Answer:
3.9230231e+12
Step-by-step explanation:
15! is 15 times 14 times 13 times 12 times 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 then times 3 because 3 shades of blue.
The number of different ways to purchase 3 shades of blue from 15 options is 455 different ways.
The question is asking about the number of combinations of 3 shades of blue that can be chosen from a total of 15 different shades. To calculate this, we use the formula for combinations without repetition, which is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items to choose from, and k is the number of items to choose. In this case, n is 15 and k is 3.
First, we calculate the factorial of n, which is 15! (15 factorial), then the factorial of k, which is 3!, and finally the factorial of n - k, which is 12!. Putting these into the formula gives us:
C(15, 3) = 15! / (3! * 12!) = (15 * 14 * 13) / (3 * 2 * 1) = 455
Therefore, there are 455 different ways to choose 3 shades of blue from 15 different shades.
300 million people live in forests worldwide. 60 million of those humans are indigenous who are completely dependent on native woods.
What is the percentage of the people who live in the forests that are indigenous?
Answer:
The percentage of the people who live in the forests that are indigenous is [tex]20\%[/tex]
Step-by-step explanation:
we know that
To find the percentage of the people who live in the forests that are indigenous, divide the number of people that are indigenous by the total number of people that live in forests
so
[tex]\frac{60}{300} =0.2[/tex]
Convert to percentage
Multiply by 100
[tex]0.2*100=20\%[/tex]
Answer: INTRODUCTION
"Of all the environmental impacts of the study projections, deforestation probably poses the most serious problems for the world, particularly for the developing world."
Global 2000
"It has been predicted that within the next 25-30 years, most of the humid tropical forest as we know it, will be transformed into unproductive land, and the deterioration of the savannah into desert will continue at ever-increasing speed."
As you read this sentence, 50 to 100 acres of primary tropical forests will be eliminated, disrupted, degraded or impoverished. Yearly, an area of tropical forest the size of Great Britain is "converted" from an area equal to the size of Europe.
If present trends continue, by the year 2000, all tropical forests, with the exception of two areas - the western Brazilian Amazon and Central Africa - will have been destroyed. Since 1950, according to the U.N. Food and Agriculture Organization (FAO), half of the world's forests have disappeared. Latin America has lost 37 percent of its tropical forests; Central America, 66 percent; Southeast Asia, 38 percent; Central Africa, 52 percent. Nearly 20 million acres are destroyed annually.
As areas of tropical forests are destroyed or degraded, tribal groups are forced to change their resource base. In some cases they move into areas occupied by other groups, straining the area's resources. In other cases they are forced to relocate outside of forests, permanently altering their way of life by converting to agriculture or to cash employment. Rarely are the rights of these groups to the lands they occupy recognized. Further, their intimate knowledge of the area's resources and how to manage them are nearly always ignored.
Millions of indigenous people live in tropical moist forests which cover some 3.6 million square miles in 70 countries. More than 80 percent of these forests are found in Bolivia, Brazil, Colombia, Gabon, Indonesia, Malaysia, Peru, Venezuela, and Zaire, while 30 additional countries contain sufficient tracts to have significant ecological and biotic values. If these areas are to be managed effectively into the next century, the indigenous peoples that inhabit them should be consulted.
ow many solutions can be found for the equation 4x + 5 = 10? (4 points) Zero One Two Infinitely Many
Answer:
One
Step-by-step explanation:
4x + 5 = 10
Subtract 5 from both sides.
4x = 5
Divide both sides by 4.
x = 5/4
There is one solution.
The cost in dollars to manufacture x pairs of shoes is given by 12,000 + 19x. This month, the manufacturer produced 1000 more pairs of shoes than last month. The average cost per pair dropped by $0.43.
a) Write an expression for the average cost per pair of shoes. Use this expression to write an equation to represent there situation.
b) Solve your equation
c) Are there any mathematical restrictions on the domain? Explain.
d) Determine reasonable domain in the context of the problem. Use your answers to parts I and II to answer the question.
