Please help by showing me how to answer this

Y = 5 cos(0.3185x) + 10 is the volume of oxygen as a function of time can be modeled by this cosine function.

What is  cosine function ?

The cosine function is a mathematical function that relates the ratio of the sides of a right triangle. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In trigonometry, the cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. The cosine function is periodic, meaning it repeats itself at regular intervals, and has a range of values between -1 and 1. It is commonly used in mathematics, physics, and engineering to model periodic phenomena such as sound waves, electromagnetic waves, and oscillations.

According to the question:
To find the equation of the cosine function, we need to identify the amplitude, period, phase shift, and vertical shift of the function based on the given information.

Since point M is the minimum value of the function, the vertical shift is 10. This means that the equation of the function is of the form:

Y = A cos(Bx - C) + D

where D = 10.

To find the amplitude, we need to find the distance between the maximum and minimum values of the function. Since the graph of a cosine function oscillates between its maximum and minimum values, the amplitude is half the distance between these values.

From the graph, we can see that the maximum value of the function is 20, so the distance between the maximum and minimum values is:

20 - M = 20 - 10 = 10

Therefore, the amplitude is:

A = 10/2 = 5

To find the period, we need to find the distance between two consecutive peaks or troughs of the function. From the graph, we can see that the distance between two consecutive peaks is approximately 6.28 units. This means that the period is:

P = 6.28

To find the phase shift, we need to find the horizontal shift of the function from its standard form. Since the minimum value of the function occurs at x = 0, the phase shift is:

C =

Therefore, the equation of the cosine function is:

Y = 5 cos((2π/P)x - C) + D

Y = 5 cos((2π/6.28)x) + 10

Simplifying,

Y = 5 cos(0.3185x) + 10

So, the volume of oxygen as a function of time can be modeled by this cosine function.

To know more about cosine function visit:
https://brainly.com/question/23437509
#SPJ1

Answer:

1x

Step-by-step explanation:

Firstly lets expand the brackets for the equation

(x - 7 )2 = 36

If we multiply what's in the brackets by 2 we get this:

2x - 14 = 36

Add 14 to both sides:

2x = 50

Divide both sides by 2:

x = 25

Answer = 1x (Only possible solution

Answer:

The two solutions to the given equation are x = 13 and x = 1.

Step-by-step explanation:

To solve the given equation (x - 7)² = 36, begin by square rooting both sides:

[tex]\implies \sqrt{(x-7)^2}=\sqrt{36}[/tex]

[tex]\implies x-7=\pm6[/tex]

Now add 7 to both sides of the equation:

[tex]\implies x-7+7=\pm6+7[/tex]

[tex]\implies x=7\pm6[/tex]

Therefore, the two solutions are:

[tex]\implies x=7+6=13[/tex]

[tex]\implies x=7-6=1[/tex]

The slope for the given point through which it passes through the graph is [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] = infinity .

What about slope?

The slope of a line is a measure of its steepness. It is defined as the ratio of the vertical change (rise) between two points on the line to the horizontal change (run) between the same two points. In other words, it is the rate at which the line rises or falls as we move along it in the horizontal direction.

Define graph:

In mathematics, a graph is a visual representation of a set of data or mathematical relationships between variables. Graphs can be used to display and analyze data in a variety of formats, including line graphs, bar graphs, scatter plots, and pie charts.

According to the given information:

As, we know that the slope =  [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex]

⇒ In which the point given are (-1,8) and (-1,-4)

By putting the value of the given point we have that

⇒ [tex]\frac{-4-(8)}{-1-(-1)}[/tex] = infinity.

To know more about slope visit:

https://brainly.com/question/3605446

#SPJ1

The equation's only solution is x = -17.

Simplifying the left-hand side of the equation:

-5 + x - 2 = -7 + x

Simplifying the right-hand side of the equation:

3x + 10 - x = 2x + 10

Substituting into the original equation, we get:

-7 + x = 2x + 10

Subtracting x from both sides, we get:

-7 = x + 10

Subtracting 10 from both sides, we get:

-17 = x

So the only solution to the equation is x = -17. Therefore, the correct statement about the solution(s) of the equation is:

• The equation's only solution is x = -17.

Learn more about equation here

https://brainly.com/question/29657992

#SPJ1

Answer:

C

.........................

