Find all angles 0 < theta < 360 that satisfy the equation to the nearest 10th
4cos2x + cosx -3= -5cosx - 8
The angles that satisfy the trigonometric equation 4 · cos 2x + cos x - 3 = - 5 · cos x - 8 are described below:
x₁₁ ≈ 104.5°, x₁₂ ≈ 255.5° x₂₁ = 120°, x₂₂ ≈ 240°How to find the solution of a trigonometric equation
In this problem we find a trigonometric equation involving both cos 2x and cos x, we need to simplify the expression such that only cos x is present. This can be done by means of trigonometric formulas. First, write the entire expression:
4 · cos 2x + cos x - 3 = - 5 · cos x - 8
Second, use trigonometric formulas to eliminate cos 2x:
4 · (2 · cos² x - 1) + cos x - 3 = - 5 · cos x - 8
8 · cos² x - 4 + cos x - 3 = - 5 · cos x - 8
8 · cos² x + 6 · cos x + 1 = 0
Third, use the quadratic formula to find all possible roots:
cos x₁ ≈ - 1 / 4, cos x₂ ≈ - 1 / 2
Fourth, use inverse trigonometric functions to determine all possible angles between 0° and 360°:
Case 1: - 1 / 4
x₁₁ ≈ 104.5°, x₁₂ ≈ 255.5°
Case 2: - 1 / 2
x₂₁ = 120°, x₂₂ ≈ 240°
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How to turn x = 3y -1 into slope intercept form?
Answer:
y=1/3(x)+1/3
Step-by-step explanation:
x=3y-1
x+1=3y
(x+1)/3=(3y)/3
y=1/3x+1/3
34) An airplane pilot is planning a flight to Hawaii. In his flight plan, he will include fuel estimates for the trip.
Explain the way he can use estimates in his plan without causing any danger to the flight
The airplane pilot should include and follow the relation y > [D][x] for his fuel estimates.
What is a mathematical function, equation and expression?
Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given is that an airplane pilot is planning a flight to Hawaii. In his flight plan, he will include fuel estimates for the trip.
Assume that the distance of Hawaii from the initial location of pilot is [D] miles. If the amount of fuel burnt per miles is [x] liters/miles. Let the total amount of fuel needed to reach Hawaii is [y] liters. Than, his estimates should include and follow the given relation as -
y > [D][x]
Therefore, the airplane pilot should include and follow the relation y > [D][x] for his fuel estimates.
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IN PHOTO⬇️
PLEASE HELP!!!
The simple interest is $2160 and the amount after 30 years is $6160.
Given that, principal =$4000, rate of interest =1.8%, and time period =30 years.
What is the simple interest?Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time.
Simple interest is calculated with the following formula: S.I. = (P × R × T)/100, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years.
a) Now, S.I =(4000×1.8×30)/100
= 40×1.8×30
= $2160
b) Amount = Principal + Interest
= 4000 + 2160
= $6160
Therefore, the simple interest is $2160 and the amount after 30 years is $6160.
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Every other weekend, Bronwyn’s brother Daniel mows the lawn. He can mow 15,000 ft2 in 3/4 fourths hour. Who mows the lawn in less time? Explain. PLEASE HELP ME
The person who mows the lawn in less time is given as follows:
Daniel.
How to decide who mows the lawn in less time?To obtain who mows the lawn in less time, we calculate the hourly rates for each person, applying the proportion, which is the division of the amount of area mowed by the time needed.
Daniel can mow 15,000 ft³ in 3/4 = 0.75 hours, hence his hourly rate is calculated as follows:
Daniel hourly rate = 15000/0.75 = 15000 x 4/3 = 20,000 ft² per hour.
Bronwyn can mow 12,000 ft² in hour, hence her rate is of:
12,000 ft² per hour.
20,000 > 12,000, hence Daniel can mow the law in less time, as he has a higher rate.
Missing InformationThe missing sentence is of:
"Bronwyn can mow 12,000 ft² in hour".
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if a line equation 2x +y =4 what is the gradie value
The gradient value of the line 2x + y = 4 is 2. This can be found by rearranging the equation into slope-intercept form, which is y = -2x + 4. The coefficient of the x term, -2, is the slope of the line.
solve the given differential equation by undetermined coefficients. 1 4 y'' + y' + y = x2 − 3x
The general solution will be:
[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]
[tex]$$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$$[/tex]
Consider the homogenous equation [tex]$14 y^{\prime \prime}+y^{\prime}+y=0$[/tex]
The auxiliary equation is [tex]$14 m^2+m+1=0$[/tex]
[tex]m=-\frac{1}{28}+i \frac{1}{28} \sqrt{55}[/tex]
The homogenous solution is
[tex]$$y_c(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right) \text {. }$$[/tex]
Consider the non-homogenous de [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex] by using the method of undetermined coefficients
consider [tex]$y_p=A x^2+B x+C \Rightarrow y_p^{\prime}=2 A x+B \Rightarrow y_p^{\prime \prime}=2 A$[/tex]
by plugging the values in [tex]$14 y^{\prime \prime}+y^{\prime}+y=x^2-3 x$[/tex]
[tex]$$\begin{aligned}& 28 A+(2 A x+B)+A x^2+B x+C=x^2-3 x \\& \Rightarrow x^2(A)+x(2 A+B)+28 A+B+C=x^2-3 x\end{aligned}$$[/tex]
On comparing both sides we get
A=1.
2A+B =-3
=>B=-3-2A
=>B=-5
28A+B+C=0
=>C=-28A-B
=>C=-23
by plugging the values [tex]y_p=x^2-5 x-23[/tex].
The general solution will be:
[tex]$$y(x)=e^{-x / 28}\left(c_1 \cos \left(\frac{\sqrt{55}}{28} x\right)+c_2 \sin \left(\frac{\sqrt{55}}{28} x\right)\right)+x^2-5 x-23$$[/tex]
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The area of a sector is 135π and the arc measure is 216°. What is the radius of the circle?
The radius of the sector of area 135π is 15.
What is a sector?A sector is a geometric figure bounded by two radii and the included arc of a circle.
To calculate the radius of the sector, we use the formula below
Formula:
r = √(360A/π∅).............. Equation 1Where:
r = Radius of the sectorA = Area of the sector∅ = Angle of the arc at the center of the circleπ = pieFrom the question,
Given:
A = 135π∅ = 216°Substitute these values into equation 1
r = √(360×135π/π×216)r = √225r = 15Hence, the radius of the sector is 15.
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An online music store is offering a promotion. When you spend $40 or more, you receive 3 songs for free. You buy an album for $8. Write and solve an inequality that represents how much more money you must spend to receive the three free songs.
Answer:[tex]x+8\geq 40[/tex]
Step-by-step explanation:
[tex]x+8\geq 40[/tex]
subtract 8 from both sides to isolate x
[tex]x\geq 32[/tex]
You must spend 32 more dollars to receive three free songs.
Please help me it’s asap
Answer:
215
Step-by-step explanation:
218+212/2
A number is such that it is as much greater than 112 as it is less than it. find the number.
Answer:
Answer. it is -111 because it is greater than 112 and in positive form it is smaller and in negative it is bigger .
10. What is the side length of a regular
36-gon that has an area of 164,592
square meters and an apothem of
228.6 m?
Answer:
39.95m
Step-by-step explanation:
To find the side length of a regular 36-gon with a given area and apothem, we need to use the formula for the area of a regular polygon and the formula for the apothem of a regular polygon.
The formula for the area of a regular polygon is given by:
A = (1/2) * a * p
where A is the area of the polygon, a is the apothem of the polygon, and p is the perimeter of the polygon.
The formula for the apothem of a regular polygon is given by:
a = r * cos(180/n)
where a is the apothem of the polygon, r is the radius of the polygon, and n is the number of sides of the polygon.
In this case, we are given that the area of the 36-gon is 164,592 square meters, the apothem of the 36-gon is 228.6 m, and the number of sides of the 36-gon is 36.
We can use these values to solve for the side length of the 36-gon. First, we can use the formula for the area of a regular polygon to solve for the perimeter of the 36-gon:
A = (1/2) * a * p
164,592 = (1/2) * 228.6 * p
164,592 = 114.3 * p
p = 1438.3
Next, we can use the formula for the apothem of a regular polygon to solve for the radius of the 36-gon:
a = r * cos(180/n)
228.6 = r * cos(180/36)
r = 561.8
Finally, we can use the formula for the perimeter of a regular polygon to solve for the side length of the 36-gon:
p = n * s
1438.3 = 36 * s
s = 39.95
Therefore, the side length of the 36-gon is 39.95 meters.
in class a there are 20 students 14 of them are girls in class b there 25 students 15 of them are girls (a) find the percentages of girls in each class
To find the percentage of girls in class A, we need to divide the number of girls by the total number of students and then multiply by 100. This gives us:
14 / 20 * 100 = 70%
To find the percentage of girls in class B, we need to divide the number of girls by the total number of students and then multiply by 100. This gives us:
15 / 25 * 100 = 60%
Thus, the percentage of girls in class A is 70%, and the percentage of girls in class B is 60%.
