Answers
Answer:
Option A: [tex]\{0,\frac{\pi}{2},\frac{3\pi}{2}\}[/tex]
Step-by-step explanation:
[tex]sin^2(\theta)=2sin^2(\frac{\theta}{2}), [0,2\pi)[/tex]
[tex]sin^2(\theta)=2sin^2(\frac{\theta}{2})[/tex]
[tex]sin^2(\theta)=2sin(\frac{\theta}{2})sin(\frac{\theta}{2})[/tex]
[tex]sin^2(\theta)=2(\sqrt{\frac{1-cos(\theta)}{2}})(\sqrt{\frac{1-cos(\theta)}{2}})[/tex]
[tex]sin^2(\theta)=2(\frac{1-cos(\theta)}{2})[/tex]
[tex]sin^2(\theta)=\frac{2-2cos(\theta)}{2}[/tex]
[tex]sin^2(\theta)=1-cos(\theta)[/tex]
[tex]1-cos^2(\theta)=1-cos(\theta)[/tex]
[tex]-cos^2(\theta)=-cos(\theta)[/tex]
[tex]cos^2(\theta)=cos(\theta)[/tex]
[tex]cos^2(\theta)-cos(\theta)=0[/tex]
[tex]cos(\theta)[cos(\theta)-1]=0[/tex]
[tex]cos(\theta)=0[/tex]
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}[/tex]
[tex]cos(\theta)-1=0[/tex]
[tex]cos(\theta)=1[/tex]
[tex]\theta=0[/tex]
Therefore, the solutions contained within the interval are [tex]\{0,\frac{\pi}{2},\frac{3\pi}{2}\}[/tex]
Helpful Tips:
Half-Angle Formula: [tex]sin(\frac{\theta}{2})=\pm\sqrt{\frac{1-cos(\theta)}{2}}[/tex]
Pythagorean Identity: [tex]sin^2(\theta)+cos^2(\theta)=1,sin^2(\theta)=1-cos^2(\theta),cos^2(\theta)=1-sin^2(\theta)[/tex]
[tex]\sin^2 \theta = 2 \sin^2 \left(\dfrac{\theta}2 \right)\\\\\implies \sin^2 \theta = 1- \cos 2 \cdot \dfrac{\theta}2\\\\\implies \sin^2 \theta = 1- \cos \theta \\\\\implies 1-\cos^2 \theta = 1 - \cos \theta \\\\\implies -\cos^2 \theta - \cos \theta = 0\\\\\implies \cos^2 \theta - \cos \theta = 0\\\\\implies \cos \theta( \cos \theta -1) = 0\\\\\\\text{Now,}\\\\\cos \theta = 0\\\\\implies \theta = n\pi + \dfrac{\pi}2\\\\\text{For n = 0,1 and}~ [0.2\pi)\\\\[/tex]
[tex]\theta = \dfrac{\pi}2, \dfrac{3\pi}2[/tex]
[tex]\text{Again,} \\\\\cos \theta -1= 0\\\\\implies \cos \theta = 1\\\\\implies \theta = 2n\pi\\\\\text{For n= 0 and}~ [0,2\pi)\\\\\theta = 0\\\\\text{Combine solutions,}\\\\\theta = 0, \dfrac{\pi}2, \dfrac{3\pi}2[/tex]
Related Questions
- z > 8 equivalent inequality
Answers
Answer: z < -8
Step-by-step explanation:
Since z is negative, divide both sides by -1, which leaves you with z > -8.
Multiplying or dividing an inequality by a negative number flips the sign, thus the answer is z < 8.
Correct me if I am incorrect.
shirt original price is $100 and is now 50% off what is the discount price now
Answers
Answer:
$50.00 is the discount price
a manufacturer has the following quality control check at the end of a production line. if at least 8 of 10 randomly picked articles meet all specifications, the whole shipment is approved. if in reality, 85% of a particular shipment meets all specifications, what is the probability that the shipment will make it through the control check?
Answers
Using the binomial distribution, it is found that there is a 0.8202 = 82.02% probability that the shipment will make it through the control check.
For each article, there are only two possible outcomes, either it meets the specifications, or it does not. The probability of an article meeting the specifications is independent of any other article, hence, the binomial distribution is used to solve this question.
