solve 3(x+1)=7(x-2)-3

Answer:

∠C and ∠R

Step-by-step explanation:

The ASA postulate represents angle / side / angle

where the included side is the side between the vertices of the two angles.

∠H and ∠I and the sides HC and IR are the AS

The required angles would be ∠C and ∠R for ASA

Answer:

Decimal point

Definition:

A decimal point is the symbol that is used to make the separation between the whole part and the fractional part of a number. Currently, not only is a point used to separate both parts, but the comma can also be used.

Example:

The number 39.847, in which 39 is the whole number, while 847 is the decimal part where 8 represent tenths, 4 hundredsths, and 7 thousandths.

Answer:

2/9

Step-by-step explanation:

Prob(consonant on first spin)

= 1/3

Prob(vowel on 2nd spin)

= 2/3

prob(event as stated)

= (1/3)(2/3)

= 2/9

Answer:

[tex]BC=11\ cm[/tex]

Step-by-step explanation:

step 1

Find the measure of the arc DC

we know that

The inscribed angle measures half of the arc comprising

[tex]m\angle DBC=\frac{1}{2}[arc\ DC][/tex]

substitute the values

[tex]60\°=\frac{1}{2}[arc\ DC][/tex]

[tex]120\°=arc\ DC[/tex]

[tex]arc\ DC=120\°[/tex]

step 2

Find the measure of arc BC

we know that

[tex]arc\ DC+arc\ BC=180\°[/tex] ----> because the diameter BD divide the circle into two equal parts

[tex]120\°+arc\ BC=180\°[/tex]

[tex]arc\ BC=180\°-120\°=60\°[/tex]

step 3

Find the measure of angle BDC

we know that

The inscribed angle measures half of the arc comprising

[tex]m\angle BDC=\frac{1}{2}[arc\ BC][/tex]

substitute the values

[tex]m\angle BDC=\frac{1}{2}[60\°][/tex]

[tex]m\angle BDC=30\°[/tex]

therefore

The triangle DBC is a right triangle ---> 60°-30°-90°

step 4

Find the measure of BC

we know that

In the right triangle DBC

[tex]sin(\angle BDC)=BC/BD[/tex]

[tex]BC=(BD)sin(\angle BDC)[/tex]

substitute the values

[tex]BC=(22)sin(30\°)=11\ cm[/tex]

ANSWER

5

EXPLANATION

The equation that expresses the approximate height h, in meters, of a ball t seconds after it is launched vertically upward from the ground is

[tex]h(t) = - 4.9 {t}^{2} + 25t[/tex]

To find the time when the ball hit the ground,we equate the function to zero.

[tex] - 4.9 {t}^{2} + 25t = 0[/tex]

Factor to obtain;

[tex]t( - 4.9t + 25) = 0[/tex]

Apply the zero product property to obtain,

[tex]t = 0 \: or \: \: - 4.9t + 25 = 0[/tex]

[tex]t = 0 \: \: or \: \: t = \frac{ - 25}{ - 4.9} [/tex]

t=0 or t=5.1 to the nearest tenth.

Therefore the ball hits the ground after approximately 5 seconds.

Answer:

The ball will hit the ground after 5 seconds ⇒ first answer

Step-by-step explanation:

* Lets study the information in the problem

- The ball is lunched vertically upward from the ground with an initial

  velocity 25 meters per second

- The ball will reach the maximum height when its velocity becomes 0

- The ball will fall down to reach the ground again

- The equation of the height (h), in meters of the ball t seconds after

  it is lunched from the ground is h = -4.9t² + 25t

- When the ball hit the ground again the height of it is equal 0

∵ h = 0

∴ 0 = -4.9t² + 25t ⇒ Multiply the two sides by -1 and reverse them

∴ 4.9t²  - 25t = 0 ⇒ factorize it by taking t as a common factor

∴ t(4.9t - 25) = 0 ⇒ equate each factor by 0

∵ t = 0 ⇒ the initial time when the ball is lunched

∵ 4.9t - 25 = 0 ⇒ add 25 to both sides

∴ 4.9t = 25 ⇒ divide each side by 4.9

∴ t = 5.102 ≅ 5 seconds

* The ball will hit the ground after 5 seconds

Answer:

