city?
Answers
1. When you travel 3 miles in New York City, the cost of The Red Cabs and The Green Cabs would be the same.
2. The difference in cost between the two companies if you need to travel 8 miles is $5.
3. It is cheaper to travel with Red Cabs than Green Cabs if the total distance is more than 3 miles.
How is the difference in cost determined?First, for the costs of both companies to be equal, we set the two cost equations to be equal to each other and solve for the unknown variable (miles).
Based on the equations for the two companies, we can determine the total costs of using any option.
The difference between the total costs is obtained by subtracting.
Let cost of Green Cab = 2 + 2m
Let cost of Red Cab = 5 + m
For the two companies costs to be equal, 2 + 2m = 5 + m
= 2m - m = 5 - 2
m = 3
The costs of traveling 8 miles:Green Cab = 2 + 2m
= 2 + 2(8)
= 18
= $18
Red Cab = 5 + m
= 5 + 8
= 13
= $13
The difference in cost = $5 ($18 - $13)
Thus, based on the solution to the cost equations, cost-conscious travelers should prefer Red Cabs to Green Cabs when they travel in New York City for a distance exceeding 3 miles.
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A square shape garden has a side of 6 feet. What is its area?
Answers
Sari put $1230 in a savings account at a simple interest rate of 4.5% per year. Avi put $1105 in a savings account at a simple interest rate of 5.5% per year. Who will have earned more interest after 4 years? How much more?
Answers
Answer:
Avi
$21.61
Step-by-step explanation:
I = Prt
Sari: I = Prt = $1230 × 0.045 × 4 = $221.40
Avi: I = Prt = $1105 × 0.055 × 4 = $243.01
$243.01 - $221.40 = $21.61
Answer:
Avi
$21.61
A cyclist cycles for x hours at 12 km/hour and y hours at 18km/hour.All together she cycles 78km in 5 hours.Find x and y 08106764242
Answers
The values of x = 2 & y = 3.
A cyclist cycles for x hours at 12 km/hour and y hours at 18km/hour.
To find x and y:
Given kilometers = 78km
Time = 5 hours
So, we know that the
Distance Equation is 12x + 18y = 78 km
Time Equation is x + y = 5 hours
Multiply the equation through the Time Equation by 12:
12x + 18y = 78
12x + 12y = 60
Subtract and solve for "y":
6y = 18
y = 3
So now we have to Solve for "x" using x + y = 5
x + 3 = 5
x = 2
Hence the answer is the values of x = 2 & y = 3.
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Divide. Write your answer on the simplest form.
7/15 ÷ 3/10
Answers
Answer:
14/9
Step-by-step explanation:7/15 divided by 3/10 equals 14/9
Consider a collection of envelopes consisting of two red envelopes three blue envelopes one green envelope and one yellow envelope if two envelopes are selected at random with replacement determine the probability that at least one envelope is red envelope
Answers
The probability that at least one envelope is red envelope is 2/7.
Given:
A collection of envelopes consisting of two red envelopes three blue envelopes one green envelope and one yellow envelope if two envelopes are selected at random with replacement.
Total = 2 + 3 + 1 + 1
= 5 + 2
= 7
Number of possible outcomes = 7
Number of favorable outcomes = 2
Probability that at least one envelope is red envelope is = 2/7
= 0.2857
Therefore the probability that at least one envelope is red envelope is 2/7.
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can someone pls help me i don’t get this
Answers
The perimeter of a quadrilateral (four-sided polygon) is 29 inches. The longest side is twice as long as the shortest side. The other two sides are equally long and are 2 inches longer than the
shortest side. Find the length of all four sides.
The length of the shortest side is ____?
the length of each of the two equal sides is____?
and the length of the longest side is___?
Answers
The length shortest side of the quadrilateral is 5 inches
The length longest side of the quadrilateral is 10 inches;
The length of the other two sides of the quadrilateral will be 7 inches.
What is the perimeter of a quadrilateral?
The length of a quadrilateral's boundary, or what remains after joining all four of its sides to form a single line segment, is referred to as the quadrilateral's perimeter. As a result, a quadrilateral's perimeter is measured in the same linear units as its sides, such as meters, inches, centimeters, etc. We are aware that a quadrilateral's perimeter can be calculated by summing all of its side lengths. This can be stated using a straightforward formula. For instance, the formula for a quadrilateral ABCD's perimeter can be written as,
AB + BC + CD + DA = The perimeter
Let the length shortest side of the quadrilateral be x;
So, According to the question,
length longest side of the quadrilateral will be: 2x;
length of the other two sides of the quadrilateral will be (x +2) inches.
We have given that:
The perimeter of the quadrilateral is 29 inches.
So,
⇒ 2x + (x+2) +(x+2)+x = 29
⇒5x + 4 = 29
⇒5x = 29-4
⇒5x = 25
⇒x =5;
Hence,
the length shortest side of the quadrilateral is 5 inches
length longest side of the quadrilateral is 10 inches;
length of the other two sides of the quadrilateral will be 7 inches.
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Find the value of x so that the ratios are equivalent.
9:x and 54:8
Answers
Answer:
1 1/3
Step-by-step explanation:
9*6=54
So 6x=8
x=1 1/3
Which pairs of fractions include a fraction equivalent to 1/2 and a fraction equivalent to 3/4 help me PLEASE
Answers
Answer:
4/8 and 8/16
Step-by-step explanation:
Multiply 1/2 and 3/4 to find a common fraction between them (4/8) then multiply that fraction by any number greater than 1 to find another fraction equivalent
. What is the present value of an annuity that pays $100 per month for 10 years if money worth 6%
compounded monthly?
Answers
The present value of an annuity is $736.01
We know that the formula for Present Value of Annuity:
PV = P * [1 - (1 + r)^(-n)]/r
where, PV = present value of an ordinary annuity
P = value of each payment
r = interest rate per period
n = number of periods
From given data:
P = $100, n = 10 and r = 0.06
So, the present value would be,
PV = 100 * [1 - (1 + 0.06)^(-10)]/0.06
PV = 100 * 7.3601
PV = $736.01
Therefore, the present value of an annuity is $736.01
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