Prove this please
Optional Math




Cos2A =
[tex] \frac{( \cot\alpha - \tan\alpha ) }{( \cot\alpha + \tan\alpha ) } [/tex]

​

[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Let's solve ~

Calculate discriminant :

[tex]\qquad \sf  \dashrightarrow \: 3 {x}^{2} + 6x - 1[/tex]

[tex]\qquad \sf  \dashrightarrow \: discriminant = {b}^{2} - 4ac[/tex]

[tex]\qquad \sf  \dashrightarrow \: d = (6) {}^{2} - (4 \times 3 \times 1)[/tex]

[tex]\qquad \sf  \dashrightarrow \: d = 36 - 12[/tex]

[tex]\qquad \sf  \dashrightarrow \: d = 24[/tex]

[tex]\qquad \sf  \dashrightarrow \: \sqrt {d} = 2 \sqrt{6} [/tex]

Now, let's calculate it's roots ( x - intercepts )

[tex]\qquad \sf  \dashrightarrow \: x = \cfrac{ - b \pm \sqrt{d} }{2a} [/tex]

[tex]\qquad \sf  \dashrightarrow \: x = \cfrac{ - 6\pm 2 \sqrt{6} }{2 \times 3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: x = \cfrac{ - 6\pm 2 \sqrt{6} }{6} [/tex]

So, the intercepts are :

[tex]\qquad \sf  \dashrightarrow \: x = \cfrac{ - 6 - 2 \sqrt{6} }{6} [/tex]

and

[tex]\qquad \sf  \dashrightarrow \: x = \cfrac{ - 6 + 2 \sqrt{6} }{6} [/tex]

Answer:

[tex]\left( \dfrac{ -3 + 2\sqrt{3}}{ 3}, \ 0\right), \ \left(\dfrac{ -3 - 2\sqrt{3}}{ 3}, \ 0\right)[/tex]

Explanation:

Given expression:

f(x) = 3x² + 6x - 1

Use quadratic formula:

[tex]\sf x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{2a} \ where \ ax^2 + bx + c = 0[/tex]

Here after finding coefficients:

Applying formula:

[tex]x = \dfrac{ -6 \pm \sqrt{6^2 - 4(3)(-1)}}{2(3)}[/tex]

[tex]x = \dfrac{ -6 \pm \sqrt{48}}{6}[/tex]

[tex]x = \dfrac{ -6 \pm 4\sqrt{3}}{6}[/tex]

[tex]x = \dfrac{ -6 \pm 4\sqrt{3}}{2 \cdot 3}[/tex]

[tex]x = \dfrac{ -3 \pm 2\sqrt{3}}{ 3}[/tex]

[tex]x = \dfrac{ -3 + 2\sqrt{3}}{ 3}, \ \dfrac{ -3 - 2\sqrt{3}}{ 3}[/tex]

Answer:

1

Step-by-step explanation:

The slope of a function is equal to the coefficient of x when the equation is in standard form (y=mx+c)

In this case the equation is already in standard form and thusly we can see that the coefficient of x is 1 (the constant has no effect on the slope)

Answer:

E

Step-by-step explanation:

line I slopes downwards from left to right and has a negative slope.

line K is a horizontal line and has a slope of zero

line J slopes upwards from left to right and has a positive slope

slopes from least to greatest are therefore I, K, J

Answer:

Value of a is 16.

Step-by-step explanation:

Solution Given:

we know that

Geometric mean=[tex]\sqrt{a*b}[/tex]

By using this formula

20=[tex]\sqrt{a*25}[/tex]

20=5[tex]\sqrt{a}[/tex]

dividing both side by 5, we get

20/5=5/5*  [tex]\sqrt{a}[/tex]

4=[tex]\sqrt{a}[/tex]

squaring both side

4²=a

:. a=16

The solution for the system of linear equations 2x- y = 3 and y - x = 1 are x = 4 and y = 5

Linear equations are equations that have constant average rates of change, slope or gradient

A system of linear equations is a collection of at least two linear equations.

In this case, the system of equations is given as

2x- y = 3

y - x = 1

Make y the subject in the second equation, by adding x to both sides of the equation

y - x + x = x + 1

This gives

y = x + 1

Substitute y = x + 1 in 2x- y = 3

2x- x - 1 = 3

Evaluate the like terms

x = 4

Substitute x = 4 in y = x + 1

y = 4 + 1

Evaluate

y = 5

Hence, the solution for the system of linear equations 2x- y = 3 and y - x = 1 are x = 4 and y = 5

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Answer:

y = 2(1/2)^x --> Second choice

y = (1/4)^x --> First choice

y = 3^x --> Third choice

Step-by-step explanation:

Take a look at the images uploaded for the reasons why.

y = 2(1/2)ˣ is y intercept (0,2) second choice, y = (1/4)ˣ is y intercept (0,1) first choice, y = 3ˣ is y intercept (0,1), fourth choice.

Let's break down each of the given exponential functions and understand their y-intercepts:

y = 2(1/2)ˣ :

When x = 0, the value of (1/2)ˣ  is 1, since any number raised to the power of 0 is 1. Therefore, y = 2 * 1 = 2 when x = 0. This gives us the y-intercept (0, 2).

y = (1/4)ˣ :

Similarly, when x = 0, the value of (1/4)ˣ  is 1, since any number raised to the power of 0 is 1. Therefore, y = 1 when x = 0. This gives us the y-intercept (0, 1).

y = 3ˣ :

Again, when x = 0, the value of 3ˣ  is 1. Therefore, y = 1 when x = 0. This gives us the y-intercept (0, 1).

The y-intercept is the point where the graph of the function intersects the y-axis, which occurs when x is zero. In all three cases, when x = 0, the exponent becomes zero, resulting in the base being raised to the power of zero, which is always equal to 1. The graph is given below.

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The sum of the complex number in standard form is 8 + 3i

Complex numbers are square roots of negative numbers. The square root of -1 for example is equal to "i" as shown;

√-1 = i

The complex number is written as x + iy. Given the expression

(9 + 6i) − (1 + 3i)

Expand to have:

9 + 6i − 1 - 3i

Collect the like terms

9-1 + 6i - 3i

Group into real and imaginary parts

(9-1) + (6i-3i)

8 + 3i

Hence the sum of the complex number in standard form is 8 + 3i

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Answer:

the order is Paco, Tim, Jenny, Maria

Step-by-step explanation:

Answer:

b.

Paco, Tim, Jenny, Maria

Step-by-step explanation:

The expressions which demonstrates the identity property of multiplication is; 25(1) = 25 option B

Identity Property of Multiplication states that any number multiplied by 1 does not change, that is, it is constant or remains the same

Check all options

25(0) = 25

0 = 25

Not true

25(1) = 25

25 = 25

True (identity property of multiplication holds)

25 + 0 = 25

25 = 25

True (Not identity property of multiplication)

25 + 1 = 25

26 = 25

Not true

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Answer:

The answer is 1

Step-by-step explanation:

Formula for finding the slope: [tex]\frac{y2-y1}{x2-x1}[/tex]

Take y2 and y1 from (4,4) (1,1)

Take x2 and x1 from (4,4) (1,1)

[tex]\frac{y2-y1}{x2-x1} = \frac{4-1}{4-1} = \frac{3}{3} = 1[/tex]

This is how the answer is 1

Hope it helps :)

Answer:

Exponential function

[tex]y=1.25(2)^x[/tex]

Step-by-step explanation:

Definitions

Asymptote: a line that the curve gets infinitely close to, but never touches.

Hole: a point on the graph where the function is not defined.

[tex]f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_0[/tex]

An equation containing variables with non-negative integer powers and coefficients, that involves only the operations of addition, subtraction and multiplication.

A continuous function with no holes or asymptotes.

[tex]f(x)=\dfrac{h(x)}{g(x)}[/tex]

An equation containing at least one fraction whose numerator and denominator are polynomials.

A rational function has holes and/or asymptotes.

[tex]f(x) =\log_ax[/tex]

A continuous function with a vertical asymptote.

A logarithmic function has a gradual growth or decay.

[tex]f(x)=ab^x[/tex]

The variable is the exponent.

A continuous function with a horizontal asymptote.

An exponential function has a fast growth or decay.

Answer: exponential

Step-by-step explanation:

Answer:

16.8

Step-by-step explanation:

here is that kite on a graph. you can see there are 2 sides of length 5, and 2 sides of length 3, so closest perimeter is 16.8

a. θ = 36. 9°

b. θ = 114. 2°

a. We use the cosine identity

cos θ = adjacent/hypotenuse

Adjacent = 4cm

Hypotenuse = 5cm

cos θ = 4/ 5

cos θ = 0. 8

Now, we have the inverse of the cosine value

θ = cos^-1 (0. 8)

θ = 36. 9°

b. We use the tangent identity

tan a = opposite / adjacent

tan a = 3. 8/1. 7

tan a = 2. 23

a = tan^-1 ( 2. 23)

a = 65. 8

We have;

a + θ = 180 °; angles on a straight line

θ = 180 - 65. 9

θ = 114. 2°

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Answer:

[tex]\sin\left(x+\frac{\pi}{7} \right)[/tex]

Step-by-step explanation:

[tex]\sin\left(x+\frac{\pi}{7} \right)=\sin \frac{\pi}{7}\cos x +\cos \frac{\pi}{7} \sin x[/tex]

Answer:

3 cm

Step-by-step explanation:

The formula for the area of a circle is:

[tex]A = \pi r^2[/tex]

Where "r" represents the radius of the circle. We can substitute the given value for Area into the equation to solve for the radius.

[tex]A=\pi r^2\\9\pi = \pi r^2[/tex]

Divide both sides by π

[tex]9=r^2[/tex]

Take the square root of both sides

[tex]\sqrt9=\sqrt {r^2}\\r=3[/tex]

The radius of the circle is 3 cm.

In (A) statistics class Wilson's students earn a higher ratio of a's to b's.

To find the higher ratio:

Given -

As we can observe that 3:5 < 11:15.

Therefore, in (A) statistics class Wilson's students earn a higher ratio of a's to b's.

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The value of x in given linear equation is x = 3.

According to the statement

we have given that the equation and we have to find the solution of that equation.

So, For this purpose,

The given equation is :

[tex]\sqrt{ x+1} = \sqrt{ x+6} -1[/tex]

From this equation it is clear that the it is a linear equation.

So, for find the value of x

To solve this equation squaring on both sides then equation become

x+1 = x+6 +1 -2√x+6.

Then

-2√x+6 = -6

then

[tex]\sqrt{x + 6} = 3[/tex]

Now remove the square by S.B.S.

Then the equation become

x +6 = 9

x = 3.

In this equation the value of x is 3.

So, The value of x in given linear equation is x = 3.

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Answer:

In standard form x^2/64 + y^2/100 = 1

Step-by-step explanation:

The major axis is the y axis because 10 has a larger value than 8.  

8x8 = 64

10 x10 = 100

The perimeter of the shape is 52 cm

The shape is given as:

Regular pentagon

The length of the sides are

Length = 12 cm

So, the perimeter of the regular pentagon is

P = 5* Length

This gives

P = 5 * 12cm

Evaluate

P = 60 cm

When the square of sides 6cm is removed from the regular pentagon, the perimeter becomes

P = 60cm - 6cm

Evaluate the difference

P =52 cm

Hence, the perimeter of the shape is 52 cm

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Answer:

416 people

Step-by-step explanation:

The way you get the answer is by adding up the number of people from the sample, which gives you 175 people. After finding that you take the number of people who like pretzels (26) and divide it by 175, giving you 0.16 or 16%.
This means 16% of people will like pretzels, so you apply this to the actual number of people coming, 2600, and find 16% of 2600 (0.16 x 2600) which is 416 people. I hope this makes sense :)

Six different sized rectangles that have an area of 64 square units:

1) 64 units x 1 unit
2) 32 units x 2 units

3) 16 units x 4 units

4) 40 units x 1.4 units

5) 20 units x 3.2 unit

6) 10 units x 6.4 units

For given question,

Let l be the length of the rectangle and w be the width of the rectangle.

We know that, the area of a rectangle is equal to

A = l × w

Here, A = 64 square units

So, we get an equation

l × w = 64                     ...............................(1)

let's assume different values of l to get the different values of w.

Case 1) For l = 64 units

substitute in the equation 1

⇒ 64 × w = 64

⇒ w = 64/64

⇒ w = 1 units

The dimensions of the rectangle are 64 units x 1 unit

See the draw in the attached figure 1

Case 2) For l = 32 units

substitute in the equation 1

⇒ 32 × w = 64

⇒ w = 64/32

⇒ w = 2 units

The dimensions of the rectangle are 32 units x 2 unit

See the draw in the attached figure 2

Case 3) For l = 16 units

substitute in the equation 1

⇒ 16 × w = 64

⇒ w = 64/16

⇒ w = 4 units

The dimensions of the rectangle are 16 units x 4 unit

See the draw in the attached figure 3

Case 4) For l = 40 units

substitute in the equation 1

⇒ 40 × w = 64

⇒ w = 64/40

⇒ w = 1.4 units

The dimensions of the rectangle are 40 units x 1.4 unit

See the draw in the attached figure 4

Case 5) For l = 20 units

substitute in the equation 1

⇒ 20 × w = 64

⇒ w = 64/20

⇒ w = 3.2 units

The dimensions of the rectangle are 20 units x 3.2 unit

See the draw in the attached figure 5

Case 6) For l = 10 units

substitute in the equation 1

⇒ 10 × w = 64

⇒ w = 64/10

⇒ w = 6.4 units

The dimensions of the rectangle are 10 units x 6.4 unit

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There are 1.40 groups of 5/7 in 1.

8 divided by 5/7 is  11 1/5.

Here is the completed table:

Length (cm)       Percentage

  12                          60% of Ehecatl's old hair

   4                         20% of Ehecatl's old hair

  20                           100% of Ehecatl's old hair

Division is the process of grouping a number into equal parts using another number. The sign used to denote division is ÷. Division is one of the basic mathematical operations.

In order to determine how many groups of 5/7 are in 1, divide 1 by 5/7

1 ÷ 5/7

1 x 7/5 = 7/5  = 1.40

8 ÷  5/7

8 x 7/5

= 56 / 5

= 11 1/5

The first step is to determine the length of her hair before she cut it

Length of the hair before it was cut = new length / 60%

12 / 0.6 = 20

60% x 20 = 12cm

20% x 20 = 4cm

100% x 20 = 20cm

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The tessellated plane has translational symmetry option (A) translational symmetry is correct.

It is defined as the change in coordinates and the shape of the geometrical body. It is also referred to as a two-dimensional transformation. In the geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, and glide translation.

A geometric figure is given:

As we know, from geometric transformation, geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, etc.

Without any further changes, the tessellated plane is moved a few units to the right. If the translated plane is again moved left by the same amount of units, we will once more obtain the original plane. Translational symmetry is present as a result.

Thus, the tessellated plane has translational symmetry option (A) translational symmetry is correct.

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The distance from the center of the ellipse to where the spots are located will be 2 feet.

What is Distance?

The distance from the center of the ellipse to where the spots are located will be -

= 5 - 3

= 2 feet.

Therefore, the distance from the center of the ellipse is 2 feet.

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The table represent y as a function of c, Yes because every x-value has at least one y-value.

According to the question,

X       -3          0          5            11

Y        0           2          -3          -2

In order to find the table represent y as a function of c, When x = -3; y=0; x= 0 ;y=2; x= 0 y=2; x= 5 y=-3; x= 11 y=-2.

⇒ Above option is wrong, because a number in the input is not same as a number in the output.

⇒ Above option is wrong, because there are many x-values with more than one y-value.

⇒ Above option is correct, because every x-value has at least one y-value.

⇒ Above option is wrong,  because there are not two inputs that have the same output

Hence, the table represent y as a function of c, Yes, because every x-value has at least one y-value. Above option is correct, because every x-value has at least one y-value.

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Answer:

a7=4/3 or a7=-4/3

Step-by-step explanation:

using the geometric sequences formula

an=ar^n-1

a2=ar

a4=ar³

when a2=324 and a4=36

324=ar...........(1)

36=ar³............(2)

from equation (1) a=324/r substitute in equation (2)

we have :

36=324/r *r³

36=324r²

r²=36/324

r²=1/9

r=±1/3

substitute when r=±1/3 in (1)

324=a(±1/3)

a=±972

so the 7th term is

when r=±1/3

we have

a7=ar^6

a7=±972(±1/3)^6

a7=972/729

a7=4/3 or a7=-4/3

The 5.25 m combined height of John, Jim, and Joe, the 12 cm height difference between John and Jim and the 0.09 m difference in height between Jim and Joe, indicates that solution to the word problem is Joe was 1.73  meters tall

A word problem is a presentation of a math problem using verbal description rather than numbers, variables and operators.

The combined height of John, Jim and Joe = 5.25 m

John's height = Jim's height - 12 cm = Jim's height - 0.12

Jim's height = Joe's height + 0.09 m

Let h represent Joe's height, we get;

Jim's height = h + 0.09

John's height = h + 0.09 - 0.12 = h - 0.03

John's height = h - 0.03

The sum of the heights is therefore; h + h + 0.09 + h - 0.03 = 3·h + 0.06

The sum of their heights = Their combined height = 5.25 meters

Therefore; 3·h + 0.06 = 5.25

h = (5.25 - 0.06)/3 = 1.73

Joe's height, h = 1.73 meters

Jim's height = 1.73 + 0.09 = 1.82

Jim's height = 1.82 meters

John's height = 1.73 - 0.03 = 1.7

John's height is 1.7 meters

Joe was 1,73 meters tall

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The first scatter plot does not have a zero correlation.

Option(a) is correct.

A statistic called correlation gauges how much two variables change in connection to one another.

Correlation quantifies correlation but cannot determine whether x causes y or vice versa, or whether a third component is responsible for the association.

A scatterplot may make it easier to spot correlation, particularly when the variables have a non-linear but nevertheless significant association.

Zero means there is no correlation between the two variables under comparison.

A 0 correlation indicates that there is no relationship between the two variables according to the correlation statistic. This merely indicates that there isn't a linear relationship, not that there isn't any link at all. The first scatter plot does not represent a linear relationship, thus, it has zero correlation.

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I assume the given series is

[tex]\displaystyle 2 + \frac4{2^2} + \frac8{3^2} + \frac{16}{4^2} + \cdots = \frac{2^1}{1^2} + \frac{2^2}{2^2} + \frac{2^3}{3^2} + \frac{2^4}{4^2} + \cdots = \sum_{n=1}^\infty \frac{2^n}{n^2}[/tex]

By the ratio test, the series diverges, since

[tex]\displaystyle \lim_{n\to\infty} \left|\frac{2^{n+1}}{(n+1)^2} \cdot \frac{n^2}{2^n}\right| = 2 \lim_{n\to\infty} \frac{n^2}{(n+1)^2} = 2 > 1[/tex]

[tex]~~~~~~~~~~~~\textit{quadratic formula} \\\\ 0=\stackrel{\stackrel{a}{\downarrow }}{3}x^2\stackrel{\stackrel{b}{\downarrow }}{+2}x\stackrel{\stackrel{c}{\downarrow }}{-5} \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ x= \cfrac{ - (2) \pm \sqrt { (2)^2 -4(3)(-5)}}{2(3)} \implies x = \cfrac{ -2 \pm \sqrt { 4 +60}}{ 6 } \\\\\\ x= \cfrac{ -2 \pm \sqrt { 64 }}{ 6 }\implies x=\cfrac{ -2 \pm 8}{ 6 }\implies x= \begin{cases} ~~ 1\\ -\frac{5}{3} \end{cases}[/tex]

Answer:

-5/3 and -1

Step-by-step explanation:

I am not sure what you mistake is, but here is my solution.  I hope that it helps.  I am sorry that you are frustrated.  We have all been there.