Answers
Answer:
x = {π/4, 7π/6, 5π/4, 11π/6} +2kπ . . . for any integer k
Step-by-step explanation:
[tex]\dfrac{\sin^3{x}}{1+\cos{x}}+\dfrac{\cos^3{x}}{1+\sin{x}}=\cos{2x}+2\cos{x}-1\\\\\dfrac{\sin{x}(1-\cos^2{x})}{1+\cos{x}}+\dfrac{\cos{x}(1-\sin^2{x})}{1+\sin{x}}=\cos^2{x}-\sin^2{x}+2\cos{x}-1\\\\\sin{x}(1-\cos{x})+\cos{x}(1-\sin{x})=\cos^2{x}-\sin^2{x}+2\cos{x}-1\\\\\sin{x}+\cos{x}-2\sin{x}\cos{x}=2\cos{x}-2\sin^2{x}\qquad\text{use $1=s^2+c^2$}\\\\\sin{x}+2\sin^2{x}-\cos{x}-2\sin{x}\cos{x}=0\\\\\sin{x}(1+2\sin{x})-\cos{x}(1+2\sin{x})=0\\\\(\sin{x}-\cos{x})(1+2\sin{x})=0[/tex]
This will have solutions where the factors are zero.
sin(x) -cos(x) = 0
Dividing by cos(x), we have ...
tan(x) -1 = 0
x = arctan(1) = π/4, 5π/4
1 +2sin(x) = 0
sin(x) = -1/2
x = arcsin(-1/2) = 7π/6, 11π/6
The four solutions in the interval [0, 2π] are x = {π/4, 7π/6, 5π/4, 11π/6}. Solutions repeat every 2π radians.
_____
Additional comment
We have made use of the factoring of the difference of squares:
(1 -a^2) = (1 -a)(1 +a)
and we have made use of the cosine double angle identity:
cos(2x) = cos(x)^2 -sin(x)^2
The "Pythagorean" identity for sine and cosine was used several times:
1 = sin(x)^2 +cos(x)^2
Related Questions
Suppose y varies jointly as x & z. If y = -180 when z = 15and x = -3,then find y when x = 7 and z = -5. 1 point
Answers
Answer:
y = - 140
Step-by-step explanation:
The statement
y varies jointly as x & z is written as
y = kxzto find y when x = 7 and z = -5 we must first find the relationship between the variables
when y = - 180
z = 15
x = - 3
- 180 = k(15)(-3)
-180 = - 45k
Divide both sides by - 45
k = 4
The formula for the variation is
y = 4xzwhen
x = 7
z = -5
y = 4(7)(-5)
y = 4(-35)
y = - 140Hope this helps you
Need help with mark brainlist.
Nam worked on a job for 10 days. On each of the last 2 days, he worked 2 hours more than the mean number of hours he worked per day during the first 8 days. If he worked 69 hours in all, how many hours did he work during the last 2 days together?
Answers
Answer: 17 hours
Step-by-step explanation:
Given that On each of the last 2 days, he worked 2 hours more than the mean number of hours he worked per day during the first 8 days. That is he worked additional 4 hours for the two days.
Let the total hours for the 8 days = E
The mean = E/8 = 0.125E
For the two last days, he worked
( 0.125E + 2 ) × 2 = 0.25E + 4
If he worked 69 hours in all, then
E + 0.25E + 4 = 69
Collect the like terms
1.25E = 69 - 4
1.25E = 65
E = 65/1.25
E = 52.
Now find the mean of the first 8 days
Mean = 52 / 8 = 6.5 hours
Nam works during the last 2 days together for:
(6.5 + 2)×2
8.5 × 2 = 17 hours
Weight of Sea Glass in Jars
2 2 2 1 11 12 12 2 24 22 24 3 37 22
Weight in Pounds
Mason loves collecting sea glass from the beach. He recorded the weight of
each jar full of sea glass. If he decided to distribute the sea glass equally among
the jars, how much would each Jar weigh?
pounds
(Write mixed numbers as x y/z.)
