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3 years ago

Complete the following to make a true identity. 1. sin^2 3x + cos^2 3x = ?

Mathematics

2 answers:

Andru [333]3 years ago

8 0

Answer:

Step-by-step explanation:

sin^2+cos^2 A=1 is an indentity and is true for all A, including A= 3x and hence sin^2+cos^2 3x=1

However, let us try is using values of sin3x and cos3x, but before this as sin^2x + cos^2x =1, squaring sin^4 x + cos^4 x =2sin^2 x cos^2 x=1 or sin^4 x +cos^4 x= 1 - 2 sin^2 x and sin^6 x + cos^6 x=1 -3 sin^2 x cos^2 x (sin^2 x + cos^2 x) =1 -3 sin^2 x cos^2 x -as a^3 =b^3= (a+b)^3 -3ab(a+b)

Now coming to proof as

sin3x= 3 sin x- 4 sin^3 x and cos x= 4 cos^3 x - 3 sin x

Therefore

sin^2 3x +cos^2 3x=(3 sin x- 4 sin^3 x)^2 +(4cos^3 x - 3 cos x)^2

9sin^2 x = 16 sin^6 x -24 sin^4 x + 16 cos^6 x + 9 cos^2 x- 24 cos^4 x

9(sin^2 x +cos^2 x) + 16(sin^6 x + cos^6 x) - 24 (sin^4 x + cos^4 x)

9*1 +16(1-3 sin^2 x cos^2 x) - 24 (1-2 sin^2 x cos^2 x)

9+16-48sin^2 x cos^2 x -24+48 sin^2 x cos^2 x

andrey2020 [161]3 years ago

5 0

Answer:

Step-by-step explanation:

I have done this by a Calculator

Red

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2 years ago

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Suppose you are interested in the effect of skipping lectures (in days missed) on college grades. You also have ACT scores and h

DIA [1.3K]

Answer:

a) For this case the intercept of 2.52 represent a common effect of measure for any student without taking in count the other variables analyzed, and we know that if HSGPA=0, ACT= 0 and skip =0 we got $colGPA=2.52$

b) This value represent the effect into the ACT scores in the GPA, we know that:

$\hat \beta_{ACT} = 0.015$

So then for every unit increase in the ACT score we expect and increase of 0.015 in the GPA or the predicted variable

c) If we are interested in analyze if we have a significant relationship between the dependent and the independent variable we can use the following system of hypothesis:

Null Hypothesis: $\beta_i = 0$

Alternative hypothesis: $\beta_i \neq 0$

Or in other wouds we want to check if an specific slope is significant.

The significance level assumed for this case is $\alpha=0.05$

Th degrees of freedom for a linear regression is given by $df=n-p-1 = 45-3-1 = 41$ , where p =3 the number of variables used to estimate the dependent variable.

In order to test the hypothesis the statistic is given by:

$t=\frac{\hat \beta_i}{SE_{\beta_i}}$

And replacing we got:

$t = \frac{-0.5}{0.0001}=-5000$

And for this case we see that if we find the p value for this case we will get a value very near to 0, so then we can conclude that this coefficient would be significant for the regression model .

Step-by-step explanation:

For this case we have the following multiple regression model calculated:

colGPA =2.52+0.38*HSGPA+0.015*ACT-0.5*skip

Part a

(a) Interpret the intercept in this model.

For this case the intercept of 2.52 represent a common effect of measure for any student without taking in count the other variables analyzed, and we know that if HSGPA=0, ACT= 0 and skip =0 we got $colGPA=2.52$

(b) Interpret $\hat \beta_{ACT}$ from this model.

This value represent the effect into the ACT scores in the GPA, we know that:

$\hat \beta_{ACT} = 0.015$

So then for every unit increase in the ACT score we expect and increase of 0.015 in the GPA or the predicted variable

(c) What is the predicted college GPA for someone who scored a 25 on the ACT, had a 3.2 high school GPA and missed 4 lectures. Show your work.

For this case we can use the regression model and we got:

$colGPA =2.52 +0.38*3.2 +0.015*25 - 0.5*4 = 26.751$

(d) Is the estimate of skipping class statistically significant? How do you know? Is the estimate of skipping class economically significant? How do you know? (Hint: Suppose there are 45 lectures in a typical semester long class).

If we are interested in analyze if we have a significant relationship between the dependent and the independent variable we can use the following system of hypothesis:

Null Hypothesis: $\beta_i = 0$

Alternative hypothesis: $\beta_i \neq 0$

Or in other wouds we want to check if an specific slope is significant.

The significance level assumed for this case is $\alpha=0.05$

Th degrees of freedom for a linear regression is given by $df=n-p-1 = 45-3-1 = 41$ , where p =3 the number of variables used to estimate the dependent variable.

In order to test the hypothesis the statistic is given by:

$t=\frac{\hat \beta_i}{SE_{\beta_i}}$

And replacing we got:

$t = \frac{-0.5}{0.0001}=-5000$

And for this case we see that if we find the p value for this case we will get a value very near to 0, so then we can conclude that this coefficient would be significant for the regression model .

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2 years ago

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1/4 divided by 2 is the answer

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3 years ago

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Black_prince [1.1K]

Answer:

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Step-by-step explanation: