Given:
[tex]=2-2\sqrt{3}i[/tex]Find-:
Convert complex numbers to a polar representation
Explanation-:
Polar from of the complex number
[tex]z=a+ib=r(\cos\theta+i\sin\theta)[/tex]Where,
[tex]\begin{gathered} r=\sqrt{a^2+b^2} \\ \\ \theta=\tan^{-1}(\frac{b}{a}) \end{gathered}[/tex]Given complex form is:
[tex]\begin{gathered} z=a+ib \\ \\ z=2-i2\sqrt{3} \\ \\ a=2 \\ \\ b=-2\sqrt{3} \end{gathered}[/tex][tex]\begin{gathered} r=|z|=\sqrt{a^2+b^2} \\ \\ r=|z|=\sqrt{2^2+(2\sqrt{3})^2} \\ \\ =\sqrt{4+12} \\ \\ =\sqrt{16} \\ \\ =4 \end{gathered}[/tex]For the angle value is:
[tex]\begin{gathered} \theta=\tan^{-1}(\frac{b}{a}) \\ \\ \theta=\tan^{-1}(\frac{-2\sqrt{3}}{2}) \\ \\ =\tan^{-1}(-\sqrt{3}) \\ \\ =-60 \\ \\ =-\frac{\pi}{3} \end{gathered}[/tex]So, the polar form is:
[tex]\begin{gathered} z=r(\cos\theta+i\sin\theta) \\ \\ z=4(\cos(-\frac{\pi}{3})+i\sin(-\frac{\pi}{3})) \end{gathered}[/tex]Use the formula:
[tex]\begin{gathered} \sin(-\theta)=-\sin\theta \\ \\ \cos(-\theta)=+\cos\theta \end{gathered}[/tex]Then value is:
[tex]\begin{gathered} z=4(\cos(-\frac{\pi}{3})+i\sin(-\frac{\pi}{3})) \\ \\ z=4(\cos\frac{\pi}{3}-i\sin\frac{\pi}{3}) \end{gathered}[/tex]The missing side length in the right triangle is __ cm.
364
1) Assuming this is a right triangle, the side whose length is 365 is the hypotenuse, so we can write out the Pythagorean Theorem:
365² = 27² +c²
133225 =729+ c²
133225 -729 = c²
132496= c² Take the square root on both sides
√132496 = √c²
c = 364
2) Hence, the missing leg is 364 cm
The length of a rectangle is given by the function l(x)=12x2+2x+4. The width of the rectangle is given by the function w(x)=3x−1.Which function represents the area of the rectangle?
The area of a rectangle is given by the next equation:
Area = Length * Width
Where
Length =1/2x²+2x+4.
and
Width = 3x−1
Replacing:
Area = (1/2x²+2x+4)* (3x-1)
Solve the operation:
(1/2x²* 3x) + (1/2x²*-1)+(2x*3x)+(2x*-1)+(4*3x)+(4*-1)
Simplify:
3/2x³-1/2x²+6x²-2x+12x-4
= 3/2x³+11/2x²+10x-4
Hence, the correct answer is a(x)=3/2x³+11/2x²+10x-4
Answer:a(x)=3/2x³+11/2x²+10x-4
Step-by-step explanation:
The legs of a right triangle measure 29 centimeters and 95 centimeters. How long is the hypotenuse in centimeters?
In this question, the two legs of a right angle triangle are:
29 centimeters and 95 centimetres
To find the hypotenuse, let's use Pythagoras Theorem.
[tex]c^2=a^2+b^2[/tex]where ,
a = 29 centimeters
b = 95 centimeters
c = hypotenuse
Therefore,
[tex]c^2=29^2+95^2[/tex][tex]c^2\text{ = 841 + 9025}[/tex][tex]c^2\text{ = 9866}[/tex]Now take the square root of both sides
[tex]\sqrt{c^2\text{ = }}\sqrt{9866}[/tex][tex]c\text{ = 99.327}[/tex]The hypotenus in centimeter is 99.3 centimeters
The table gives the temperature( in Fahrenheit) in five cities at 6 am on the same day please zoom in pic so its not blurry
(a)
Temp. in fairbanks is -29 if the temp. risen by 17 then temp. is:
[tex]\begin{gathered} \text{Present temp. =initial temp. }+\text{ change in temp.} \\ =-29+17 \\ =-12 \end{gathered}[/tex]In Noon the temp in fairbanks is -12 degree fehrebheit.