Complete each congruence statement by naming
the corresponding angle or side.
ABCA = AFGA
B
AB = ?
C
? = side
A
side type your answer...
F
G
If the congruence statement ABCA = AFGA is true, then the following must also be true:
B = G, since corresponding angles are congruent
AB = FG, since corresponding sides are congruent
C = F, since corresponding angles are congruent
Thus, the complete congruence statement is ABCA = AFGA, where B = G, AB = FG, and C = F.
the value in dollars of recycling varies with the type of material. what is the weighted mean of this mixture of recyclables if compost is now worth twice as much?
The weighted mean of the mixture of recyclables would be (4 x $1 + 2 x $2) / 6 = $1.33.
To calculate the weighted mean of the mixture of recyclables, first identify the number of each material and their respective values. In this case, there are 4 items worth $1 each and 2 items worth $2 each. To calculate the weighted mean, multiply each item's value by the number of items and then divide by the total number of items
In this case,
4 x $1 + 2 x $2 = $6
and 6 items total, so the weighted mean is
$6 / 6 = $1.33.
This means that if compost is now worth twice as much, the weighted mean would be
(4 x $1 + 2 x $2) / 6 = $1.33.
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a five-foot prep casts a shadow that is 40 feet long while standing 200 feet from a streetlight. how high above the ground is the lamp?
30ft high above the ground is the lamp.
What is an angle?
An angle is a shape created by two rays that share a terminus and are referred to as the angle's sides and vertices, respectively. Angles created by two rays are in the plane where the rays are located. The meeting of two planes also creates angles. We refer to these as dihedral angles.
Here, we have
BC⊥AD and DE⊥AD
∠ABC = ∠ADE(=90°)
Also,∠A=∠A
ΔABC ΔADE (AA similarity)
AB/AD = BC/DE
40/(200 + 40) = 5ft/DE
DE = 5ft×6 = 30ft
Hence, 30ft high above the ground is the lamp.
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match the parametric equations with the correct graph. x = t cos t, y = t, z = t sin t, t ≥ 0
[tex]x^{2} + y^{2} = t^{2}[/tex] represent the correct graph which is the first figure graph.
Equations with Parameters If x and y are continuous functions of t on an interval I, the equations x = x(t) and y = y(t) are referred to as parametric equations, and t is also known as the parameter. The graph of the parametric equations is the set of points (x, y) obtained as t varies over the interval I.
Since x = tsint
And z= tcost
We have a circle in the x-z plane a
[tex]x^{2} + z^{2} = t^{2}[/tex]
Now, as y = t
The graph will be a helix with movement along the y-axis. Furthermore, the circle made on the x-z plane will have a variable radius since the radius is equal to t which by itself is changing.
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Let P(n) be the statement that 13 +23 +···+n3 = (n(n+1)/2)2 for the positive integer n.a) What is the statement P(1)?b) Show that P(1) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step, identifying where you use the inductive hypothesis.f) Explain why these steps show that this formula is true whenever n is a positive integer.
P(n) = 1³ +2³ +···+n³ = (n(n+1)/2)²
P(1) = 1³ = 1
According to statement P(n) = (n(n+1)/2)²
Checking for n = 1,
(n(n+1)/ 2)² = (1(1+1)/ 2)²
P(1) = (1 × 2/ 2)²
P(1) = 1² = 1
P(1) = 1³ = 1
So P(n) holds for n = 1.
The inductive hypothesis is that the statement holds for P(n). To prove this the inductive step under the assumption that the inductive hypothesis is true, we prove it is true for P(n + 1)
Let us assume P(n) = 1³ + 2³ + ... + n³ = (n(n+1)/ 2)²
Then P(n + 1) = 1³ + 2³ + ... + n³ + (n + 1)³ = (n(n+1)/ 2)² + (n + 1)³
P(n + 1) = (n(n+1)/ 2)² + n³ + 3n² + 3n + 1
P(n + 1) = ((n + 1)((n + 1) + 1)/ 2)²
As assuming P(n) holds, it is also true for P(n + 1) so it will be true for all values of n as it is proved already that P(1) is true.
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Find two number uch that one of them i five time the other and their difference i 200
Based on the relation between two numbers considering their difference and multiple, the numbers are 50 and 250.