 [tex]y_1=-\frac{1}{2}\\\\y_2=-\frac{3}{2}[/tex]

Step-by-step explanation:

The Quadratic formula is:

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

Given the equation [tex]4y^2+ 8y +7 =4[/tex], you need to subtract 4 from both sides:

[tex]4y^2+ 8y +7 -4=4-4[/tex]

[tex]4y^2+ 8y +3 =0[/tex]

Now you can identify that:

[tex]a=4\\b=8\\c=3[/tex]

Then you can substitute these values into the Quadratic formula. Therefore, you get these solutions:

[tex]y=\frac{-8\±\sqrt{8^2-4(4)(3)} }{2(4)}[/tex]

[tex]y_1=-\frac{1}{2}\\\\y_2=-\frac{3}{2}[/tex]

Answer:

The solutions are y=-1/2 and y=-3/2

Step-by-step explanation:

Ok, for this problem we need to use the quadratic formula:

For [tex]ax^{2} +bx+c=0[/tex]

The values of x which are the solutions of the equation:

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

In this case your variable is y, so:

[tex]ay^{2} +by+c=0[/tex]

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

So, a=4, b=8 and c=3

[tex]y=\frac{-(8)+-\sqrt{(8)^{2}-4(4)(3) } }{2(4)}[/tex]

[tex]y=\frac{-(8)+-\sqrt{(16)}}{8}[/tex]

[tex]y=\frac{-8+4}{8}[/tex] and [tex]y=\frac{-8-4}{8}[/tex]

The solutions are

[tex]y=\frac{-1}{2}[/tex] and [tex]y=\frac{-3}{2}[/tex]

Answer:

18.85

Step-by-step explanation:

you divide 37.7 by 2

Answer:

360

Step-by-step explanation:

Write and solve an equation:

(1/6)(1/3)(1/4)n = 5

Multiplying the left side by (6)(3)(4), or 72, an

Multiply both sides by 6.  The resulting equation will still be true, but not have 6 in the denominator:

(1)(1/3)(1/4)n = 5(6) = 30

Mult both sides by 12.  The resulting equation will still be true, but not have (3)(4) in the denominator:

   (1)(1)(1)n = 360

Then n = 360.

Check:  Does (1/6)(1/3)(1/4)(360) = 5?

Yes.  This confirms that the unknown number n is 360.

Let us find the number step by step.
Let the unknown number be represented by 'x'.
We are given that one-sixth of one-third of one-fourth of this number is equal to five.
Let's break this down:
One-fourth of x is represented by x/4.
Now, we take one-third of that amount, which means we multiply (x/4) by 1/3, giving us (x/4) * (1/3).
Finally, we take one-sixth of the result from the previous step, so we then multiply by 1/6, yielding (x/4) * (1/3) * (1/6).
The expression we have is:
(x/4) * (1/3) * (1/6) = 5
To solve for x, we must clear the fractions by performing the multiplications:
(x/4) * (1/3) * (1/6) can be simplified before we do anything else:
* 1 * 1 = x
--
4 * 3 * 6   72
So we have:
x/72 = 5
To solve for x, multiply both sides of the equation by 72:
x = 5 * 72
Now, let's do the multiplication:
x = 360
Therefore, one-sixth of one-third of one-fourth of 360 is equal to five.

Answer:

• 800 pavilion seats

• 800 lawn seats

Step-by-step explanation:

Let p represent the number of pavilion seats sold. Then 1600-p is the number of lawn seats sold. Total revenue is ...

25p +20(1600 -p) = 36000

5p + 32000 = 36000

5p = 4000

p = 800 . . . . . . . . . . number of pavilion seats sold

1600-p = 800 . . . . . number of lawn seats sold

_____

You can work these problems in your head. Consider that all seats were sold at the lower price. Then revenue would be 1600·20 = 32000. It was 36000 -32000 = 4000 more than that. The difference in ticket price is 25 -20 = 5 dollars, so there must have been 4000/5 = 800 higher-priced tickets sold. (Compare this working to the math above. You will find it substantially similar.)

Answer:

about 27% likely I think......

Answer:

13%

Step-by-step explanation:

You would take 100% and subtract 87%. That will equal 13%. So it is 13% likely that we will win.