(b)
6 A.M temp in Santa =74
6 A.M. temp in toronto =-19
[tex]\begin{gathered} \text{change in temp.= high temp. - low temp.} \\ =74-(-19) \\ =74+19 \\ =93 \end{gathered}[/tex]In 6 A.M. temp 93 fehre
Can someone help me with this geometry question I don’t know if I am right.
Let us find out if the given two triangles ABC and DEF are similar triangles or not.
Triangle ABC is a right-angled triangle so we can apply the Pythagorean theorem to find the missing side.
[tex]a^2+b^2=c^2[/tex]Where a and b are the shorter sides and c is the longest side (hypotenuse)
[tex]\begin{gathered} 20^2+21^2=c^2 \\ 400+441=c^2 \\ 841=c^2 \\ \sqrt[]{841}=c \\ 29=c \\ c=29 \end{gathered}[/tex]Similarly, we can apply the Pythagorean theorem to triangle DEF to find the missing side.
[tex]\begin{gathered} d^2+e^2=f^2 \\ 40^2+e^2=58^2 \\ e^2=58^2-40^2 \\ e^2=3364-1600 \\ e^2=1764 \\ e=\sqrt[]{1764} \\ e=42 \end{gathered}[/tex]Now, recall that two triangles are similar if the ratio of the corresponding sides is equal.
The corresponding sides are
AB = DE
BC = EF
AC = DF
[tex]\begin{gathered} \frac{DE}{AB}=\frac{EF}{BC}=\frac{DF}{AC} \\ \frac{40}{20}=\frac{42}{21}=\frac{58}{29} \\ \frac{2}{1}=\frac{2}{1}=\frac{2}{1} \end{gathered}[/tex]As you can see, the ratio of the corresponding sides of the two triangles is equal.
Hence, the triangles ABC and DEF are similar.
Use the unit circle to identify the reference angle for 155°.
SOLUTION
Given the question in the question tab, the following are the solution steps to answer the question.
STEP 1: Define Unit circle
[tex]undefined[/tex]Solve the equation: √ t − 13 = 4 Answer: t =
Taking the given equation to the power of 2 we get:
[tex](\sqrt{t-13})^2=4^2.[/tex]Simplifying the above result we get:
[tex]t-13=16.[/tex]Adding 13 to the above equation we get:
[tex]\begin{gathered} t-13+13=16+13, \\ t=29. \end{gathered}[/tex]Answer: t=29.
the volume of a recrangular prism is 72 centimeters.the prism is 2 centimeters wide and 4 centimeters high. what is the length of the prism
The volume of a rectangular prism is given by:
[tex]V=w\cdot l\cdot h[/tex]We know that the volume is 72 cub cm, the width is 2 cm and the height is 4 cm. Plugging this value in the equation we have:
[tex]72=2\cdot l\cdot4[/tex]Solving the equation for l we have:
[tex]\begin{gathered} 72=2\cdot l\cdot4 \\ 72=8l \\ l=\frac{72}{8} \\ l=9 \end{gathered}[/tex]Therefore the lenght of the prism is 9 cm.
J(7, -2), K(-4, 9), L(-3,-1)
Find the sum of 5 even positive integers
sum of 5 even positive integers
[tex]n+(n+2)+(n+4)+(n+6)+(n+8)[/tex]here, n is the first even number,
let's simplify this,
[tex]\begin{gathered} =n+n+2+n+4+n+6+n+8 \\ =5n+20 \end{gathered}[/tex]Thus the expression to find the sum of 5 consecutive even positive integers is, 5*n + 20 , where n is the 1st even positive integer.
let's use it when n = 2 or to sum 2, 4, 6, 8 and 10
[tex]5*2+20=10+20=30[/tex]which is the same as,
[tex]2+4+6+8+10=30[/tex]Two birds spot a cat trying to jump up to their cage Find the angle of depression of the cat.