Answer:
Imagination is more important than knowledge. --Clifford big red dog
Step-by-step explanation:
a) To find the average cost per pair of shoes, we need to divide the total cost by the number of pairs produced. Since the manufacturer produced x pairs of shoes, the total cost is 12,000+19x dollars. Therefore, the average cost per pair is given by:
Average cost per pair = (12,000+19x) / x
b) Using the given information that this month, the manufacturer produced 1000 more pairs of shoes than last month, we can represent the number of pairs produced last month as x−1000. Since the average cost per pair dropped by $0.43, the equation representing the situation is:
(12,000+19(x)−43(x)) / x=Average cost per pair
Solving this equation will provide us with the value of x.
c) Yes, there is a mathematical restriction on the domain. Since the number of pairs produced cannot be negative (it doesn't make sense to produce a negative number of shoes), the domain must be restricted to x>0.
d) In the context of the problem, the reasonable domain would be x>0 since the number of pairs of shoes produced cannot be negative. Additionally, since the manufacturer produced 1000 more pairs of shoes this month than last month, it's reasonable to assume that the number of pairs produced last month is also positive, leading to a positive change of 1000 pairs. Therefore, the domain would be x>1000.
a) The expression for the average cost per pair of shoes is given by:
Average cost per pair = (12,000+19x) / x
To represent the situation, we use this expression to equate the average cost per pair before and after the change in production.
b) Substituting x+1000 for x in the expression, we get:
(12,000+19(x+1000)) / (x+1000) = Average cost per pair
We solve this equation to find the value of x.
c) Yes, there's a restriction on the domain. Since the number of pairs of shoes cannot be negative, the domain must be x>0.
d) Considering the context, the reasonable domain is x>1000, as the manufacturer produced 1000 more pairs this month than last month. Also, x must be positive, so x>0. Thus, the domain is x>1000.
Complete Question:
The cost in dollars to manufacture x pairs of shoes is given by 12,000 + 19x. This month, the manufacturer produced 1000 more pairs of shoes than last month. The average cost per pair dropped by $0.43.
a) Write an expression for the average cost per pair of shoes. Use this expression to write an equation to represent their situation.
b) Solve your equation
c) Are there any mathematical restrictions on the domain? Explain.
d) Determine reasonable domain in the context of the problem. Use your answers to parts I and II to answer the question.
This beach toy is a square pyramid. It has a height of 4 inches and the base has side lengths of 4 inches. What is the volume?
Check the picture below.
[tex]\bf \textit{volume of a pyramid}\\\\ V=\cfrac{1}{3}Bh~~ \begin{cases} B=&area~of\\ &its~base\\ h=&height\\ \cline{1-2} B=&\stackrel{4\times 4}{16}\\ h=&4 \end{cases}\implies V=\cfrac{1}{3}(16)(4)\implies V=\cfrac{64}{3}\implies V=21.\overline{3}[/tex]
Answer:
21.03^3 this is the answer.
A school attendance clerk wants to determine if there is a relationship between the number of times a student arrives to school late and the distance the student lives from the school. The clerk gathered data for the month of October and made the graph shown below.
Which statement BEST describes the data?
There is no clear relationship between the number of times students arrive late and the distances they live from school.
Students who arrived late more than six times live the shortest distances from the school.
Students who arrived late more than six times live more than one mile from the school.
Students who arrived late more than six times live the farthest distances from the school.
Answer:
There is no clear relationship between the number of times students arrive late and the distances they live from school.
Step-by-step explanation:
By looking at the graph, there is no clear direction or association that the graph has. Therefore, there is no clear relationship between the variables being compared.
Answer:
looking at the graph, there is no clear direction or association
Step-by-step explanation:
Brainliest and lots of points
The function P(x)=2x2+3x+4, is dilated by the function I(x)=P(13x).
Which function rule represents I(x)?
Answer:
Step-by-step explanation:
I(x)=P(13x) = 2(13x)²+3(13x) +4
I(x)=P(13x).= 338x²+39x+4
Factor completely x³ - 7x² + 2x + 40 given x = -2 is a root.
Answer:
(x +2)(x -4)(x -5)
Step-by-step explanation:
When you divide the cubic by (x +2), the factor that has -2 as a root, you get a quadratic with roots 4 and 5. Thus the factors are ...
(x +2)(x -4)(x -5)
___
In the attachment, we have done the division using a graphing calculator. You can also do it by polynomial long division or by synthetic division. The resulting quadratic is ...
x^2 -9x +20 = (x -4)(x -5) . . . . . factor the quadratic
What is the inverse of the function below? f(x) = 5x - 2
A. f -1(x) = x - 2/ 5
B. f -1(x) = x + 2 / 5
C. f -1(x) = -2 + 5x
D. f -1(x) = -5x + 2
Answer:
B. f -1(x) = x + 2 / 5
[tex]f^{-1}(x)=\frac{x+2}{5}[/tex]
Step-by-step explanation:
To find the inverse of a function we need to interchange x and y an solve for y.
Since [tex]f(x)=y[/tex], then
[tex]f(x)=5x-2[/tex]
[tex]y=5x-2[/tex]
[tex]x=5y-2[/tex]
Add 2 to both sides
[tex]x+2=5y-2+2[/tex]
[tex]x+2=5y[/tex]
Divide both sides by 5
[tex]\frac{x+2}{5}=\frac{5y}{5}[/tex]
[tex]\frac{x+2}{5}=y[/tex]
[tex]y=\frac{x+2}{5}[/tex]
[tex]f^{-1}(x)=\frac{x+2}{5}[/tex]
We can conclude that the correct answer is B. f -1(x) = x + 2 / 5
Miguel has a piece of string that is 3 m 25cm long he cut a piece off and used it to tie Around his sleeping bag now his oringinol piece is only 1 m 46 cm long how long is the piece of string Miguel cut off
Answer:
1 m 79 cm
Step-by-step explanation:
Subtract 1.46 m from 3.25 cm: 1.79 cm, or 1 m 79 cm
Michelle borrowed $10,125 at 12.5 percent for 10 years. What is her monthly payment
Answer:
$148.21
Step-by-step explanation:
A suitable financial calculator, web site, or spreadsheet can figure this for you. Or you can use the formula given in your reference material (text or web site).
Answer:
Michelle's monthly payment will be $148.21.
Step-by-step explanation:
The EMI formula is =
[tex]\frac{p\times r\times(1+r)^{n} }{(1+r)^{n}-1}[/tex]
Here,
p = $10125
r = [tex]12.5/12/100=0.010417[/tex]
n = [tex]10\times12=120[/tex]
Putting all these values in the formula we get,
[tex]\frac{10125\times0.010417\times(1+0.010417)^{120} }{(1+0.010417)^{120}-1}[/tex]
=>[tex]\frac{10125\times0.010417\times(1.010417)^{120} }{(1.010417)^{120}-1}[/tex]
=$148.21
So, Michelle's monthly payment will be $148.21.
Please help me out with this
Answer:
48.3 cm²
Step-by-step explanation:
The area (A) of the yellow region = area of square - area of quarter circle, that is
A = 15² - ([tex]\frac{1}{4}[/tex] πr² )
= 225 -( [tex]\frac{1}{4}[/tex] × π × 15² )
= 225 - 176.71 ≈ 48.3
show work please.
solve system of equations using matrices.
Answer:
(t, t -1, t)
Step-by-step explanation:
You have three unknowns but only 2 equations, so you can't really SOLVE this...you can get a solution with a variable still in it (I forget what this is called. I think it refers to infinite many solutions). Here's how it works:
Set up your matrix:
[tex]\left[\begin{array}{ccc}1&-2&1\\2&-1&-1\\\end{array}\right] \left[\begin{array}{ccc}2\\1\\\end{array}\right][/tex]
You want to change the number in position 21 (the 2 in the scond row) to a 0 so you have y and z left. Do this by multiplying the top row by -2 then adding it to the second row to get that 2 to become a 0. Multiplying in a -2 to the top row gives you:
[tex]\left[\begin{array}{ccc}-2&4&-2\\2&-1&-1\\\end{array}\right]\left[\begin{array}{ccc}-4\\1\\\end{array}\right][/tex]
Then add, keeping the first row the same and changing the second to reflect the addition:
[tex]\left[\begin{array}{ccc}-2&4&-2\\0&3&-3\\\end{array}\right] \left[\begin{array}{ccc}-4\\-3\\\end{array}\right][/tex]
The second equation is this now:
3y - 3z = -3. Solving for y gives you y = z - 1. Let's let z = t (some random real number that will make the system true. Any number will work. I'll show you at the end. Just bear with me...)
lf z = t, and if y = z - 1, then y = t - 1. So far we have that y = t - 1 and z = t. Now we solve for x:
From the first equation in the original system,
x - 2y + z = 2. Subbing in t - 1 for y and t for z:
x - 2(t - 1) + t = 2. Simplify to get
x - 2t + 2 + t = 2 and x - t = 0, and x = t. So the solution set is (t, t - 1, t). Picking a random value for t of, let's say 2, sub that in and make sure it works. If:
x - 2y + z = 2, then t - 2(t - 1) + t = 2 becomes t - 2t + 2 + t = 2, and with t = 2, 2 - 2(2) + 2 + 2 = 2. Check it: 2 - 4 + 4 = 2 and 2 = 2. You could pick any value for t and it will work.
Alberto has 92 stamps in one large álbum and 38 stamps in anotaré small álbum. How can he use mental math to find how many more stamps are in the large album use drawings to show your answer
Answer:
The answer would be the large album has 54 more stamps than the small album.
Step-by-step explanation:
... 92 - 38 = 54
Do you need a drawing done though?
[Trigonometric Graphs]
Use the following information to write an equation of the graph described:
7. sin; Amp = 2, per = π, vertical shift = down 3.
8. cos; Amp = 1; per = π/4, phase shift = left π.
Explain.
Answer: 7. y = 2 sin (2x) - 3
[tex]\bold{8.\quad y=sin\bigg(8x + \dfrac{8}{\pi}\bigg)}[/tex]
Step-by-step explanation:
The general form of a sine equation is: y = A sin (Bx - C) + D where
Amplitude = |A|Period (P) = [tex]\dfrac{2\pi}{B}[/tex]Phase Shift = [tex]\dfrac{C}{B}[/tex]Vertical Shift = D (positive is up, negative is down)7. Given:
A = 2P = π = [tex]\dfrac{2\pi}{B}[/tex] → B = 2C = 0 (none)D = -3--> y = 2 sin (2x) - 3
8. Given:
A = 1P = [tex]\dfrac{\pi}{4}[/tex] = [tex]\dfrac{2\pi}{B}[/tex] → B = 8[tex]\text{Phase Shift = }-\pi = \dfrac{C}{8}\implies C = -8\pi[/tex]D = 0 (none)[tex]\implies y=sin\bigg(8x + \dfrac{8}{\pi}\bigg)[/tex]
A man who weighs approximately 140 pounds has two 12 ounce beers and 1.5 ounce shot of liquor in two hours. How many standard drinks did the man consume
The answer is:
The man did consume two standard drinks in two hours.
Why?According to the standard drink conversion, we know that each 12 ounce beer we have one (1) standard drink, and each 1.5 ounce shot of liquor, we have another standar drink.
Calculating we have:
[tex]DrinksConsumed=12Oz(Beer)+1.5Oz(Liquor)=1StandardDrink+1StandardDrink\\\\DrinksConsumed=1StandardDrink+1StandardDrink=2StandardDrinks[/tex]
We have that the man did consume two standard drinks in two hours.
Have a nice day!
A man who consumed two 12 ounce beers and a 1.5 ounce shot of liquor has consumed three standard drinks according to the National Institute on Alcohol Abuse and Alcoholism's definition of a standard drink in the United States.
Explanation:According to the National Institute on Alcohol Abuse and Alcoholism (NIAAA), a standard alcoholic drink in the United States is equivalent to 14 grams (0.6 ounces) of pure alcohol. This typically is found in a 12-ounce beer, a 5-ounce glass of wine, or a 1.5 ounce shot of distilled spirits or liquor. The man in the question consumed two 12 ounce beers and one 1.5 ounce shot of liquor. Given that both the beer and the shot each lay within the standard drink size, we can conclude that the man consumed three standard drinks in total. Alcohol consumption and understanding standard drinks is important to monitor one's drinking habits for overall health.
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PLEASE HELP!!!
9/10 divided by (-3/5)
Write the answer as a mixed number
[tex]
\frac{\frac{9}{10}}{\frac{-3}{5}}=\frac{45}{-30}=-\frac{9}{6}=\boxed{\frac{3}{2}}
[/tex]
Hope this helps.
r3t40
James bought two T-shirts and one pair of jeans at an online store and paid $40, not including taxes, for his purchase. A month later, the same store sold the T-shirts and jeans at a 50% discount from their original prices. James bought two T-shirts and five pairs of jeans for $60, not including taxes.
Answer:
The T-shirts cost $10 and the jeans cost $20.
Step-by-step explanation:
I guess you want to find the cost of the T-shirts and jeans at his first buy,
Let the cost be $t and $j for T-shirts and jeans respectively.
Then we have the system:
2t + j = 40...................(1)
and a month later:
0.5*2t + 0.5*5j = 60
t + 2.5j = 60................(2)
Multiply by - 2:
-2t - 5j = -120.............(3)
Adding equations (1) and (3):
-4j = -80
j = -80 / -4
j = $20.
Substituting j = 20 into equation (2):
t + 2.5(20) = 60
t = 60 - 50
t = $10.
t = $55
Identify the value of x and the length of each secant segment. HELP ASAP!!
Answer:
[tex]x=10, NL=8, NP=12[/tex]
Step-by-step explanation:
we know that
Applying the Intersecting secant Theorem
[tex]NL*NM=NP*NO[/tex]
substitute the given values and solve for x
[tex](3+5)*3=(2+x)*2[/tex]
[tex](8)*3=(2+x)*2[/tex]
[tex]24=(2+x)*2[/tex]
[tex]12=(2+x)[/tex]
[tex]x=12-2=10[/tex]
Find the value of NP
[tex]NP=(2+x)=2+10=12[/tex]
therefore
[tex]x=10, NL=8, NP=12[/tex]
Answer:
x = 7; NL = 12; NP = 20
Step-by-step explanation:
ur welcome
The frequency table was made using a deck of cards in which each card is numbered 1, 2, 3, or 4.
Create a bar graph by dragging the sliders on the horizontal axis to represent the probability distribution.
Answer:
Step-by-step explanation:
So the table means that there are 20 1's, 120 2's, 40 3's, and 20 4's.
The total number is 20+120+40+20 = 200.
So the probability of each is:
1: 20/200 = 0.10
2: 120/200 = 0.60
3: 40/200 = 0.20
4: 20/200 = 0.10
A bar graph that represents the probability distribution of each card is shown in the image below.
In Mathematics and Statistics, a frequency table can be used for the graphical representation of the frequencies or relative frequencies that are associated with a categorical variable or data set.
Based on the frequency, the total number of deck of cards can be calculated as follows;
Total number of deck of cards = 20 + 120 + 40 + 20
Total number of deck of cards = 200 cards.
Next, we would determine the probability distribution of each card as follows:
P(X = 1) = 20/200 = 0.10.
P(X = 2) = 120/200 = 0.60.
P(X = 3) = 40/200 = 0.20.
P(X = 4) = 20/200 = 0.10.
the plot line shows the number of miles the individual members in a group of runners run each day
How many runners run at most 2 miles per day?
The number of runners run at most 2 miles per day is 13.
The calculation is as follows;2 miles = 6
3 miles = 4
4 miles = 2
5 miles = 1
Now if we add this,
So,
= 6 + 4 + 2 + 1
= 13
Learn more: brainly.com/question/17429689
Answer:
13
Step-by-step explanation:
Its easy, but its a trick question into getting you to click 6
Determine if the Mean Value Theorem for Integrals applies to the function f of x equals 2 times the square root of x on the interval [0, 4]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.