Let the one number be x. So, the other number, according to the information in question, will be 5x. Now representing the equation depending on the stated details -
5x - x = 200
Performing subtraction on Left Hand Side of the equation
4x = 200
Writing the equation again to find the value of x
x = 200/4
Performing division on Right Hand Side of the equation
x = 50
First number = 50
Another number = 50×5
Performing multiplication on Right Hand Side of the equation
Second number = 250
The two numbers according to multiple and difference are 50 and 250.
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Find the value of k that would make the left side of the equation a perfect square trinomial.
9x² - kx + 4 = 0
Answer: k=±`12
Step-by-step explanation:
[tex]ax^2-bx+c=0\\\\9x^2-kx+4=0\\\\D=b^2-4ac\\\\a=9\ \ \ \ b=-k\ \ \ \ c=4\\\\D=0\\\\Hence,\\\\D=(-k)^ 2-4(9)(4)=0\\\\k^2-144=0\\\\k^2=144\\\\k^2=12^2[/tex]
Extract the square root of both parts of the equation:
[tex]k=б\sqrt{12^2} \\\\k=б12[/tex]
Answer:
k = - 12 or k = 12---------------------------------------------
A perfect square trinomial is:
(a ± b)² = a² ± 2ab + b²Given trinomial:
9x² - kx + 4 = 0Show this as:
(±3x)² - kx + (±2)² = 0Compare and find possible values of k:
k = - 2(±3)(±2) = ± 12fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. his leash is exactly long enough to reach the midpoint of each side of his yard. if the fraction of the area of fido's yard that he is able to reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of the product $ab$?fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. his leash is exactly long enough to reach the midpoint of each side of his yard. if the fraction of the area of fido's yard that he is able to reach while on his leash is expressed in simplest radical form as $\frac{\sqrt{a}}{b}\pi$, what is the value of the product $ab$?
Product ab has a value of 18 and is shaped like a regular hexagon.
What is area?The quantity area indicates the extent of a region on a planar or curved surface. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas area of a plane region or plane area refers to the area of a form or planar lamina. The area is the region defined by an object's form. The area of a form is the space covered by a figure or any two-dimensional geometric shape in a plane.
Here,
Let the side of the hexagon be 2.
so the radius of the circle is sqrt(3),
area of hexagon=sqrt(3) * 2 * 3
= 6 sqrt(3)
area of circle is pi (sqrt(3))^2 = 3 pi
3 pi / 6 sqrt(3) = sqrt(3)/ 6
3 * 6 = 18
The value of product ab is 18 which is in the shape of a regular hexagon.
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Mary bought an antique clock.
In the first year, its value increased by 30%
In the second year, its value decreased by 30%
Work out whether the clock increased in value, decreased in value or remained the
same value over the two years.
You must show your working clearly.
The overall price considering the percentages is decreased.
What are percentages?Percentage is defined as "out of 100." Similar to fractions and decimals, percentages are used in mathematics to represent subsets of a whole. When expressing a sum as a percentage, 100 equal components are regarded to make up the whole. A percentage can be shown by the symbol % or, less frequently, by the abbreviation 'pct'.
In stores, online, in commercials, and in the media, you can see percentages practically anywhere. Understanding what percentages signify is a crucial skill that could help you save time and money and raise your employability.
Calculation:Let us suppose mary bought that antique clock for 100$.
So now in the first year its value is increased by 30% which is 100+30 = 130$ after the 1 st year.
Now the clock price is 130$
So now in the second year its value is decrease by 30% which is 130- 39 = 91 $
So the overall clock price has been decreased by $9 since mary bought it.
The overall price considering the percentages is decreased.
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© {2, 4, 6, 8}
(3, 4, 5, 6}
(2, 3), (4, 4), (6, 5), (8, 6)
{2, 3, 4, 5, 6, 8)
X
2
4
6
8
у
3
4
5
6
In many countrie, cigarette moking ha decreaed in the 21" century. The percentage of moker in Ruia can be
repreented by the equation 36. * 100 y = 346, and the percentage of moker in Poland i repreented by the
year ince 20000
equation 18 r 50 y = 176. The equation ue y for the percentage of adult who moke and * for the number of year ince 2000,
Determine the olution to the ytem and how your work
Based on the provide system of linear equations, the solution is that there is no solution.
A system of linear equations refers to a collection of one or more linear equations involving the same variables. The solution to a system of linear equations refer to the point at which the lines representing the linear equations intersect. If the lines are parallel, they will never intersect hence, the system has no solution.