Answers
Answer:
[tex]Average = 12\frac{4}{7}\ lb[/tex]
Step-by-step explanation:
Given
Number of jars = 14
Weights = 2, 2, 2, 1, 11, 12, 12, 2, 24, 22, 24, 3, 37, 22 in pounds
Required
What is the weight of each jar if the sea glass is evenly distributed
What this question implies is that, the average weight of the sea glass is calculated;
[tex]Average = \frac{Sum\ of\ weights}{Number\ of\ jars}[/tex]
[tex]Average = \frac{2+ 2+ 2+ 1+ 11+ 12+ 12+ 2+ 24+ 22+ 24+ 3+ 37+ 22}{14}[/tex]
[tex]Average = \frac{176}{14}[/tex]
[tex]Average = 12\frac{8}{14}[/tex]
Divide the numerator and denominator by 2
[tex]Average = 12\frac{4}{7}[/tex]
Hence, the weight of each jar would be [tex]12\frac{4}{7}\ lb[/tex]
pls help asap .........................
Answers
Answer:
40
Step-by-step explanation:
We can split this figure into two figures, a square and a trapezoid. The square, as shown has a side length of 4. That means that the area of the square is 16. The trapezoid, has base lengths of 2 and 4, and a height of 8. We can plug that into the area of a trapezoid formula:
[tex]\frac{2+4}{2}(8)[/tex]
Two plus 4 is 6, and 6 divided by 2 is 3. Three times 8 is 24.
24 plus 16 (the area of the square) is equal to 40, which is the area of the shape.
Donavan and Jones are randomly choosing chalices to drink from. 4 of the chalices contain purified water, 5 contain spoiled milk, and 6 contain diluted flat soda. Jones gets to choose a chalice from the remaining ones after Donavan drinks from 3 of them. If the chances of Jones choosing a chalice with purified water is then 1/3, how many of the chalices that Donavan drank out of contained purified water?
Answers
Answer:
0
Step-by-step explanation:
Chalice with purified water = 4
With spoiled milk = 5
With flat soda = 6
Probability of Jones drinking from a chalice with purified water after Donovan had drank from 3 = 1/3
Probability = required outcome / Total possible outcomes
Total possible outcomes = (4 + 5 + 6) = 15
After Donovan drinks 3 :
Total possible outcomes = (15 - 3) = 12
If the probability of Jones choosing a chalice with purified water is 1/3 then :
(Required outcome / Total possible outcomes) =1 /3
(Required outcome / 12) = 1 / 3
Required outcome * 3 = 12
Required outcome = 12 / 3
Required outcome = 4
Therefore, since initial number of chalice with purified water is 4,
4 - 4 = 0
Then Donovan did not drink from a chalice containing purified water.
pls help will give you good rating
Answers
Answer:
-1
Step-by-step explanation:
The width of Estelle's rectangular living room is 4 meters, and the distance between opposite corners is 7 meters. What is the length of Estelle's living room? If necessary, round to the nearest tenth.
Answers
Answer:
5.7 meters
Step-by-step explanation:
Image a rectangle being cut diagonally in half, that would lead to two triangles. We can use the pythagorean theorem to find the missing side.
7 is the hypotenuse and 4 is the side length
4^2+x^2=7^2
16+x^2=49
sqrx=sqr(49-16)
x=sqr33
x=5.74
Rounded to the nearest tenth is 5.7
sloving polynomial (2x+8)(-3y-4)
Answers
Answer:
Step-by-step explanation:
2x ( -3y - 4) + 8 ( -3y - 4 )
= -6xy - 8x - 24y - 32
Hope this helps
plz mark as brainliest!!!
Answer:
Step-by-step explanation:
Use FOIL method
(2x + 8)(-3y -4) = 2x*(-3y) + 2x*(-4) + 8*(-3y) + 8*(-4)
= -6xy - 8x - 24y - 32
help?!?!??!!?!?????!
Answers
Answer:
[tex]n=10[/tex]
Step-by-step explanation:
[tex]\frac{4^8}{(4^2)^-^3}\ / 4^4=4^n[/tex]
Simplify, remember, an exponent raised to the power of another exponent is equal to the product of the two exponents,
[tex]\frac{4^8}{(4^2)^-^3}\ / 4^4=4^n[/tex]
[tex]\frac{4^8}{4^-^6}\ / 4^4=4^n[/tex]
Now bring the term in the denominator (number under the fraction bar) to the numerator (number over the fraction bar). This can be done by multiplying the exponent of the denominator by (-1).