Answer: the answer is A

Step-by-step explanation: The process of nuclear fusion is the union of two light atomic nuclei into one heavier one while releasing enormous quantities of energy.

Answer:

Step-by-step explanation:

It will take 50 minutes for the turtle and snail to meet.

because they are all moving at feet per minute we can create a formula

total feet = (turtle feet per minute) + (snail feet per minute)

300 = 5m +1m

combine like terms

300 = 6m

divide both sides by 6

50=m

Answer:

See below.

Step-by-step explanation:

Let's denote the distance the turtle walks by x. Then the distance the snail crawls would be 300 − x.

We can now set up an equation to represent the situation. Since distance = rate × time, we have

x/5 = (300 - x)/1

Solving for x, we get

x = 250

So the turtle walks 250 feet before meeting the snail, and the snail crawls the remaining 50 feet.

To find the time it takes for them to meet, we can use either of the two distances and its corresponding rate:

time = distance/rate

For example, using the turtle's distance

time = 250/5 = 50 minutes

Therefore, it takes 50 minutes for the turtle and the snail to meet.

The graph represents a proportional relationship.

What is proportional relationship?

A proportional relationship is a relationship between two variables where their ratio remains constant. This means that as one variable increases or decreases, the other variable changes proportionally in order to maintain a constant ratio. In other words, the two variables are directly proportional to each other. This relationship can be represented by a straight line passing through the origin on a graph. Proportional relationships are commonly used in various fields such as physics, finance, and engineering to analyze and predict outcomes.

Explaining the table, equation and graph to know which represents a proportional relationship :

The relationship between the values in the given table is not proportional. If it was a proportional relationship, then the ratio of y to x would be constant. However, in this case, the ratios are not equal. For example, the ratio of y to x for the first row is 3.6/3 = 1.2, but the ratio of y to x for the second row is 6/5 = 1.2, and the ratio of y to x for the third row is 7.2/8 = 0.9. Therefore, the ratios are not equal and the relationship is not proportional.

The equation y=2x+5 does not represent a proportional relationship. In a proportional relationship, there is a constant ratio between the two variables. However, in this equation, the ratio between y and x is not constant, but rather it increases as x increases.

The graph represents a proportional relationship as the ratio between y and x is  constant.

To know more about proportional relationship visit :

brainly.com/question/29765554

#SPJ1

The differentiation of the variable y is equal to  [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex]  for the differential equation.

Given equation: [tex](x^{2} +y^{2} )^{2} = 2xy^{2}[/tex]

differentiate with respect to x

2 [tex](x^{2} + y^{2})[/tex] [ [tex]2x+2y.\frac{dy}{dx}[/tex] ] =[ (1).[tex]y^{2}[/tex] + [tex]x (2y) + \frac{dy}{dx}[/tex] ]

4 [tex](x^{2} + y^{2})[/tex] [ [tex]x+y \frac{dy}{dx}[/tex] ] = [tex]y^{2}[/tex] + [tex]2xy \frac{dy}{dx}[/tex]

4( [tex]x^{3} + x^{2} y \frac{dy}{dx} + xy^{2} + y^{3} \frac{dy}{dx}[/tex] ) = [tex]y^{2}[/tex] + [tex]2xy \frac{dy}{dx}[/tex]

[tex]4x^{3} +4 x^{2} y \frac{dy}{dx} +4 xy^{2} +4 y^{3} \frac{dy}{dx}[/tex] = [tex]y^{2}[/tex] +  [tex]2xy \frac{dy}{dx}[/tex]

[tex]4x^{2}y \frac{dy}{dx}[/tex] + [tex]4y^{3}[/tex] [tex]\frac{dy}{dx}[/tex] - [tex]2xy\frac{dy}{dx}[/tex] = [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex]

[tex](4x^{2}y + 4y^{3} - 2xy )[/tex]  [tex]\frac{dy}{dx}[/tex] =  [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex]

[tex]\frac{dy}{dx}[/tex]  = [tex]y^{2} - 4x^{3} - 4xy^{2}[/tex] / [tex](4x^{2}y + 4y^{3} - 2xy )[/tex]

[tex]\frac{dy}{dx}[/tex] = [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex]

Hence solved. The differentiation for the given differential equation is done using the technique of implicit differentiation. The differentiation of the variable y is equal to  [tex]\frac{1}{2} ( \frac{y^{2}-4x^{3}-4xy^{2} }{2x^{2} y+2y^{3}-xy } )[/tex]  for the differential equation.

Learn more about Implicit Differentiation:

https://brainly.com/question/2292940

#SPJ1

We will use sine function, and length of the opposite side is [tex]\frac{25\sqrt{3}}{2}$[/tex] feet.

To solve for the opposite side of the right triangle, we would use the sine function. In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

In this problem, we are given that one angle of the right triangle is 60 degrees and the hypotenuse is 25 feet. Let's label the opposite side as x and the adjacent side as y. Then we can use the sine function:

[tex]$\sin 60^\circ = \frac{x}{25}$[/tex]

Simplifying the left side, we have:

[tex]$\frac{\sqrt{3}}{2} = \frac{x}{25}$[/tex]

Multiplying both sides by 25, we get:

[tex]$x = \frac{25\sqrt{3}}{2}$[/tex]

Therefore, the length of the opposite side is [tex]\frac{25\sqrt{3}}{2}$[/tex] feet.

To know more about Function visit:

brainly.com/question/28193995

#SPJ1

Answer:

7

Step-by-step explanation:

if you add 7 4's you get 28.

4+4=8+4=12+4=16+4=20+4=24+4=28

now if you count all the single 4's then you will get 7

The exact value of tan(2θ) in simplest radical form is 58√(793) / 48 which has been calculated through Pythagorean theorem.

The Pythagorean Theorem can be used to find the correct angled triangle's missing length. The triangle contains three sides: the hypotenuse, this same opposite, which will always be the longest, and the adjacent side, which really doesn't touch the hypotenuse. The Pythagorean equation is: a² + b² = c².

We know that sin(θ) = 29/4 and that θ is in Quadrant I, which means that all three trigonometric functions (sine, cosine, and tangent) are positive in this quadrant.

Using the identity:

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

We can find the value of tan(2θ) by first finding tan(θ) and then using it to calculate tan(2θ).

To find tan(θ), we can use the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

cos²(θ) = 1 - sin²(θ)

cos(θ) = ± √(1 - sin²(θ))

Since θ is in Quadrant I, we know that cos(θ) is positive, so we take the positive square root:

cos(θ) = √(1 - (29/4)²) = √(793) / 4

Now we can find tan(θ) as:

tan(θ) = sin(θ) / cos(θ) = (29/4) / (√(793) / 4) = 29 / √(793)

Substituting this into the formula for tan(2θ), we get:

tan(2θ) = 2tan(θ) / (1 - tan²(θ))

tan(2θ) = 2(29 / √(793)) / (1 - (29 / √(793))²)

tan(2θ) = 2(29 / √(793)) / (1 - 841/793)

tan(2θ) = 58√(793) / 48

Therefore, the exact value of tan(2θ) in simplest radical form is 58√(793) / 48.

To know more about Pythagorean visit:

brainly.com/question/343682

#SPJ1

Answer:

4/8 in lowest terms is 1/2.

Step-by-step explanation:

Answer: 1/2

4/8 is equivalent to 1/2

Kenny will spend $4,593.75 on fencing the dilated rectangle.

What is rectangle ?

A rectangle is a quadrilateral with four right angles (90-degree angles) and opposite sides of equal length. It is a type of parallelogram in which both pairs of opposite sides are parallel and congruent (of equal length). The opposite sides of a rectangle are also perpendicular (form a right angle) to each other. The area of a rectangle is equal to the product of its length and width, while its perimeter is equal to the sum of the lengths of all its sides. Rectangles are commonly used in geometry, architecture, engineering, and many other fields.

According to the question:
Since each unit on the graph represents 25 feet, the dimensions of the original rectangle ABCD are:

AB = 4 units, which represents 4 x 25 = 100 feet

BC = 3 units, which represents 3 x 25 = 75 feet

Therefore, the perimeter of ABCD is 2(AB + BC) = 2(100 + 75) = 350 feet.

When dilating by a scale factor of 2.5, each dimension of ABCD will be multiplied by 2.5. Therefore, the dimensions of the dilated rectangle A'B'C'D' are:

A'B' = AB x 2.5 = 100 x 2.5 = 250 feet

B'C' = BC x 2.5 = 75 x 2.5 = 187.5 feet

The perimeter of A'B'C'D' is 2(A'B' + B'C') = 2(250 + 187.5) = 875 feet.