Answer: fixed costs of production

The business application that would utilize the slope-intercept form of a linear equation is 1."break-even analysis." therefore, option 1.break-even analysis is correct.

In break-even analysis, you are trying to find the point at which your total revenue equals your total costs, resulting in zero profit (break-even point). This analysis often involves linear equations, and the slope-intercept form (y = mx + b) is commonly used to represent cost and revenue functions. The slope (m) represents the variable cost per unit, and the y-intercept (b) represents the fixed costs.

The other options, "fixed cost of production," "scheduling employee vacation," and "scheduling of employee," may involve mathematical modeling and equations, but they do not typically use the slope-intercept form of a linear equation for the primary analysis. Instead, these applications may involve other types of equations or mathematical models.

for such more question on break-even analysis

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Answer:

2i√3

Step-by-step explanation: Simplify the radical by breaking the radicand up into a product of known factors.

Hope this helps! :) ~Zane

For this case we must find an expression equivalent to:

[tex]\sqrt {-12}[/tex]

Then, rewriting:

[tex]\sqrt {-1 (12)} =\\\sqrt {-1} * \sqrt {12} =[/tex]

We know that:

[tex]i = \sqrt {-1}\\12 = 4 * 3 = 2 ^ 2 * 3[/tex]

Then we have:

[tex]i * \sqrt {2 ^ 2 * 3} =\\i * 2 \sqrt {3} =[/tex]

Finally we have:

[tex]2i \sqrt {3}[/tex]

Answer:[tex]2i \sqrt {3}[/tex]

Answer:

The mean of the worker's salary would be $26,667

The standard deviation is $2915.43

Step-by-step explanation:

Add the salaries together and divide by the number of workers for the mean

Use the formula for standard deviation s=√∑(x¹-x⁻)²/n-1

Answer:

(g ° h)(-3) = 8/5 ⇒ first answer

Step-by-step explanation:

* Lets explain the meaning of the composition of functions

- Composition of functions is when one function is inside of an another

 function

# If g(x) and h(x) are two functions, then (g ° h)(x) means h(x) is inside

  g(x) and (h ° g)(x) means g(x) is inside h(x)

* Now lets solve the problem

∵ g(x) = [tex]\frac{x+1}{x-2}[/tex]

∵ h(x) = 4 - x

∵ (g ° h)(x) means h(x) is inside g(x)

- Find (g ° h)(-3) means find h(-3) at first and then replace the value of

 h(-3) by the x of g(x)

* Lets find h(-3)

∵ h(x) = 4 - x

- Replace the x by -3

∴ h(-3) = 4 - (-3) = 4 + 3 = 7

- Now find g(7), means replace x by 7

∵ g(x) = [tex]\frac{x+1}{x-2}[/tex]

∴ g(7) = [tex]\frac{7+1}{7-2}=\frac{8}{5}[/tex]

∴ (g ° h)(-3) = 8/5

Answer:

-1154

Step-by-step explanation:

(-554) - 600 = -1154

Answer:

y = 3x - 14

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = 3x - 4 is in this form with slope m = 3

• Parallel lines have equal slopes, hence

y = 3x + c ← is the partial equation of the parallel line

To find c substitute (4, - 2) into the partial equation

- 2 = 12 + c ⇒ c = - 2 - 12 = - 14

y = 3x - 14 ← equation of parallel line

The line that is parallel to y=3x-4 and passes through the point (4,-2) has the equation y=3x - 14. This is determined by knowing that parallel lines have the same slope and using the point-slope form of a line equation.

In Mathematics, two lines are parallel if they have the same slope. The line provided in this case is y=3x-4. The slope is 3, which we identify by taking the coefficient of x. Thus, any line parallel to this one will also have a slope of 3.

We also know the line we're seeking passes through the point (4,-2). We will use the point-slope form of a line equation, which is y - y1 = m(x - x1), where m is the slope, and (x1,y1) is a point on the line. Substituting the known values, we get: y - (-2) = 3(x - 4).

Simplifying this equation gives us the equation of the line that is parallel to y = 3x - 4 and passes through the point (4,-2) as: y = 3x -14.

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We can rewrite the equation of the circle as

[tex](x-0)^2+(y-0)^2=10^2[/tex]

so that we can be in the form

[tex](x-h)^2+(y-k)^2=r^2[/tex]

When you write the equation of a circle in this form, then the center is [tex](h,k)[/tex] and the radius is [tex]r[/tex].