Binomial probability distribution
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes. n is the number of trials. p is the probability of a success on a single trial.In this problem:
10 articles are picked, hence [tex]n = 10[/tex].85% of the articles meets all specifications, hence [tex]p = 0.85[/tex]The probability is:
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)[/tex]
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 8) = C_{10,8}.(0.85)^{8}.(0.15)^{2} = 0.2759[/tex]
[tex]P(X = 9) = C_{10,9}.(0.85)^{9}.(0.15)^{1} = 0.3474[/tex]
[tex]P(X = 10) = C_{10,10}.(0.85)^{10}.(0.15)^{0} = 0.1969[/tex]
[tex]P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) = 0.2759 + 0.3474 + 0.1969 = 0.8202[/tex]
0.8202 = 82.02% probability that the shipment will make it through the control check.
For more on the binomial distribution, you can check https://brainly.com/question/24863377
The vertices of quadrilateral PQRS are listed.
P(3,7), Q(6,-2), R(0,-4), S(-3,5)
Which of the following is the strongest classification that identifies quadrilateral PQRS
A.
Quadrilateral PQRS is a square.
B.
Quadrilateral PQRS is a trapezoid.
C.
Quadrilateral PQRS is a rectangle.
D.
Quadrilateral PQRS is a parallelogram.
Answers
Answer:
It's C. for plato. It's a rectangle
Step-by-step explanation:
FILE BELOWAnswer:
its a rectangle .
Step-by-step explanation:
Write an algebraic expression for the given scenario and define the variables.
Answers
Answer:
n($6.50) + m($5.50) + k($6.00) = p
Step-by-step explanation:
n = matinee ticket
m = drink
k = popcorn
p = total cost
Without knowing the exact amount that was bought we must put a variable to show an unknown number. All of this together makes an algebraic expression.
The function v(t) is the velocity in m/sec of a particle moving along the x-axis. Use analytic methods to do each of the following: (a) Determine when the particle is moving to the right, to the left, and stopped. (b) Find the particle's displacement for the given time interval. If s(0) = 3, what is the particle's final position? (c) Find the total distance traveled by the particle. v(t) = 5 (sint)^2(cost); 0 ≤ t ≤ 2π
Answers
Answer:
(a) The particle is moving to the right in the interval [tex](0 \ , \ \displaystyle\frac{\pi}{2}) \ \cup \ (\displaystyle\frac{3\pi}{2} \ , \ 2\pi)[/tex] , to the left in the interval [tex](\displaystyle\frac{\pi}{2}\ , \ \displaystyle\frac{3\pi}{2})[/tex], and stops when t = 0, [tex]\displaystyle\frac{\pi}{2}[/tex], [tex]\displaystyle\frac{3\pi}{2}[/tex] and [tex]2\pi[/tex].
(b) The equation of the particle's displacement is [tex]\mathrm{s(t)} \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ 3[/tex]; Final position of the particle [tex]\mathrm{s(2\pi)} \ = \ 3[/tex].
(c) The total distance traveled by the particle is 9.67 (2 d.p.)
Step-by-step explanation:
(a) The particle is moving towards the right direction when v(t) > 0 and to the left direction when v(t) < 0. It stops when v(t) = 0 (no velocity).
Situation 1: When the particle stops.
[tex]\-\hspace{1.7cm} v(t) \ = \ 0 \\ \\ 5 \ \mathrm{sin^{2}(t)} \ \mathrm{cos(t)} \ = \ 0 \\ \\ \-\hspace{0.3cm} \mathrm{sin^{2}(t) \ cos(t)} \ = \ 0 \\ \\ \mathrm{sin^{2}(t)} \ = \ 0 \ \ \ \mathrm{or} \ \ \ \mathrm{cos(t)} \ = \ 0 \\ \\ \-\hspace{0.85cm} t \ = \ 0, \ \displaystyle\frac{\pi}{2}, \ \displaystyle\frac{3\pi}{2} \ \ \mathrm{and} \ \ 2\pi[/tex].
Situation 2: When the particle moves to the right.
[tex]\-\hspace{1.67cm} v(t) \ > \ 0 \\ \\ 5 \ \mathrm{sin^2(t) \ cos(t)} \ > \ 0[/tex]
Since the term [tex]5 \ \mathrm{sin^{2}(t)}[/tex] is always positive for all value of t of the interval [tex]0 \ \leq \mathrm{t} \leq \ 2\pi[/tex], hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is positive in the first and third quadrant or when [tex]\mathrm{t} \ \epsilon \ (0, \ \displaystyle\frac{\pi}{2}) \ \cup \ (\displaystyle\frac{3\pi}{2}, \ 2\pi)[/tex] .
*Note that parentheses are used to demonstrate the interval of t in which cos(t) is strictly positive, implying that the endpoints of the interval are non-inclusive for the set of values for t.
Situation 3: When the particle moves to the left.