Therefore, Kenny will need to fence a perimeter of 875 feet. At a cost of $5.25 per foot, the total cost of fencing will be:

875 feet x $5.25/foot = $4,593.75

Therefore, Kenny will spend $4,593.75 on fencing the dilated rectangle.

To know more about rectangle visit:
https://brainly.com/question/15238546
#SPJ1

Answer:y=0.5

Step-by-step explanation:Comme tu peux le voir, y est égal à au tiers de 3, se qui équivaut à 1. Si 2x=y, cela signifie que x=y/2, soit 0,5

Answer:

X=-1 or 9

Step-by-step explanation:

using the quadratic formula:

[tex]X=\frac{-b+-\sqrt{b^{2} -4ac} }{2a}[/tex]

X=-1

X=9

Answer:1 and 2

Step-by-step explanation:

so, they asked 50 students and then that was later broke down to smaller groups.

10,15,25

so, for every 10 students will be one. 1 vote= 10 students.

also, the dinosaur had the greatest number of votes so that explains answer 2.

All the true statements, if each interior angle measure of a regular polygon is (2x + 20)° include the following:

A. If x = 60, then the regular polygon is a nonagon.

C. If x = 77, then the regular polygon is a 60-gon.

D. If x = 65, then the regular polygon is a dodecagon.

D. If x = 44, then the regular polygon is a pentagon.

In Geometry, the measure of each interior angle of a regular polygon can be calculated by using this mathematical expression:

Interior angle = [180 × (n - 2)]/n

Where:

n represents the number of sides of a regular polygon.

When x = 60, the number of sides of the regular polygon is given by:

(2x + 20)° = [180 × (n - 2)]/n

(2(60) + 20)° = [180 × (n - 2)]/n

140 = [180 × (n - 2)]/n

140n = 180n - 360

360 = 40n

n = 9 sides.

When x = 77, the number of sides of the regular polygon is given by:

(2x + 20)° = [180 × (n - 2)]/n

(2(77) + 20)° = [180 × (n - 2)]/n

174 = [180 × (n - 2)]/n

174n = 180n - 360

360 = 6n

n = 60 sides.

When x = 65, the number of sides of the regular polygon is given by:

(2x + 20)° = [180 × (n - 2)]/n

(2(65) + 20)° = [180 × (n - 2)]/n

150 = [180 × (n - 2)]/n

150n = 180n - 360

360 = 30n

n = 12 sides.

When x = 41, the number of sides of the regular polygon is given by:

(2x + 20)° = [180 × (n - 2)]/n

(2(41) + 20)° = [180 × (n - 2)]/n

102 = [180 × (n - 2)]/n

102n = 180n - 360

360 = 78n

n = 4.6 sides.

When x = 44, the number of sides of the regular polygon is given by:

(2x + 20)° = [180 × (n - 2)]/n

(2(44) + 20)° = [180 × (n - 2)]/n

108 = [180 × (n - 2)]/n

108n = 180n - 360

360 = 72n

n = 5 sides.

Read more on regular polygon here: brainly.com/question/20911145

#SPJ1

The probability of both gumballs not being red is 11/21.

What is the probability?

The probability of the first gumball not being red is:

P(first gumball not red) = P(blue) + P(yellow)

= (6/15) + (5/15)

= 11/15

After removing the first gumball, there are 14 gumballs left in the bag. The probability of the second gumball not being red depends on what color the first gumball was.

Case 1: The first gumball was blue

If the first gumball was blue, there are 5 blue, 4 red, and 5 yellow gumballs left in the bag. The probability of the second gumball not being red is:

P(second gumball not red | first gumball was blue) = P(blue or yellow)

= P(blue) + P(yellow)

= (5/14) + (5/14)

= 5/7

Case 2: The first gumball was yellow

If the first gumball was yellow, there are 6 blue, 4 red, and 4 yellow gumballs left in the bag. The probability of the second gumball not being red is:

P(second gumball not red | first gumball was yellow) = P(blue or yellow)

= P(blue) + P(yellow)

= (6/14) + (4/14)

= 5/7

The probability of both gumballs not being red is the product of the probabilities of each event:

P(both gumballs not red) = P(first gumball not red) * P(second gumball not red | first gumball not red)

P(both gumballs not red) = (11/15) * (5/7)

P(both gumballs not red) = 55/105

P(both gumballs not red) = 11/21

Therefore, the probability of both gumballs not being red is 11/21.

