MATHEMATICS COLLEGE

How to solve trigonometric functions such as

sin 2x sin x + cos x= 0

Answers

Answer 1
Answer:

Use the double angle identity:

sin(2x) = 2 sin(x) cos(x)

Now rewrite

sin(2x) sin(x) + cos(x) = 0

as

2 sin²(x) cos(x) + cos(x) = 0

Factor out cos(x) :

cos(x) (2 sin²(x) + 1) = 0

Consider the two cases,

cos(x) = 0   OR   2 sin²(x) + 1 = 0

Solve for cos(x) and sin²(x) :

cos(x) = 0   OR   sin²(x) = -1/2

Squaring a real number always gives a non-negative number, so the second case doesn't offer any real solutions. We're left with

cos(x) = 0

Cosine is zero for odd multiples of π/2, so we have

x = (2n + 1) π/2

where n is any integer.

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The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Price in Dollars 25 33 34 45 48
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Step 1 of 6: Find the estimated slope. Round your answer to three decimal places.
Step 2 of 6: Find the estimated y-intercept. Round your answer to three decimal places.Step 3 of 6: Find the estimated value of y when x = 34. Round your answer to three decimal places.Step 4 of 6: Determine the value of the dependent variable yˆ at x = 0.Step 5 of 6: Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ.Step 6 of 6: Find the value of the coefficient of determination.

Answers

Answer:

1) b1=5.831

2) b0=12.510

3) y(34)=210.764

4) y(0)=12.510

5) y=12.510+5.831x

6) R^2=0.85

Step-by-step explanation:

We have the linear regression model y=b_0+b_1 x.

We start by calculating the all the parameters needed to define the model:

- Mean of x:

\bar x=(1)/(5)\sum_(i=1)^(5)(2+3+4+5+7)=(21)/(5)=4.2

- Uncorrected standard deviation of x:

s_x=\sqrt{(1)/(n)\sum_(i=1)^(5)(x_i-\bar x)^2}\n\n\ns_x=\sqrt{(1)/(5)\cdot [(2-4.2)^2+(3-4.2)^2+(4-4.2)^2+(5-4.2)^2+(7-4.2)^2]}\n\n\n s_x=\sqrt{(1)/(5)\cdot [(4.84)+(1.44)+(0.04)+(0.64)+(7.84)]}\n\n\n s_x=\sqrt{(14.8)/(5)}=√(2.96)\n\n\ns_x=1.72

- Mean of y:

\bar y=(1)/(5)\sum_(i=1)^(5)(25+33+34+45+48)=(185)/(5)=37

- Standard deviation of y:

s_y=\sqrt{(1)/(n)\sum_(i=1)^(5)(y_i-\bar y)^2}\n\n\ns_y=\sqrt{(1)/(5)\cdot [(25-37)^2+(33-37)^2+(34-37)^2+(45-37)^2+(48-37)^2]}\n\n\n s_y=\sqrt{(1)/(5)\cdot [(144)+(16)+(9)+(64)+(121)]}\n\n\n s_y=\sqrt{(354)/(5)}=√(70.8)\n\n\ns_y=8.414

- Sample correlation coefficient

r_(xy)=\sum_(i=1)^5((x_i-\bar x)(y_i-\bar y))/((n-1)s_xs_y)\n\n\nr_(xy)=((2-4.2)(25-37)+(3-4.2)(33-37)+...+(7-4.2)(48-37))/(4\cdot 1.72\cdot 8.414)\n\n\nr_(xy)=(69)/(57.888)=1.192

Step 1

The slope b1 can be calculated as:

b_1=r_(xy)(s_y)/(s_x)=1.192\cdot(8.414)/(1.72)=5.831

Step 2

The y-intercept b0 can now be calculated as:

b_o=\bar y-b_1\bar x=37-5.831\cdot 4.2=37-24.490=12.510

Step 3

The estimated value of y when x=34 is:

y(34)=12.510+5.831\cdot(34)=12.510+198.254=210.764

Step 4

At x=0, the estimated y takes the value of the y-intercept, by definition.

y(0)=12.510+5.831\cdot(0)=12.510+0=12.510

Step 5

The linear model becomes

y=12.510+5.831x

Step 6

The coefficient of determination can be calculated as:

R^2=1-(SS_(res))/(SS_(tot))=1-(\sum(y_i-f_i))/(ns_y^2)\n\n\n\sum(y_i-f_i)=(25-24.17)^2+(33-30)^2+(34-35.83)^2+(45-41.67)^2+(48-53.33)^2\n\n\sum(y_i-f_i)=0.69+ 8.98+ 3.36+ 11.12+ 28.38=52.53\n\n\n ns_y^2=5\cdot 8.414^2=353.98\n\n\nR^2=1-(52.53)/(353.98)=1-0.15=0.85

hi can someone help me with is problem on how to break it down so I can explain it to my daughter. please.​

Answers

Answer:  30

==========================================================

Explanation:

Refer to the drawing below. I have a 2 by 5 grid of squares. So there are 2*5 = 10 squares total.

Four of those squares are shaded blue to represent the fraction (4)/(10)

If Susan used 4 blue tiles (instead of 12), then she'd have 10 tiles total. This would mean 10 would be the answer.

However, she's using 12 blue tiles. The jump from 4 to 12 is "times 3". So we'll need to multiply that 10 by 3 as well.

10*3 = 30

If Susan uses 12 blue tiles, then she has 30 tiles total.

Notice that the fraction (4)/(10) is the same as (12)/(30) after multiplying both top and bottom by 3.

(4)/(10)= (4*3)/(10*3)= (12)/(30)

If you wanted, you can split a round cake into 10 equal slices. If a person eats 4 slices, then that represents the fraction (4)/(10). Now imagine splitting each of those initial ten slices into three smaller pieces. If you manage to do so, then you'd have 30 very small slices. The four that the person ate would have effectively eaten 12 very small slices. So this is another way to see how (4)/(10) = (12)/(30). Personally, when it comes to fractions, I prefer using a grid of squares because it's a bit tricky sometimes to divide up circles perfectly.

Please help fast, thanks.

Answers

I believe it is m < 5 I could be wrong sorry.

Graph the function f(x) = x^3+2x^2-3

Answers

you graph will look something like this :)

Answer this fast and I’ll make you Brainly!!

Answers

Answer:

8.5 or 4.25 i think pretty sure its 8.5 tho

Step-by-step explanation:

Consider △LNM.
Which statements are true for triangle LNM? Check all that apply.

Answers

Answer:

The correct options are 1, 2 and 4.

Step-by-step explanation:

In a triangle, the side which does not make an angle is called opposite side of that angle.

From the given graph it is clear that

Opposite side of ∠L is NM.

Option 1 is correct.

Opposite side of ∠M is NL.

Opposite side of ∠N is ML.

Option 2 is correct.

The side next to an angle is called adjacent side.

Adjacent sides of ∠L is LM and LN.

Adjacent sides of ∠M is LM and MN.

Adjacent sides of ∠N is LN and MN.

In a right angled triangle, the opposite sides of a right angle is called hypotenuse.

Angle M is a right angled, so the side NM is hypotenuse. Option 4 is correct.

Therefore the correct options are 1, 2 and 4.
1 2 and 4 are correct. the others arent
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