[tex]\-\hspace{1.67cm} v(t) \ < \ 0 \\ \\ 5 \ \mathrm{sin^2(t) \ cos(t)} \ < \ 0[/tex]
Similarly, the term [tex]5 \ \mathrm{sin^{2}(t)}[/tex] is always positive for all value of t of the interval [tex]0 \ \leq \mathrm{t} \leq \ 2\pi[/tex], hence the determining factor is cos(t). Then, the question becomes of when is cos(t) positive? The term cos(t) is negative in the second and third quadrant or [tex]\mathrm{t} \ \epsilon \ (\displaystyle\frac{\pi}{2}, \ \displaystyle\frac{3\pi}{2})[/tex].
(b) The equation of the particle's displacement can be evaluated by integrating the equation of the particle's velocity.
[tex]s(t) \ = \ \displaystyle\int\ {5 \ \mathrm{sin^{2}(t) \ cos(t)}} \, dx \ \\ \\ \-\hspace{0.69cm} = \ 5 \ \displaystyle\int\ \mathrm{sin^{2}(t) \ cos(t)} \, dx[/tex]
To integrate the expression [tex]\mathrm{sin^{2}(t) \ cos(t)}[/tex], u-substitution is performed where
[tex]u \ = \ \mathrm{sin(t)} \ , \ \ du \ = \ \mathrm{cos(t)} \, dx[/tex].
[tex]s(t) \ = \ 5 \ \displaystyle\int\ \mathrm{sin^{2}(t) \ cos(t)} \, dx \\ \\ \-\hspace{0.7cm} = \ 5 \ \displaystyle\int\ \ \mathrm{sin^{2}(t)} \, du \\ \\ \-\hspace{0.7cm} = \ 5 \ \displaystyle\int\ \ u^{2} \, du \\ \\ \-\hspace{0.7cm} = \ \displaystyle\frac{5u^{3}}{3} \ + \ C \\ \\ \-\hspace{0.7cm} = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ C \\ \\ s(0) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(0)} \ + \ C \\ \\ \-\hspace{0.48cm} 3 \ = \ 0 \ + \ C \\ \\ \-\hspace{0.4cm} C \ = \ 3.[/tex]
Therefore, [tex]s(t) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(t)} \ + \ 3[/tex].
The final position of the particle is [tex]s(2\pi) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(2\pi)} \ + \ 3 \ = \ 3[/tex].
(c)
[tex]s(\displaystyle\frac{\pi}{2}) \ = \ \displaystyle\frac{5}{3} \ \mathrm{sin^{3}(\frac{\pi}{2})} \ + \ 3 \\ \\ \-\hspace{0.85cm} \ = \ \displaystyle\frac{14}{3} \qquad (\mathrm{The \ distance \ traveled \ initially \ when \ moving \ to \ the \ right})[/tex]
[tex]|s(\displaystyle\frac{3\pi}{2}) - s(\displatstyle\frac{\pi}{2})| \ = \ |\displaystyle\frac{5}{3} \ (\mathrm{sin^{3}(\frac{3\pi}{2})} \ - \ \mathrm{sin^{3}(\displaystyle\frac{\pi}{2})})| \ \\ \\ \-\hspace{2.28cm} \ = \ \displaystyle\frac{5}{3} | (-1) \ - \ 1| \\ \\ \-\hspace{2.42cm} = \displaystyle\frac{10}{3} \\ \\ (\mathrm{The \ distance \ traveled \ when \ moving \ to \ the \ left})[/tex]
[tex]|s(2\pi) - s(\displaystyle\frac{3\pi}{2})| \ = \ |\displaystyle\frac{5}{3} \ (\mathrm{sin^{3}(2\pi})} \ - \ \mathrm{sin^{3}(\displaystyle\frac{3\pi}{2})})| \ \\ \\ \-\hspace{2.28cm} \ = \ \displaystyle\frac{5}{3} | 0 \ - \ 1| \\ \\ \-\hspace{2.42cm} = \displaystyle\frac{5}{3} \\ \\ (\mathrm{The \ distance \ traveled \ finally \ when \ moving \ to \ the \ right})[/tex].
The total distance traveled by the particle in the given time interval is[tex]\displaystyle\frac{14}{3} \ + \ \displaystyle\frac{5}{3} \ + \ \displaystyle\frac{10}{3} \ = \ \displaystyle\frac{29}{3}[/tex].
Kyla earns a commission. She makes 2.5% of the amount she sells. Last week, she sold
$3,500 worth of furniture. How much was her commission?
Answers
Answer:
$87.50
Step-by-step explanation:
Alvin is 9 years older than Elga. The sum of their ages is 81. What is Elga's age?
Answers
Answer:
elga is 32 and alvin is 49
Let f(x)=x* + 14x and g(x) = 6 - X. Find the domain off f + g. Determine the domain of f + g.