To know more about probability, visit:

https://brainly.com/question/11234923

#SPJ1

If her hοuse actually 56 feet wide, then scale οf her drawing is scale = 56 feet / (drawing width)

The slοpe is a mathematical cοncept that refers tο the steepness οf a line. It is cοmmοnly denοted by the letter "m" and can be calculated using the fοrmula:      

slοpe (m) = (change in y) / (change in x)

Withοut having the actual measurements οf the drawing, we cannοt determine the scale.

Tο find the scale οf a drawing, yοu need tο knοw the actual measurements οf the οbject being drawn as well as the cοrrespοnding measurements in the drawing. Fοr example, if yοu knοw that the actual width οf Rοse's hοuse is 56 feet and the width οf the hοuse in the drawing is 8 inches, yοu can determine the scale by using the fοllοwing fοrmula:

scale = actual width/drawing width

Using this fοrmula with the given infοrmatiοn, we get:

scale = 56 feet / (drawing width)

Therefοre, we dο nοt have the drawing width, we cannοt calculate the scale.

To know more about the slope and in the given link -

https://brainly.com/question/3605446

#SPJ1

The type of transformation represented in the figure is the translation transformation i.e. (c) none of these

Given the triangles ABC and A'B'C'

The transformation between the triangles is translation

The translation transformation is a type of transformation that moves an object without changing its size, shape, or orientation.

This transformation involves sliding an object in a particular direction by a certain distance, either horizontally or vertically.

In this case, the direction is horizontally and vertically

Read more about transformation at

https://brainly.com/question/27224272

#SPJ1

The difference between f and g can be written as:

(f - g)(x) = x² - 7x - 18

Here we have two polynomials given by:

f(x) = x² - 6x - 27

g(x) = x - 9

We want to find an expression for:

(f - g)(x)

This difference can be written as:

f(x) - g(x)

Now replacing the polynomials, we will get:

x² - 6x - 27 - (x - 9)

Now we can simplify this to get.

x² - 7x - 18

That is the simplest form.

Learn more about polynomials at:

https://brainly.com/question/4142886

#SPJ1

Answer:

Step-by-step explanation:

The surface area of a sphere is given by the formula:

S = 4πr^2

where S is the surface area and r is the radius of the sphere.

We are given that the surface area of the sphere is 60 square feet. Using the formula above, we can solve for the radius:

60 = 4πr^2

Dividing both sides by 4π, we get:

15/π = r^2

Taking the square root of both sides, we get:

r = sqrt(15/π)

Using 3.14 as an approximation for π, we can evaluate this expression:

r ≈ sqrt(15/3.14)

r ≈ 2.20

Therefore, the best approximation for the radius of the sphere is 2.20 feet.

Answer:

Radius is 2.18

Step-by-step explanation:

;)

For the given function  f(x) = ⁿ√p(x), The Domain depends on the value of n and The Domain is all real numbers if n is ODD.

What is the definition of a function?

In mathematics, a function is a relation between a set of inputs (also known as the domain) and a set of possible outputs (also known as the range) with the property that each input is associated with exactly one output.

In other words, a function takes an input value, performs a specific operation or set of operations on it, and produces an output value. It can be represented by a rule, formula, or graph that relates each input to its corresponding output.

Now,

The domain of the function f(x) = ⁿ√p(x) depends on the value of n.

If n is odd, then the function is defined for all real numbers, and the domain is all real numbers.

If n is even, then the function is defined only for non-negative real numbers, and the domain is all non-negative real numbers.

Therefore, the statement "The Domain is all real numbers" is not always true for the function f(x) = ⁿ√p(x). The correct statements are:

To know more about functions visit the link

brainly.com/question/12431044

#SPJ1

Answer:

It depends on the context of the problem and the interpretation of the ratio represented by the point (7, 8).

If we interpret the point (7, 8) as representing the ratio 7:8, then Adela's claim is incorrect. Adding the same number to both coordinates of the point would change the value of the ratio. For example, adding 1 to both coordinates would give the point (8, 9), which represents the ratio 8:9, which is not equivalent to 7:8.

However, if we interpret the point (7, 8) as representing a different type of ratio, such as the ratio of the distances from the point to two fixed points or the ratio of the areas of two shapes, then it may be possible to find an equivalent ratio by adding the same number to both coordinates of the point. In this case, Adela's claim could be correct.