No, the theorem does not apply.
Yes, x = 1.
Yes, x equals 8 over 3.
Yes, x equals sixteen divided by 9.
[tex]f(x)=2\sqrt x[/tex] is continuous on [0, 4] and differentiable on (0, 4), so the MVT holds. We have
[tex]f'(x)=\dfrac1{\sqrt x}[/tex]
so that by the MVT, there is some [tex]c\in(0,4)[/tex] such that
[tex]f'(c)=\dfrac{f(4)-f(0)}{4-0}\implies\dfrac1{\sqrt c}=\dfrac{2\sqrt4}4=1[/tex]
[tex]\implies1=\sqrt c\implies \boxed{c=1}[/tex]
Answer:
x= 16/9
Step-by-step explanation:
[tex]f(x) = 2\sqrt{x}[/tex] is differentiable on [0,4] so the Mean Value Theorem For Integrals applies.
Average Value of the Integral:
[tex]\frac{1}{b-a} \int\limits^b_a {f(x)} \, dx = \frac{1}{4-0} \int\limits^4_0 {2\sqrt{x} } \, dx[/tex]
After evaluating you will get [tex]\frac{8}{3}[/tex]
Now, this is the average value so you still need to find the x-value using the original equation:
[tex]\frac{8}{3} =2\sqrt{x}\\\\\frac{4}{3} = \sqrt{x}\\\\\frac{16}{9} =x[/tex]
Kyle volunteered 5.5 hours at the library this week. He needs to volunteer a total of 12 hours. Which equation can be used to find the number of hours, h, Kyle still needs to volunteer this week?
Answer:
12-5.5=h
12-5.5=6.5
Kyle needs to volunteer 6.5 more hours.
Step-by-step explanation:
Answer:
5.5 + h = 12
Step-by-step explanation:
Here, h represents the number of hours, Kyle still needs to volunteer this week,
Given,
He volunteered 5.5 hours at the library this week.
Since, the total number of hours he needs to volunteer = 5.5 + h
According to the question,
5.5 + h = 12
Which is the required equation.
A series of steps designed to help you solve problems and answer questions
A) observation
B) scientific method
C) experiment
D) hypothesis
Answer: B Scientific Method
A function is graphed on the coordinate plane.
what is the value of the function when x = 4
Answer:
1
Step-by-step explanation:
y = mx + b
y = 1/2 x - 1
y = 1/2 (4) - 1
y = 2 - 1
y = 1
When x = 4, the function y = (1/2)x - 1 evaluates to y = 1. This linear equation has a slope of 1/2 and a y-intercept of -1.
To find the value of the function y = (1/2)x - 1 when x = 4, you simply need to substitute x = 4 into the equation and solve for y:
y = (1/2)(4) - 1
y = 2 - 1
y = 1
So, when x = 4, the value of the function y is 1.
In a more general sense, this equation represents a linear function with a slope of 1/2 and a y-intercept of -1. The slope (1/2) represents the rate of change, meaning that for every unit increase in x, y increases by 1/2. The y-intercept (-1) is the value of y when x is 0.
When x = 4, as calculated above, y is 1. This means that if you were to plot this function on a coordinate plane, the point (4, 1) would lie on the graph.
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PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
If f(x) = -3x^4 - 2x^3 + 3x^2, and g(x) = 3x^4 - 4x^3 + x^2 then f(x) + g(x) =
Answer: B) -6x³ + 4x²
Step-by-step explanation:
f(x): -3x⁴ - 2x³ + 3x²
+ g(x): 3x⁴ - 4x³ + x²
f(x) + g(x): - 6x³ + 4x²
Please help!!!!!!!!!!!!!!!!!!
A = 0.5 x 10.4 x 16.9
A = 87.88
The area of the triangle is 87.88 ft²
A polynomial has been factored, as shown below:
f(x) = (x - 3)(x + 8)(x - 11)
What are the zeros of the polynomial?