Answers
[tex]\begin{cases} f(x) = x^4+14x\\ g(x) = 6-x \end{cases}\qquad \qquad h(x) = f(x) + g(x) \\\\\\ h(x) = (x^4+14x)+(6-x)\implies h(x) = x^4+14x-x+6 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill h(x) = x^4+13x+6~\hfill[/tex]
now, if we graph h(x), Check the picture below, we can see that horizontally the line keeps on moving towards the left, going going and going towards -infinity, and it also keeps on moving towards the right, going going and going towards +infinity, and since the horizontal area used by the function is the domain of it, the domain for h(x) will be (-∞ , +∞).
How many sides does the regular polygon have if each interior angle measure is four times the (1 point)
measure of each exterior angle measure?
Answers
Answer:
The exterior and interior angles must add up to 180 degrees. Thus, 180 divided by five gives the exterior angle as 36 degrees (and hence the interior angles as 144 degrees).
Answer:
10 Sides
Step-by-step explanation:
Int Angle + Ext Angle = 180
4Ext + Ext = 180
Ext = 36 degrees
-----
360/36 = 10 sides
At the restaurant, Gordon packed 8 orders with 4 items per order
in the morning. In the afternoon, he packed 6 orders with 7 items
per order.
Answers
Answer:
what is the question?
Step-by-step explanation:
A basket of fruit contains 6 apples, 5 oranges, 3 bananas, and 2 limes.Which of the following statements about the fruits in the basket are true?Select the two correct statements.
Answers
Answer:
[tex]6 + 5 = 11 + 3 = 14 + 2 = 16[/tex]
[tex]6 \div 5 \div 3 \div 2 = 0.2[/tex]
[tex] 6 \times 5 \times 3 \times 2 = 180[/tex]
Step-by-step explanation:
If its addition add them, multiplication multiply them, division divided them.
Answer:
Step-by-step explanation:
i need help sorry no answer got u
Approximately what portion of the beaker is filled?
A. 1/2
B. 1/4 C.3/4
Answers
Answer:
B. [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
The whole beaker is 1. If you measure the beaker, you will notice you can fill up the beaker to the brim (whole, which is 1) if you use the current amount 4 times, and four times is [tex]\frac{1}{4}[/tex].
Which graph shows a linear equation?
Answers
Answer:
The bottom right is a linear equation.
Step-by-step explanation:
Answer:
right side down one
Step-by-step explanation:
as you know linear means supplementary having 180 °
3.
In parallelogram ABCD, M
m
Answers
Answer:
Please send a picture or explain properly
Adelita, Elena, Betina, and Bianca each work as a doctor, lawyer, teacher, or banker. From these clues, decide who is the doctor.
Answers
Answer: betina or adelita
Step-by-step explanation: hope it helps
Could someone help me solve this please? With explanation?
Answers
Answer:
x=109 degrees
Step-by-step explanation:
By alternate interior angles, the measure of angle ADE is the same as that of EAB, both of which are 38.
Because ADE is an isosceles triangle, the measure angle EAD is equal to that angle EDA; let that measure be x.
Because the angles of a triangle add up to 180, x+x+38=180 -> 2x+38=180 -> 2x=142 -> x=71
That means that angle EDA is 180 degrees
Because x is supplementary to angle EDA, the measure of angle x is 180-71=109 degrees
PLEASE HELP -7y + 11 = 75 + y
Answers
Answer:
y = -8
Step-by-step explanation:
-7y + 11 = 75 + y
Bring the "y" variable on one side, and rest on the other.
Rearranging, we get,
-7y - y = 75 - 11
-8y = 64
-y = 8
y = -8
Hope it helps :)
Help pleaseee????????
Answers
it will be equal to the photo
median 5
lowerHELP ASAP PLEASE!!!!
Answers
Answers:
c = 7d = 5=========================================================
Explanation:
For equation A, I'll transform the right hand side into a similar form as the left side. Throughout the steps below, the left hand side stays the same.
[tex]\sqrt{448x^c} = 8x^3\sqrt{7x}\\\\\sqrt{448x^c} = \sqrt{(8x^3)^2}\sqrt{7x}\\\\\sqrt{448x^c} = \sqrt{64x^{3*2}}\sqrt{7x}\\\\\sqrt{448x^c} = \sqrt{64x^{6}}\sqrt{7x}\\\\\sqrt{448x^c} = \sqrt{64x^{6}*7x}\\\\\sqrt{448x^c} = \sqrt{64*7x^{6+1}}\\\\\sqrt{448x^c} = \sqrt{448x^{7}}\\\\[/tex]
Therefore, c = 7
Notice that 7/2 = 3 remainder 1. The quotient 3 is the exponent for the term outside the root for [tex]8x^3\sqrt{7x}[/tex] while the remainder 1 is the exponent for the x term inside the root.