Without more information about the context and interpretation of the ratio represented by the point (7, 8), it is difficult to determine the correctness of Adela's claim.

Answer:

Step-by-step explanation:

Your sister can use the formula for calculating the present value of an annuity to determine how much she can withdraw each year.

The formula for the present value of an annuity is:

PV = Payment x (1 - (1 + r)^-n) / r

Where:

- PV is the present value of the annuity

- Payment is the amount of each withdrawal

- r is the annual interest rate

- n is the number of periods (in this case, 25 years)

We can rearrange this formula to solve for Payment:

Payment = PV x r / (1 - (1 + r)^-n)

We know that your sister has R375 000 to start with, and she wants to end up with zero in 25 years. So, her present value (PV) is R375 000, and we can assume her future value (FV) is zero.

Using the future value formula, we can calculate the interest rate she needs to earn in order to end up with zero in 25 years:

FV = PV x (1 + r)^n

0 = 375000 x (1 + r)^25

(1 + r)^25 = 1

1 + r = (1)^1/25

r = 0

This means that your sister needs to withdraw all of her money over the 25 years in order to end up with zero at the end.

Her withdrawal each year would be:

Payment = PV x r / (1 - (1 + r)^-n)

Payment = 375000 x 0.08 / (1 - (1 + 0.08)^-25)

Payment = R34,028.82 per year

Your sister can withdraw R34,028.82 at the beginning of each year for the next 25 years, and she will end up with zero at the end, assuming she earns an 8% return on her invested funds.

Felicia swims 0.43 miles, walks 0.0057 miles and runs 0.57 miles. The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

The four fundamental operations, often known as "arithmetic operations," are said to adequately describe all real numbers. The mathematical operations following division, multiplication, addition, and subtraction are quotient, product, sum, and difference.

We are given that she swims 15 laps in a pool where each lap is 50 yards.

By using arithmetic operations, we get

Total yards = 15 * 50 = 750 yard

We know that 1 mile = 1760 yards

So,

750 yards = 0.43 miles

She walks 30 feet.

We know that 1 mile = 5280 feet

So,

30 feet = 0.0057 miles

She runs 5 laps around an indoor track that is 600 feet long.

So, total feet she ran = 5 * 600 = 3000 feet

We know that 1 mile = 5280 feet

So,

3000 feet = 0.57 miles

Hence, she swims 0.43 miles, walks 0.0057 miles and runs 0.57 miles.

Learn more about arithmetic operations from the given link

brainly.com/question/30283549

#SPJ1

according the given question the exact value of given expression is [tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]

First, we need to find [tex]$\sin x$[/tex] using the identity[tex]$\cos^2x + \sin^2x = 1$:$\sin^2x = 1 - \cos^2x = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25}$[/tex]

Since  [tex]$\frac{\pi}{2} < x < \pi$[/tex], we know that  [tex]$\frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}$[/tex]. Therefore, we can use the

identity [tex]$\tan\frac{x}{2} = \frac{\sin x}{1 + \cos x}$[/tex]:

[tex]$\tan\frac{x}{2} = \frac{\sqrt{\frac{9}{25}}}{1 - \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{1}{5}} = \boxed{3}$[/tex]

[tex]If $\tan x = \frac{40}{9}$ and $\pi < x < \frac{3\pi}{2}$, find $\cos\frac{x}{2}$.[/tex]

First, we need to find [tex]$\sin x$[/tex] using the identity [tex]$\tan^2x + 1 = \sec^2x$[/tex]:

[tex]$\sin x = \frac{\tan x}{\sec x} = \frac{\frac{40}{9}}{-\frac{9}{40}} = -\frac{160}{81}$[/tex]

[tex]Since $\pi < x < \frac{3\pi}{2}$, we know that $\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}$[/tex]. Therefore, we can use the identity [tex]$\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$[/tex]:

[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}} = -\sqrt{\frac{1 + \frac{\cos^2x}{\sin^2x}}{2}} = -\sqrt{\frac{\sin^2x + \cos^2x}{2\sin^2x}} = -\sqrt{\frac{1}{2(1 - \sin^2x)}}$[/tex]

Plugging in [tex]$\sin x = -\frac{160}{81}$[/tex] , we get:

[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]

To know more about problems related to expression , click here:

https://brainly.com/question/16804733

#SPJ1