[tex]\frac{4^8}{4^-^6}\ / 4^4=4^n[/tex]
[tex]4^8(4^(^-^1^)^(^-^6^))\ / 4^4=4^n[/tex]
Simplify,
[tex]4^8(4^6)\ / 4^4=4^n[/tex]
Remember, multiplying two numbers of the same base is the same as adding their exponents, apply this property here,
[tex]4^8(4^6)\ / 4^4=4^n[/tex]
[tex]4^1^4\ / 4^4=4^n[/tex]
Dividing two numbers of the same base is the same as subtraction their exponents,
[tex]4^1^4\ / 4^4=4^n[/tex]
[tex]4^1^0=4^n[/tex]
Thus, the following statement can be made,
[tex]4^1^0=4^n[/tex]
[tex]n=10[/tex]
Solve: 3a^2-4b a= -6 b= -5 If you could also leave an explanation that would be great! Thank you for your time!
Answers
Answer:
128
Step-by-step explanation:
3a² - 4b
plug in values
3(-6)² - 4(-5)
use PEMDAS and simplify (-6)² first
3(36) -4(-5)
multiply
108 + 20
add
128
hope this helps :)
A quadratic polynomial whose zeroes are 5 and -3 is
Answers
Answer:
Well, we can write a really simple quadratic polynomial as:
P(x) = a*(x - c1)*(x - c2)
Where a is a constat, and c1 and c2 are the roots, here we have:
c1 = 5 and c2 = -3
P(x) = a*(x -5)(x + 3) = a*x^2 + a*x*3 - a*5*x - a*15
then the quadratic polynomial can be:
P(x) = a*x^2 + a*(3 - 5)*x - a*15
P(x) = a*x^2 - 2*a*x - a*15
Where a can be any real number.
MAKE EXAMPLES WHERE YOU USE THE DISCOUNT AND THE INCREASE OF PERCENTAGES
Answers
Answer:
once upon a time a dude went to a store. there was a dude jacket for 20% off.
Step-by-step explanation:
Now do the opposite. for exapmle, the price increased by 20 %.
Draw a line segment AB of length 6.6 cm. Bisect it perpendicularly at N using a
ruler and set squares
Answers
Step-by-step explanation:
If a line segment AB of length 6.6cm is drawn, bisecting it perpendicularly at N, we have two lines AN and NB of length 3.3cm each.
Because the line is bisected perpendicularly, the angles formed at the point of bisection are 90 degrees.
BRAINLIEST!! The equation of the line is Y=2.x- 1.8. Based on the graph which of the following are true?
(select all that apply)
A. If tony stays for 30 minutes in the record store it is likely her will spend $70
B. Each additional minute tony spends in the store is associated with an additional cost of $2.40
C. The correlation coefficient for the line of best fits 2.4
D. The line of best fit will have a positive correlation coefficient.
Answers
Answer:
b y=2.4x -1.8
Step-by-step explanation:
the equation y=2.4x -1.8 represents 2.4 from it
I need help super fast
Answers
Answer:
The last option
Step-by-step explanation:
When you have a fraction as a power, the denominator goes to the root, so in this case, it becomes root 4. The base, 5, goes to the outside, and the numerator 1, goes beside the x.
Hope this helps!
The National Weather Service collects data on the number of hours of consecutive rainfall and the number of minor traffic accidents in a particular city. The scatter plot shows the data it gathered and the line of best fit. For a school project, Peyton uses technology to calculate the equation for line of best fit. If Peyton's calculation is correct, which equation could represent the line of best fit for this data?
Answers
The equation that could best represent the line of best fit for the given data is; y = 0.625x
How to find a linear equation from a scatter plot?The formula for the equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept which is the point where the line intersects the y-axis
Now, in this question, we see that the line intersects the y-axis at 0. Thus;
y-intercept; c = 0
Now, let us find the slope from the formula;
m = (y₂ - y₁)/(x₂ - x₁)
Using the first and penultimate coordinate which are;
(0, 0) and (8, 5), we have;
m = (5 - 0)/(8 - 0)
m = 0.625
Thus, the equation is;
y = 0.625x
Read more about Scatter plot equation at; https://brainly.com/question/6592115
#SPJ1
Answer:
y=0.625x
Step-by-step explanation:
plato
If an arrow is shot upward on Mars with a speed of 62 m/s, its height in meters t seconds later is given by y = 62t − 1.86t². (Round your answers to two decimal places.) Estimate the speed when t = 1. Can you please show me the steps to solve this?
Answers
Answer:
Approximately [tex]58.28\; \rm m \cdot s^{-1}[/tex].