Step 1
Draw the triangle for clarity
From the diagram above it is clear, the angle of depression is equal to the angle of elevation since both angles are alternate interior angles. Therefore, we can go ahead to find the angle of elevation which is equal to the angle of depression.
Step 2
Find the angle of depression using SohCahToa
To find this angle we will use the ratio Cah written as
[tex]\begin{gathered} \cos x=\frac{adjacent}{\text{hypotenuse}} \\ \text{adjacent}=4ft \\ \text{hypotenuse}=5ft \end{gathered}[/tex][tex]\begin{gathered} \cos x=\frac{4}{5} \\ x=\cos ^{-1}(\frac{4}{5}) \\ x=36.86989765^{\circ}_{} \\ x\approx36.9^{\circ}\text{ to 1 decimal places} \end{gathered}[/tex]Hence the angle of depression of the cat, x to 1 decimal place = 36.9°
hi thank you for helping me . is this just multiplication?
ANSWER:
25.84
STEP-BY-STEP EXPLANATION:
To multiply two decimal numbers:
1. They are multiplied as if they were whole numbers.
2. The final result is a decimal number whose number of decimal places is equal to the sum of the number of decimal places of the two factors.
Therefore:
M O GEOMETRY Identifying parallelograms, rectangles, and squares Answer the questions about the figures below. 3 m Figure A Explanation 3 m 3 m (a) Which figures are squares? Mark all that apply. Figure A (b) Which figures are parallelograms? Mark all that apply. O Figure A O Figure B (c) Which figures are rectangles? Mark all that apply. O Figure A 3 m O Figure B O Figure C Figure B Check O Figure C O Figure C Figure B 2 m 5 m 4 m O None of the figures O None of the figures O None of the figures 3 m L X Figure C 5 m 5 m ☐ L ? 3 m 0/5 O2022 McGraw Hill LLC. All Rights Reserved. Terms of Use Privacy Center Nikid Ac
Given the dimensions of each figure:
• Figure A:
Length of each side = 3m
All sides have equal lengths.
• Figure B:
Length of top base = 2 m
Length of bottom base = 5m
Length of each leg = 4m
• Figure C.
Length = 5 m
Width = 3m
Since the length of all sides in figure A are equal and they meet at right angles, we can say figure A is a square.
Also, figure A has 2 pairs of parallel sides.
Since it has 2 pairs of parallel sides, we can say it is also a parallelogram.
Figure B has two parallel bases, while the opposite legs are equal.
Figure B has just one pair of parallel side.
Thus, we can figure B is a trapezoid.
Figure C has two pairs of parallel sides and the opposite sides have equal lengths.
Thus, we can say Figure B is rectangle and also a parallelogram.
Hence, we have the following:
• Squares ==> Figure A
,• Parallelogram ==> Figure A and Figure C
,• Rectangle ==> Figure C.
• ANSWER:
(a).
(c).
Jimmy's school is selling tickets to a play. On the first day of ticket sales the school sold 12 senior citizentickets and 5 child tickets for a total of $173. The school took in $74 on the second day by selling 1 seniorcitizen ticket and 5 child tickets. Find the price of a senior citizen ticket and the price of a child ticket.
SOLUTION
Let s represent senior citizen tickets
and let c represent child ticket
On the first day the school sold 12 senior tickets and 5 child's tickets for $173.
This will be represented algebraically by 12s + 5c = 173 ..... equation 1
The following day they sold 1 senior ticket and 5 child's tickets for $74.
This will be represented by s + 5c = 74 ....... equation 2
Now we will solve equations 1 and 2 simultaneously we have
[tex]undefined[/tex]1x-5x? please help me !
Solve for x. I think u have to do a portion I’m not sure
Answer:
Explanation:
Based on the given figure, we can form similar triangles:
Triangle 1:
Triangle 2:
To solve for the value of x, we use ratio.
[tex]\begin{gathered} \frac{10}{6}=\frac{15}{x} \\ \text{Simplify and rearrange} \\ x=\frac{(15)(6)}{10} \\ x=\frac{90}{10} \\ \text{Calculate} \\ x=9 \end{gathered}[/tex]Therefore, the value of x is 9.