So, in our case, the radius of the circle is 10.

To find the length of the bridge, we find the two points where the bridge crosses the lake (i.e. we solve the system between the equations of the line and the circle), and compute the distance between those points:

[tex]\begin{cases}y=x+2\\x^2+y^2=100\end{cases}\implies\begin{cases}y=x+2\\x^2+(x+2)^2=100\end{cases}[/tex]

Solving the second equation for x, we have

[tex]x^2+(x+2)^2=100 \iff x^2+x^2+4x+4=100\iff\\2x^2+4x-96=0 \iff x^2+2x-48=0\\\iff x=-8\ \lor\ x=6[/tex]

We use the first equation to compute the correspondent values of y:

[tex]x=-8\implies y=x+2=-6 \implies P_1 = (-8,-6)[/tex]

[tex]x=6\implies y=x+2=8 \implies P_1 = (6,8)[/tex]

Now, the distance between these two points is given by the pythagorean's theorem:

[tex]d = \sqrt{(-8+6)^2+(-6+8)^2} = \sqrt{4+4}=2\sqrt{2}[/tex]

The radius of the lake is 10 units. The length of the bridge is 14√2 units. We found this by solving the equations of the circle and the line representing the bridge.

The equation of the circular lake is given as  [tex]x^2 + y^2 = 100[/tex] . This equation is of the form  [tex](x -b)^2 + (y - c)^2 = r^2,[/tex] where the center of circle is at (b, c) and the radius is r. In this case, the center is at (0, 0), and the radius squared is 100. Therefore, the radius (r) is:

r = √100 = 10

So, the radius of the lake is 10 units.

The bridge is represented by the function y - x = 2, which can be rewritten as y = x + 2.

To find the points of intersection between this line and the circle, substitute y = x + 2 into the circle's equation  

[tex]x^2 +[/tex][tex]y^2 = 100:[/tex]

[tex]x^2 + (x + 2)^2 = 100[/tex]

Simplify this to:

[tex]x^2 + x^2 + 4x + 4 = 100[/tex]

[tex]2x^2 + 4x + 4 = 100[/tex]

[tex]2x^2 + 4x - 96 = 0[/tex]

Divide everything by 2:

[tex]x^2 + 2x - 48 = 0[/tex]

To solve this quadratic equation, use the quadratic formula x = (-b ± √([tex]b^2[/tex] - 4ac))/(2a), where a = 1, b = 2, and c = -48:

x = (-2 ± √([tex]2^2[/tex] - 4*1*(-48)))/(2*1)

x = (-2 ± √(4 + 192))/2

x = (-2 ± √196)/2

x = (-2 ± 14)/2

This results in two solutions:

x = (12)/2 = 6

x = (-16)/2 = -8

Thus, the points of intersection are (6, 8) and (-8, -6).

Finally, to find the length of the bridge, calculate the distance between these two points using the distance formula: d = [tex]\sqrt{ ((x2 - x1)^2 + (y2 - y1)^2)[/tex]

[tex]d = \sqrt{((6 - (-8))^2 + (8 - (-6))^2)[/tex]

[tex]d = \sqrt{((6 + 8)^2 + (8 + 6)^2)[/tex]

d = [tex]\sqrt{(14^2 + 14^2)[/tex]

d = [tex]\sqrt{(196 + 196)[/tex]

[tex]d = 14\sqrt{2}[/tex]

Therefore, the length of the bridge is [tex]14\sqrt{2}[/tex] units.

Answer:

Terms = x

Coefficients = 3

Constants = 10

Hope this helped! :)

In the expression 3x+10, there are two terms: 3x and 10. The coefficient is 3, which multiplies the variable x, and the constant is 10, which is the term without a variable.

In the expression 3x+10, the terms are 3x and 10. A term is a single mathematical expression. The coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression, so here the coefficient is 3, which is the number multiplying the variable x. The constant is a term without a variable, and in this case, it is 10.

Answer:

35 square units

Step-by-step explanation:

multiply the base by the hight

Answer:

35 square units

Step-by-step explanation:

rectangle with vertices  (2,3), (7,3), (7,10), and (2,10)

Area of a rectangle = length times width

Apply distance formula to find the distance between the sides

[tex]D= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

points (2,3) and  (7,3)

[tex]D= \sqrt{(7-2)^2+(3-3)^2}=5[/tex]

points (7,3) and (7,10)

[tex]D= \sqrt{(7-7)^2+(10-3)^2}=7[/tex]

Area of the rectangle = length times width = 5 times 7 = 35

Answer:

Option B) add 5 to both sides of the equation

Step-by-step explanation:

we have

[tex]x^{2}+18x+76=0[/tex]

step 1

Add 5 to both sides of the equation

[tex]x^{2}+18x+76+5=0+5[/tex]

[tex]x^{2}+18x+81=5[/tex]

step 2

Rewrite as perfect squares

[tex](x+9)^{2}=5[/tex]

Answer:

B) Add five to both sides of the equation

Step-by-step explanation:

I did this test on FLVS

Answer:

C

Step-by-step explanation:

Because harder problems do require equations you cannot solve E=MC square without an equation. OH wait it is an equation E represents units of energy, m represents units of mass, and c2 is the speed of light squared, or multiplied by itself.

Answer:

C

Step-by-step explanation:

If you ask me none of these are reasons to not use a table but for the sake of the question it is probably C. Both A and D are reasons to use a table, and option B is simply false, you very much can find patterns when using tables. As for why C is the most viable answer? Well thats the only one that (truthfully) describes a flaw of using a table, the rest are either untrue or they are describing the pros of using a table.

Answer:-(3x+5)^2

Step-by-step explanation:

The polynomial is factored to give (-3x -5)(3x + 5)

From the information given, we have that the polynomial is given as;

-9x - 30x-25

Now,  multiply the constant value by the coefficient of x squared in the expression, we have;

-9(-25)

225

Now, find the pair factors of 225 that adds up to -30

Then, substitute the values, we get;

-9x² - 15x - 15x - 25

Group in pairs, we get;

(-9x² - 15x ) - (15x - 25)

Factorize the expressions

-3x(3x + 5) - 5(3x + 5)

Then, we have;

(-3x -5)(3x + 5)

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Answer:

[tex]x=n\pi[/tex] or [tex]x= \frac{(2n\pm1)\pi}{2}[/tex], where [tex]n\ge0[/tex]

Step-by-step explanation:

The given trigonometric equation is:

[tex](\sin x)(\cos x)=0[/tex]

By the zero product principle;

Either [tex]\sin x=0[/tex] or [tex]\cos x=0[/tex]

When  [tex]\sin x=0[/tex],  then [tex]x=n\pi[/tex]

When [tex]\cos x=0[/tex], then [tex]x=2n\pi \pm \cos^{-1}(0)[/tex]

This implies that: [tex]x=2n\pi \pm \frac{\pi}{2}[/tex]

Therefore the general solution is  [tex]x=n\pi[/tex] or [tex]x= \frac{(2n\pm1)\pi}{2}[/tex], where [tex]n\ge0[/tex]

Answer:

B. pi divided by two plus n pi comma n pi such that n equals zero, plus or minus one, plus or minus two to infinity

Step-by-step explanation:

Given equation,

[tex](sin x)(cosx)=0[/tex]

[tex]\implies sinx=0\text{ or }cosx=0[/tex]

[tex]x=sin^{-1}(0)\text{ or }x=cos^{-1}(0)][/tex]

[tex]x=\pi, 2\pi, 3\pi,......\text{ or }x=\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}....[/tex]

[tex]\implies x = n\pi\text{ or }x=\frac{\pi+2n\pi}{2}[/tex]

Or

[tex]x=n\pi\text{ or }x= \frac{\pi}{2}+n\pi[/tex]

Where, n = 0, 1, -1, -2, ........∞

Hence, option 'B' is correct.

Answer:

The teacher can hand out pizza to 8 students.

Step-by-step explanation:

To find how many slices of pizza the teacher can hand out , you can divide 7 by 7/8 and round down the nearest whole number.

7 ÷ 7/8 = 8

8 is already a whole number. No need to round down.

The teacher can hand out pizza to 8 students.

Answer is 8 students

N= 7/(7/8)

N= (7*8)/7

N= 56/7

N= 8

d. x=-3

hope this helps!

Answer:

It's D

Step-by-step explanation:

Because the X axis is -3

Answer:

64

Step-by-step explanation:

To do this they have 4 options for each so to get the answer do 4*4*4

Answer: 64

Step-by-step explanation:

So you have a to choose a language, an elective and a science class, each of them has 4 options to choose.

The total combination is the product of all the options.

it is 4 languages * 4 electives * 4 science = 4*4*4 combinations = 64.

So you have 64 possible combinations.

Answer:

Yes. Hexagon is 10 cm

Step-by-step explanation:

Based on the analysis of the residual plots, Runner B's data is best fit by the regression line. This means that Runner B kept the most consistent pace throughout the race.

Analyze the residual plots to determine the best-fit regression line and identify the runner with the most consistent pace.

Observations from the Residual Plots:

- Runner A: The residual plot shows a clear U-shaped pattern, indicating a non-linear relationship and a poor fit for the regression line.

- Runner B: The residual plot exhibits a relatively random scatter of points around zero, suggesting a better fit for the regression line compared to Runner A.

- Runner C: The residual plot displays a distinct downward trend, also indicating a non-linear relationship and a poor fit for the regression line.

Conclusion:

Based on the analysis of the residual plots, Runner B's data is best fit by the regression line. This means that Runner B kept the most consistent pace throughout the race, as their actual pace closely aligned with the predicted pace from the linear model.

Therefore, the answer is B. Runner B.

The correct option is A. A. Runner A because if Runner A’s residual plot fits the criteria of a good linear fit.

To determine which runner kept the most consistent pace based on their residual plots, you need to understand the interpretation of residual plots in the context of linear regression.

Understanding Residual Plots

1. Residuals are the differences between the observed values and the values predicted by the regression line.

2. A residual plot displays these residuals on the vertical axis and the corresponding explanatory variable on the horizontal axis.

3.Consistency in Pace: For a runner to have a consistent pace, the residuals should be randomly scattered around zero with no discernible pattern. This indicates that the linear model is a good fit.

Interpreting Residual Plots

- Runner A: If the residuals for Runner A are randomly scattered around zero with no pattern, it means Runner A's data is well-fitted by the regression line, indicating a consistent pace.

- Runner B: If Runner B's residual plot shows a pattern (e.g., residuals increase or decrease systematically), it suggests that the regression line does not fit Runner B's data well, indicating inconsistency in pace.

- Runner C: Similarly, if Runner C's residual plot shows randomness around zero with no pattern, it means Runner C's data is well-fitted by the regression line, indicating a consistent pace.

Decision

Given the prompt, if you have residual plots for each runner:

1. Look for the plot with residuals randomly scattered around zero.

2. This runner’s data is best fit by the regression line and indicates the most consistent pace.

- Choose Runner A if Runner A’s residual plot shows randomly scattered residuals around zero.

- Choose Runner B if Runner B’s residual plot shows randomly scattered residuals around zero.

- Choose Runner C if Runner C’s residual plot shows randomly scattered

Since the question prompt explicitly mentions Runner A, the implication is that Runner A has the most consistent pace if their residual plot indeed shows a good fit with no pattern. Thus, based on the information given and assuming the residual plots for Runner A are the best fit, the answer would be:A. Runner A

The complete questionis:

Three runners competed in a race. Data were collected at each mile mark for each runner. If the runner ran at a constant pace, the data would be linear. A regression line was fitted to their data. Use the residual plots to decide which data set is best fit by the regression line, and then identify the runner that kept the most consistent pace.

A. Runner A

B. Runner B

C. Runner C

Answer:

(√2)/2

Step-by-step explanation:

A right triangle with one acute angle equal to 45° will also have the other acute angle equal to 45°. The angles being equal means the legs will be equal. If we assign each leg the length 1, then the Pythagorean theorem tells us the hypotenuse is length ...

√(1^2 +1^2) = √2

The cosine of the acute angle is the ratio of the nearest leg length (1) to the hypotenuse length (√2), so the exact value of the cosine is ...

cos(45°) = 1/√2 = (√2)/2

____

(√2)/2 is the fraction with the denominator "rationalized". Sometimes, that is the preferred presentation of this number.