Step-by-step explanation:
The velocity of an object is the rate at which its position changes. In other words, the velocity of an object is equal to the first derivative of its position, with respect to time.
Note that the arrow here is launched upwards. (Assume that the effect of wind on Mars is negligible.) There would be motion in the horizontal direction. The horizontal position of this arrow will stays the same. On the other hand, the vertical position of this arrow is the same as its height: [tex]y = 62\, t - 1.86\, t^2[/tex].
Apply the power rule to find the first derivative of this [tex]y[/tex] with respect to time [tex]t[/tex].
By the power rule:
the first derivative of [tex]t[/tex] (same as the first derivative of [tex]t^2[/tex] (same as [tex]t[/tex] to the second power) with respect toTherefore:
[tex]\begin{aligned}\frac{dy}{d t} &= \frac{d}{d t}\left[62 \, t - 1.86\, t^2\right] \\ &= 62\,\left(\frac{d}{d t}\left[t\right]\right) - 1.86\, \left(\frac{d}{d t}\left[t^2\right]\right) \\ &= 62 \times 1 - 1.86\times\left(2\, t) = 62 - 3.72\, t\end{aligned}[/tex].
In other words, the (vertical) velocity of this arrow at time [tex]t[/tex] would be [tex](62 - 3.72\, t)[/tex] meters per second.
Evaluate this expression for [tex]t = 1[/tex] to find the (vertical) velocity of this arrow at that moment: [tex]62 - 3.72 \times 1 =58.28[/tex].
Answer:
58.28 m/s
Step-by-step explanation:
y = 62t - 1.86t²
Speed, S = dy/dt = 62 - 2(1.86)t
S = 62 - 3.72t
When t = 1
S = 62 - 3.72 = 58.28 m/s
Solve for x. Round your answer to the nearest tenth if necessary.
X=
Answers
Answer:
7.2
Step-by-step explanation:
Since the angle is equal, 24/(x+8.7)=13.1/8.7. x=7.2
Answer:
x ≈ 7.2
Step-by-step explanation:
Δ MNO and Δ MKL are similar, so the corresponding sides are in proportion, that is
[tex]\frac{MN}{MK}[/tex] = [tex]\frac{MO}{ML}[/tex] , substitute values
[tex]\frac{13.1}{13.1+10.9}[/tex] = [tex]\frac{8.7}{8.7 +x}[/tex]
[tex]\frac{13.1}{24}[/tex] = [tex]\frac{8.7}{8.7+x}[/tex] ( cross- multiply )
13.1(8.7 + x) = 208.8 ← distribute left side
113.97 + 13.1x = 208.8 ( subtract 113.97 from both sides )
13.1x = 94.83 ( divide both sides by 13.1 )
x = [tex]\frac{94.83}{13.1}[/tex] ≈ 7.2 ( to the nearest tenth )
Margot is sewing a ribbon on a seam along the perimeter of a square pillow. The side length of the pillow is 23-1 inches. She plans to make a similar pillow, including the ribbon, whose side length is 4x-7 inches. What expression can be used for the length of ribbon that she needs for both pillows, and what is the length if x = 3.5?
2x14x-6: 22.0 inches
2x14x-6; 32.5 inches
4(2x+4x-6) 88.0 inches
4(2x +48-6:) 130.0 inches
Answers
Answer:
D. 4(2x² + 4x - 6); 130.0 inches
Step-by-step explanation:
Margot is sewing a ribbon on a seam along the perimeter of a square pillow. The side length of the pillow is 2x²+1 inches. She plans to make a similar pillow, including the ribbon, whose side length is 4x−7 inches. What expression can be used for the length of ribbon that she needs for both pillows, and what is the length if x = 3.5?