3, 8, 11
3,-8, 11
-3, 8, -11
-3,3, -8,8, -11, 11
Answer:
3, -8, 11
Step-by-step explanation:
The first factor is zero when x=3.
The second factor is zero when x=-8.
The third factor is zero when x=11.
Whenever any factor is zero, the polynomial is zero.
The zeros are 3, -8, 11.
Find the area. (Geometry)
Answer: 12800m^2
This shape is made up of a rectangle and a triangle.
Rectangle
= 120 x 60
= 7200
Triangle
= (200 - 60) x (120 - 40) x 0.5
= 140 x 80 x 0.5
= 5600
Total Shape
= 7200 + 5600
= 12800m^2
What is the range of the function for the domain?
The domain are the x values.
The problem is saying the x value is all numbers between 4 and 8
Replace x in the equation with 4 and 8 and solve to find the range for the y values:
y =4(4) -1 = 16-1 = 15
y = 4(8) - 1 = 32-1 = 31
So Y would be between 15 and 31.
The first answer is the correct one.
Simplify the expression. sine of x to the second power minus one divided by cosine of negative x
Answer:
the answer is the letter a) -sin x
Step-by-step explanation:
Simplify the expression.
sine of x to the second power minus one divided by cosine of negative x
(1−sin2(x))/(sin(x)−csc(x))
sin2x+cos2x=11−sin2x=cos2x
cos2(x)/(sin(x)−csc(x))csc(x)=1/sin(x)cos2(x)/(sin(x)− 1/sin(x))= cos2(x)/((sin2(x)− 1)/sin(x))sin2(x)− 1=-cos2(x)cos2(x)/(( -cos2(x))/sin(x))
=-sin(x)
Answer:
[tex]-cos \ x[/tex]
Step-by-step explanation:
First of all, we must have to understand what is the described expression in the paragraph
"sine of x to the second power minus one divided by cosine of negative x"
In this sentence, we need to identify what are the elements and operations involved in the expression.
In the sentence appears ""to the second power", "minus" and "divided by" (highlighted)
"sine of x to the second power minus one divided by cosine of negative x"
Therefore, the expression must has three operations:
"to the second power": refers to exponentiation"minus": refers to a substraction"divided by": refers to a divisionNow, we can identify what are the elements: "sine of x", "one" and "cosine of negative x"
"sine of x": refers to [tex]sin\ x[/tex]"one": refers to the number one (1)"cosine of negative x": refers to [tex]cos (-x)[/tex]Therefore, the expression is:
[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}[/tex]
In order to find the simplified expression, we must have to apply these trigonometric identities:
[tex](sin\ x)^{2} = sin^{2}x[/tex][tex]sen\x^{2}x \ +\ cos\x^{2}x=1[/tex][tex]cos(-x)=cos\ x[/tex]Applying the first and third identities, we have:
[tex]\frac{(sin\ x)^{2}-1}{cos(-x)}=\frac{sin\x^{2}x-1}{cos\ x}[/tex]
From the second trigonometric identity, we have:
[tex]cos\x^{2}x=\ 1-sin\x^{2}x[/tex]
Now, multiplying by -1 in both sides:
[tex](-1)(cos\x^{2}x)=(-1)(1-\ sin\x^{2}x)[/tex]
In the left side, multiplying by -1 the sign of the expression changes:
[tex](-1)(cos\x^{2}x)=-cos\x^{2}x[/tex]
In the right side, multiplying by -1 changes the order of the substraction:
[tex](-1)(1-\ sin\x^{2}x)=\ sin\x^{2}x-1[/tex]
Putting all together:
[tex]-cos\x^{2}x=\ sin\x^{2}x-1[/tex]
Now, replacing values we have:
[tex]\frac{sin\x^{2}x-1}{cos\ x}=\frac{-cos\x^{2}x}{cos\ x}=-\frac{cos\x^{2}x}{cos\ x}[/tex]
Finally, the property of the first trigonometric identity (property of exponentiation) can be apply in this case:
[tex]-\frac{cos\x^{2}x}{cos\ x}=-\frac{(cos\ x)^{2}}{cos\ x}=-cos\ x[/tex]