(b)Dan leaves his house on his bike. He rides at a constant speed until he reaches a lemonade stand, where he parks his bike and takes a rest. Then he turns around and bikes home as fast as he can.
Answer:
The correct answer is the second graph.
Explanation:
Dan leaves his house at a constant speed. Then, the time starts counting, and his graph is a line that starts in (0, 0)
Then, he stops, we can see this part in the graph constant. Because he is not moving, the distance from his house remains the same.
Finally, goes back and we can this as a line. Since he is going at a faster speed, the line is more steep.
This is what we can see in the second graph
A real estate agent works on a 11 % commission. What is her commission on a house that she sold for $597,000?Follow the problem-solving process and round your answer to the nearest cent, if necessary.
Given:
11% commission.
House sold is $597, 000
To find the commission on a house she sold for $597 000, simply find 11 % of that amount.
That is;
Her commision = 11% of $597,000
[tex]=\frac{11}{100}\times597000[/tex][tex]=\text{ 6567000/100}[/tex][tex]=\text{ \$65670}[/tex]Hence, her commision is $65670
If the side ratio is 4:17, the the area ratio is
The Solution:
Given the side ratio below:
[tex]4\colon17[/tex]We are asked to find the area ratio of the figures being compared.
While the ratio of sides is linear, the ratio of area is square.
This means that if:
[tex]\begin{gathered} \text{ side ratio =4:17} \\ \text{Then it follows that} \\ \text{area ratio will be} \\ \text{ 4}^2\colon17^2=16\colon289 \end{gathered}[/tex]Therefore, the correct answer is [ 16:289 ]
a pizza song head offer 6 kinds of meat toppings and 6 vegetable topping and how many different ways could you select a meat topping and in vegetable topping
In AFGH, the measure of ZH=90°, GF = 89, HG = 80, and FH = 39. What ratiorepresents the cotangent of ZF?
cotangent ZF = adjacent side / opposite side
adjacent side = 39
opposite side = 80
cotangent ZF = 39/80
Section 1.5: Mortgages and Credit Cards8. A car costs $10,500, and you're offered a loan that requires $800 down and a monthly payment of $187.53 for 60 months, how much will you pay in interest? Round your answer to the nearest dollar.$
A monthly payment of $187.53 for 60 months will give us:
[tex]187.53\times60=\text{ \$11,251.8}[/tex]The total money paid is:
[tex]\begin{gathered} \text{the down payment of \$800 + \$11,251}.8 \\ \Rightarrow800+11,251.8=12051.8 \end{gathered}[/tex]Hence, the interest paid is:
[tex]\begin{gathered} 12,051.8-10,500 \\ \Rightarrow\text{ \$}1551.8 \end{gathered}[/tex]What is the volume of this sphere?Use3.14 and round your answer to the nearest hundredth.4 mcubic metersHELPP!!!
Answer:
Explanation:
The formula for the area of a sphere is:
[tex]V=\frac{4}{3}\pi r^3[/tex]Where r is the radius of the circle.
In this case, the diameter of the circle is 6in, then;
[tex]r=\frac{6in}{2}=3in[/tex]The volume is:
[tex]undefined[/tex]Solve in R Sin (x/4) = √2/2
Trigonometric Equations
Solve for x in R:
[tex]sin\text{ }\frac{x}{4}=\frac{\sqrt{2}}{2}[/tex]There are two angles whose sine is the given value: 45° and 135°. We need to express them in radians:
[tex]45^o=45\text{ }\frac{\pi}{180}=\frac{\pi}{4}[/tex][tex]135^o=135\frac{\pi}{180}=\frac{3\pi}{4}[/tex]Thus, we have two solutions:
[tex]\begin{gathered} \frac{x}{4}=\frac{\pi}{4} \\ x=\pi \\ And: \\ \frac{x}{4}=\frac{3\pi}{4} \\ x=3\pi \end{gathered}[/tex]Both solutions point to the same terminal angle, so we only have one solution in the first rotation of the angle:
x = π
Since it's required to find the solution for all the real numbers, we must account for all the possible angles in any number of rotations clockwise or counterclockwise as follows:
x = π + 2kπ
Where k is an integer number. For example, for k = 1, we have the already-found solution above x = 3π
A triangle with side lengths 8, 15, and 17 is a right triangle by theconverse of thePythagorean Theorem. What are the measures of the other 2 angles?Round your answers to the nearest whole number.HINT: Draw a diagram of this problem and label your triangle.The méasure of the smaller acute angle is ____degreesand the larger acute angle measures_______degrees.