2x2+4x−6; 22.0 inches
2x2+4x−6; 32.5 inches
4(2x2+4x−6;) 88.0 inches
4(2x2+4x−6;) 130.0 inches
Perimeter of a square = 4 * side length
1- First square pillow
length = 2x²+1 in
Perimeter of first pillow = 4 * (2x²+1) in
Second square pillow
length = 4x-7 in
Perimeter of second pillow = 4(4x-7) in
Total ribbon length required = perimeter of first square pillow + perimeter of second square pillow
Total ribbon length required = 4(2x²+1) + 4(4x-7)
Factorise
4 is the common factor
=4(2x²+1)+(4x-7)
=4(2x²+1+4x-7)
=4(2x²+4x-6)
Total ribbon length required=4(2x²+4x-6)
If x=3.5
Total ribbon length required=4(2x²+4x-6)
=4{2(3.5)²+4(3.5)-6}
=4{2(12.25)+14-6)
=4(24.5+14-6)
=4(32.5)
=130.0 inches
Given an angle of a triangle and the opposite side length; which trigonometric function would you use to find the hypotenuse? a TAN b COS c SIN d Not enough information
Answers
Answer:
Sin
Step-by-step explanation:
Sin < = opposite/hypotenuse
What is the simple interest at the end of two year for a loan of rupees 20000 at 10% intrest annum
Answers
Answer:
4000 rupees
Step-by-step explanation:
20000×10×2/100
= 4000 rupees
I want you to help me with this
Answers
Answer:
$44.1
Step-by-step explanation:
March>>>>$40
for April: $40 × 5/100= $2 which is the 5% of $40.
Meaning for April Siti got $42
Then for May: $42 × 5 /100= $2.1 meaning she got $44.1 for May.
a number when divided by 10 leaves the remainder 5. If the same number is doubled, and divided by 10, the new remainder is _____
Answers
Answer:
10
Step-by-step explanation
50 divided by 10 is 5 so
50 times 2 which is 100 divided by 10 is 10
Roselyn is driving to visit her family, which lives 150150150 kilometers away. Her average speed is 606060 kilometers per hour. The car's tank has 202020 liters of fuel at the beginning of the drive, and its fuel efficiency is 666 kilometers per liter. Fuel costs 0.600.600, point, 60 dollars per liter.
Answers
Answer:
see explanation (please confirm if I read you problem correctly)
Step-by-step explanation:
150 kilometers
60 kph
tank is 20 liters
6 km/l
150/6 = 25 liters needed
$60/l
you will need to get 5 more liters of fuel
total cost 25 x .60 = $15
Answer:
AHHHHH I think its two hours
Step-by-step explanation:
she...? can drive 120 Kilometers before she runs out of fuel since the car's tank has 20 liters of fuel at the beginning of the drive, and its fuel efficiency is 6 kilometers per liter. and that means she can drive 2 hours
Tell me the answer
69:?::89:36
Answers
?????????????????????????????????????
Find the domain and range of the function. Use a graphing utility to verify your results. (Enter your answers using interval notation.)
Answers
Answer:
Step-by-step explanation:
sorry it should be the second one for the interval notations
two years ago a woman wad 7 times as old as her daughter, but in 3 years time she would be x times as old as the girl. how old are they now?
Answers
Answer:
The present age of the woman is 37 years and her daughter is 7 years
Step-by-step explanation:
two years ago a woman was 7 times old as her daughter, but in 3 years time she would be 4 times old as the girl. how old are they now
Two years ago a woman wad 7 times as old as her daughter
Let her daughter=x-2
The woman=y-2
x-2
7(y-2)=7y-14
x-2=7y-14
x-7y=-14+2
x-7y= -12 (1)
but in 3 years time she would be 4 times as old as the girl.
x+3
y+3
x+3
4(y+3)=4y+12
x+3=4y+12
x-4y=12-3
x-4y=9. (2)
x-7y= -12 (1)
x-4y=9. (2)
Subtract (1) from (2)
4y-(-7y)=9-(-12)
-4y+7y=9+12
3y=21
y=21/3
y=7
Substitute
y=7 into (1)
x-7y= -12
x-7(7)=-12
x-49=-12
x= -12+49
=37
The present age of the woman is 37 years and her daughter is 7 years
NEED HELP ASAP!!!!!!!
Answers
Solve |2x+3/4 |=5 1/2 Please help!!!!
Answers
Answer:
=2x+3/4=5.50
x=19/8 or 2 3/8
hope this helps
Step-by-step explanation:
PLESE HELP YOU WILL GET ALOT OF POINTS
Answers
Answer:
It's a (space) lot,... haha na I just kidding,... you're good,... I mean I'm not kidding cause it is correct,... but you are good,... like totally cool,... anyway,...
Step-by-step explanation:
By x the just mean what number,... so,...
what number + 6 = 7,
6 + what number = 7,
2 + what number + 5 = 7,
1 + what number = 7,
what number + 4 + what number = 7,
So that is what it mean,... hope it helps, alot LOL,... and,... Chow