---------------------------------------
We do the same idea for equation B.
[tex]\sqrt[3]{576x^{d}} = 4x\sqrt[3]{9x^{2}}\\\\\sqrt[3]{576x^{d}} = \sqrt[3]{(4x)^3}\sqrt[3]{9x^{2}}\\\\\sqrt[3]{576x^{d}} = \sqrt[3]{64x^3}\sqrt[3]{9x^{2}}\\\\\sqrt[3]{576x^{d}} = \sqrt[3]{64x^3*9x^{2}}\\\\\sqrt[3]{576x^{d}} = \sqrt[3]{64*9x^{3+2}}\\\\\sqrt[3]{576x^{d}} = \sqrt[3]{576x^{5}}\\\\[/tex]
This must mean d = 5
Note: 5/3 = 1 remainder 2, which means [tex]\sqrt[3]{x^5} = x^1\sqrt[3]{x^2} = x\sqrt[3]{x^2}[/tex]
write 321.51 as word form
Answers
Answer:
three hundred twenty one and fifty one hundredths
One side of a square garden is 8 feet long. How can you find the area of That is the
the garden?
Answers
Answer:
64 ft
Step-by-step explanation:
A square has to be equal on all 4 sides for it to be considered a square. Considering that it is 8 feet on one side it would be 8 feet on all the others as well. From there you multiply 8*8 and receive the solution of 84 feet for the area of the garden.
anyone please help this hard
Answers
Answer:
a. 6
b. 21
c. 78
d. 13.5
e. 14
f. 24
Hope that helps!
Approximately what portion of the box is shaded blue?
A.2/3. B.9/10
C.3/5
Answers
what is the slope of the line?
Answers
Answer:
1/2
Step-by-step explanation:
We can use the slope formula to find the slope
m = ( y2-y1)/(x2-x1)
We have two points on the line
(-1,3) and ( 1,4)
m = ( 4-3)/(1 - -1)
= (4-3)/(1+1)
= 1/2
Answer:
Start where the line meets a point. Then go up and over until it meets another.
The answer is 1/2
So go up one and over two. Then other problems such as this one should be pretty straight forward
What is 10³?
10
100
1000
Answers
Answer:
1000
Step-by-step explanation:
Answer: 1000, hope it helps.
Step-by-step explanation:
need help with solving this please
Answers
cos60 = opposite / hypothenuse = x / sqrt3
cos60 = 1/2
x = sqrt3/2
Answer:
3/2
Step-by-step explanation:
Since the shape is an equilateral triangle, all the angles are equal measure, 60° and all the sides are also of equal measure that was given, root3. So half of the triangle has length (root3)/2. The perpendicular drawn in the interior is also an angle bisector. The triangles created are 30°-60°-90° triangles. The sides of this special right triangle are in the ratio
s : 2s : sroot3
The longest side of the 30-60-90 triangle is given. The shortest side is half the length of the longest side. The length of the long leg is the short leg × root3
In this diagram the short leg is (root3)/2 .
(root3)/2 × root3 = 3/2
See image.
Two polygons have a similarity ratio of 4:5. If the perimeter of the first one is 10 inches, then what is the perimeter of the second?
Group of answer choices
11 inches
8 inches
12.5 inches
15 inches
Answers
Luis goes out to lunch. The bill, before tax and tip, was $14.60. A sales tax of 9% was added on. Luis tipped 19% on the amount after the sales tax was added. How much tip did he leave? Round to the nearest cent.
Answers
Add the sales tax to the bill:
Bill with tax = 14.60 x 1.09 = $15.95
Multiply the total bill by the tip percentage:
Tip = 15.95 x 0.19 = 3.02
Tip = $3.02
Make x the subject of the formula
t=
[tex] \sqrt{2(x - ut) \div a}[/tex]
Answers
[tex]t = \sqrt \frac{ {2(x - ut)} }{a} \\ = > t = \sqrt{ \frac{2x - 2ut}{a} } \\ = > {t}^{2} = \frac{2x - 2ut}{a} \\ = > a {t}^{2} = 2x - 2ut \\ = > \frac{ { - at}^{2} }{2ut} = 2x \\ = > \frac{ - at}{2u} = 2x \\ = > \frac{ - at}{2u \times 2} = x \\ = > \frac{ - at}{4u} = x[/tex]
Hope you could get an idea from here.
Doubt clarification - use comment section.