We are given a right-angle triangle with side lengths 8, 15, and 17.
Since it is a right triangle, one angle must be 90°
Let us find the other two angles of this right triangle.
With respect to angle x, the opposite side is 15 and the hypotenuse side is 17.
Recall from the trigonometric ratios,
[tex]\begin{gathered} \sin (x)=\frac{\text{opposite}}{\text{hypotenuse}} \\ \sin (x)=\frac{15}{17} \\ x=\sin ^{-1}(\frac{15}{17}) \\ x=61.9\degree \end{gathered}[/tex]So, the second angle is 61.9°
Recall that the sum of angles inside a triangle must be equal to 180°
So, the third angle can be found as
[tex]\begin{gathered} y+61.9\degree+90\degree=180\degree \\ y=180\degree-90\degree-61.9\degree \\ y=28.1\degree \end{gathered}[/tex]So, the third angle is 28.1°
The measure of the smaller acute angle is 28.1 degrees and the larger acute angle measures 61.9 degrees.
Which of the following is NOT a level of measurement?Choose the correct answer below.A) OrdinalB) NominalC) RatioD)Quantitative
Given:
A) Ordinal
B) Nominal
C) Ratio
D)Quantitative
Required:
We want to find that Which of the given is NOT a level of measuremen
Explanation:
Levels of Measurement Nominal, Ordinal, Interval and Ratio
but the Quantitative is not the Levels of Measurement because it is
Solve Step 3 onlyTherefore, the solutions of the original equation are the following. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)
ANSWER:
[tex]x=\pi n, \frac{3\pi}{2}+2\pi n[/tex]EXPLANATION"
Given:
[tex]\sin x(\sin x+1)=0[/tex]Having solved Step 1 and Step 2 as seen above, we can go ahead and write the solutions of the equation as seen below;
[tex]\begin{gathered} If\text{ }\sin x=0 \\ \therefore x=\pi \\ \\ If\text{ }\sin x=-1 \\ then\text{ }x=\frac{3\pi}{2} \\ \\ So\text{ }the\text{ }solution\text{ }will\text{ }be; \\ x=\pi n,\frac{3\pi}{2}+2\pi n \end{gathered}[/tex]Find the slope of the line that passes through the points (15,-2) and (5,-4).Write answer as an integer or a reduced fraction
1) To find the slope of the line that passes through those points, we'll need to use the Slope Formula
[tex]\begin{gathered} m=\frac{y_2-y_1_{}}{x_2-x_1} \\ m=\frac{-4-(-2)}{5-15}=\frac{-4+2}{-10}=\frac{-2}{-10}=\frac{-1}{-5}=\frac{1}{5} \end{gathered}[/tex]Note that the slope is the measure of how steep is the line between those points.
2) That is the answer.
write the sum in unit form. 4 fifths + 3 fifths =
We write it as follows:
[tex]\frac{4}{5}+\frac{3}{5}=\frac{7}{5}=1\frac{2}{5}[/tex]It is 1 unit and 2/5.
Frank has a vintage comic book worth $454. According to a dealer, the value of this particular comic book will increase by 15% each year. How much will the comic book be worth in 2 years?If necessary, round your answer to the nearest cent.
Given:
Current worth = $454
Rate of increase = 15% = 0.15
Time = t
To find how much the book will be worth in 2 years, apply the exponential growth formula:
[tex]y=a(1+r)^t[/tex]Where:
a is the current worth = 454
r is the rate of increase = 0.15
t is the time = 2 years
Hence, we have:
[tex]\begin{gathered} y=454(1+0.15)^2 \\ \\ y=454(1.15)^2 \\ \\ y=454(1.3225) \\ \\ y=600.42 \end{gathered}[/tex]Therefore, the worth of the comic book in 2 years is $600.42
ANSWER:
$600.422 ye