According to the provided information, the percentage of smokers in Russia is given by the equation 36x+100y=346 and the percentage of smokers in Poland is given by the equation 18x+50y=176. In order to determine the solution of the system of linear equations, multiply the Poland equation by 2 and subtract it from the Russian equation. Hence,
2*(18x+50y=176)
36x + 100y = 352
-(36x+100y=346)
0 = 6
It means the original equations are parallel. As both the variables are eliminated, the system has no solution.
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Juan claims that y=4x^3+1 is a function, but not a linear function. Select the statement that supports Juan's claim.
A. The function does not contain the point (0,0)
B. The coefficient of x^3 is not 1
C. The graph of the function does not form a straight line
D. The function increases
Answer:
C. The graph of the function does not form a straight line
The cash price of a refrigerator is Rs 34 500. The hire purchase price is a deposit of Rs 6500 and 10 % simple interest on the outstanding balance, to be paid in 24 monthly installments. Calculate the amount of each monthly instalment
Answer:
Rs 1400
Step-by-step explanation:
You want the monthly installment payment if the balance of a Rs 34500 purchase after a 6500 deposit earns 10% simple interest and is paid in 24 months.
OutstandingThe outstanding balance is ...
Rs 34500 -6500 = Rs 28000
InterestThe interest on this balance for 24 months is ...
I = Prt
I = Rs 28000(0.10)(2) = Rs 5600
Monthly paymentThe monthly payment will be 1/24 of the total due:
(28000 +5600)/24 = 1400
Each monthly installment will be Rs 1400.
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It takes 73 pounds of seed to completely plant an 11 -acre field. How many pounds of seed are needed per acre?
Step-by-step explanation:
It takes 73 pounds of seed to completely plant an 11 -acre field. How many pounds of seed are needed per acre?
73 / 11 = 6.6363636 pounds per acre
rounded to two decimal places: 6.64 pounds per acre
Use the given information to find the new price. Round to the nearest cent Original Price: $28 Discount Rate: 25%
Answer:
To find the new price after a discount, you can use the formula: new price = original price * (1 - discount rate). In this case, the new price would be $28 * (1 - 0.25) = $28 * 0.75 = $<<28*0.75=21>>21. So the new price, rounded to the nearest cent, would be $21.
Step-by-step explanation:
Which of the following rational functions is graphed below? 5 -5 5 -5
Answer:
b
Step-by-step explanation:
Infer from domain and range
(2+√-4) (3 +5i)(-4-i)
Answer:
[tex]32-60i[/tex]----------------------------------------
Simplify the expression in below steps:
[tex](2+\sqrt{-4} ) (3 +5i)(-4-i)=[/tex][tex](2+\sqrt{-1*4} ) (3 +5i)(-4-i)=[/tex][tex](2+2i ) (3 +5i)(-4-i)=[/tex][tex]2(1+i ) (3 +5i)(-4-i)=[/tex][tex]2(1+i ) (3(-4) -3i+5i(-4)-5i*i)=[/tex][tex]2(1+i ) (-12 -3i-20i-5(-1))=[/tex][tex]2(1+i ) (-12 -23i+5)=[/tex][tex]2(1+i ) (-7 -23i)=[/tex][tex]2(-7 -23i -7i-23i*i)=[/tex][tex]2(-7 -30i +23)=[/tex][tex]2(16 -30i)=[/tex][tex]32-60i[/tex]Answer:
[tex]32-60i[/tex]
Step-by-step explanation:
Given expression:
[tex](2+\sqrt{-4})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Rewrite $\sqrt{-4}$ as $\sqrt{4 \cdot -1}$}:[/tex]
[tex]\implies (2+\sqrt{4 \cdot -1})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Apply radical rule} \quad \sqrt{ab}=\sqrt{a}\sqrt{b}:[/tex]
[tex]\implies (2+\sqrt{4}\sqrt{-1})(3 +5i)(-4-i)[/tex]
[tex]\implies (2+2\sqrt{-1})(3 +5i)(-4-i)[/tex]
[tex]\textsf{Apply imaginary number rule} \quad \sqrt{-1}=i:[/tex]
[tex]\implies (2+2i)(3 +5i)(-4-i)[/tex]
Multiply:
[tex]\implies (6+16i+10i^2)(-4-i)[/tex]
[tex]\implies -24-6i-64i-16i^2-40i^2-10i^3[/tex]
[tex]\implies -24-70i-56i^2-10i^3[/tex]
[tex]\textsf{Apply imaginary number rule} \quad i^2=-1:[/tex]
[tex]\implies -24-70i-56(-1)-10(-1)i[/tex]
Simplify:
[tex]\implies -24-70i+56+10i[/tex]
[tex]\implies 32-60i[/tex]