prove that -
\( \sin {}^{2} (\theta) + \cos { }^{2} (\theta ) = 1\)
thankyou! ~
Answers
Answer:
Depends.
If you defined \(cos\ \theta\) and \(sin\ \theta\) based on the unit circle, as in the coordinates of the point at a distance of \(\theta\) from the point \((1;0)\) measured counterclockwise along the circle, then. it's proven by the fact that since that points sit on a circle of center the origin and radius 1, it's coordinates needs to satisfy the equation \(x^2+y^2 = 1\).
If you define them based on right triangles, as in the ratios between adjacent (cosine) or oppsite (sine) sides and the hypotenuse, then it's simply applying the Pythagorean theorem. In particular let a be the hypotenuse of a right triangle of which one of the angles measures \(\theta\), let b the adjacent side to said angle, and c the opposite side.
By definition, \(\frac ba = cos\ \theta ; \frac ca = \sin\ \theta\). clearing denominators we get \(b= a\cdot\cos\ \theta; c= a\cdot sin\ \theta\), and applying the Pythagorean theorem:
\(a^2= b^2+c^2 = (a\cdot\cos\ \theta)^2 + (a\cdot sin\ \theta)^2\) And, taking the first and last step in this chain, dividing everything by \(a^2\) which is obviously non-zero we're left with \(1= cos^2\theta + sin^2\theta\), QED.
Related Questions
Multiply.
(2x+6)²
NEED ANSWER ASAPP
Answers
Answer:
4x² + 24x + 36
Step-by-step explanation:
(2x + 6)² = (2x + 6)(2x + 6) = 4x² + 12x + 12x + 36 = 4x² + 24x + 36
Find the volume of the
triangular prism.
8 ft
6 ft
The volume of the triangular prism is ft³.
15 ft.
Answers
The volume of the triangular prism will be 400 cubic feet.
Given that:
Width, W = 15 feet
Height, h = 8 feet
Base length, b = 6²/₃ feet
Volume is a measurement of three-dimensional space that is utilized. The concept of length is linked to the notion of capacity.
The volume of the triangular prism is calculated as,
V = 1/2 · b · h · W
V = 0.5 · 6²/₃ · 8 · 15
V = 400 cubic feet
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Which is the vertex of x^2 + 10x = - 17?
Answers
Answer:
(-5, -8)
Step-by-step explanation:
Parabolas always have a lowest point (or a highest point, if the parabola is upside-down). This point, where the parabola changes direction, is called the "vertex". If the quadratic is written in the form y = a(x – h)2 + k, then the vertex is the point (h, k).
x^2 + 10x = - 17
x^2+10x+17=0
x^2+2*5x+25 - 8=0
(x+5)^2-8=0
h=-5, k= -8
vertex is (-5, -8)
Problem
(a) Expand and simplify \((x^2+2)^2\).
(b) Hence fully factorise \(x^4 +4x+3\).
Answers
a. \(x^4 + 4x^2 + 4\)
b. x = ∛-1 and -3
How to expand the expressiona. Given
(x² + 2)²
First, we open up the bracket
\((x^2 + 2) (x^2 + 2)\)
Multiply through
\(x^4 + 2x^2 + 2x^2 + 4\)
Collect like terms
\(x^4 + 4x^2 + 4\)
b. Let's factorize
\(x^4 + 4x + 3\)
Find the factors of 3 that are products of 4, they are 3 and 1
Use it to replace 4x as 3x and x
\(x^4 + 3x + x + 3\)
Let's make the expression two
\((x^4 + 3x) (x+ 3)\)
Find the common factor
\(x^3(x + 3)+ 1(x+ 3)\)
So,
\(x^3 + 1 = 0\) and \(x + 3 = 0\)
\(x = 3\sqrt{-1}\) and x = -3
Thus, x = ∛-1 and -3
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Use the cards below to create a list of steps, in order, that will solve the following equation.
3(x+6)2 = 75
Solution steps:
Add 6 to both sides
Divide both sides by 3
1
Divide both sides by
3
Subtract 6 from both sides
ay
23
Square both sides
Take the square root of both sides
Answers
2. Take the square root of both sides
3. Subtract 6 from both sides
Which equation is correct?
427 × 3 = 400 × 3 + 20 × 3 + 7 × 3
427 × 3 = 400 + 20 × 3 + 7 × 3
427 × 3 = 400 × 3 + 20 × 3 + 7
427 × 3 = 4,000 × 3 + 200 × 3 + 70 × 3
Answers
Answer:
A) 427 × 3 = 400 × 3 + 20 × 3 + 7 × 3
Step-by-step explanation:
\(427*3=(400+20+7)*3=(400*3)+(20*3)+(7*3)\)
Therefore, the first option is correct
pls help and explain how you got the answer
Answers
The class of the equation and the cross section equation are;
The equation is a hyperboloid of two sheets
The cross sections are;
Equation at z = 0 is; y²/8 - x²/8 = 1
Equation at y = -4 is; x²/8 + z² = 1
Equation at y = 0 is; x²/8 + z² = -1
Equation at y = 4 is; x²/8 + z² = 1
Equation at x = 0 is; y²/8 - z² = 1
What is an equation ?An equation is a mathematical statement which indicates the equivalence two expressions by joining them with the '=' sign.
The equation can be presented as follows;
y² = x² + 8·z² + 8
y² - x² - 8·z² = 8
y²/8 - x²/8 - z² = 1
The above equation is the equation of an hyperboloid of two sheets, where;
a² = 8, b² = 1, and c² = 8
Please find attached the diagram of the surface, created with an online 3D graphing tool.
The equation for the cross section at z = 0, can be obtained by plugging in z = 0, into the equation as follows;
y²/8 - x²/8 - 0 = 1
y²/8 - x²/8 = 1
The above equation is the equation of an hyperbola in the xy plane
The equation for the cross section at y = -4, can be obtained by plugging in y = -4, into the equation as follows;
4²/8 - x²/8 - z² = 1
- x²/8 - z² = 1 - 4²/8 = -1
- x²/8 - z² = -1
x²/8 + z² = 1
The above is an equation of an ellipse in the xz plane
The equation for the cross section at y = 0, can be obtained by plugging in y = 0, into the equation as follows;
0²/8 - x²/8 - z² = 1
- x²/8 - z² = 1
Therefore; x²/8 + z² = -1
The equation for the cross section at y = 4, can be obtained by plugging in y = 4, into the equation as follows;
4²/8 - x²/8 - z² = 1
- x²/8 - z² = 1 - 2 = -1
x²/8 + z² = 1
The above equation is the equation of an ellipse in the xz plane
The equation for the cross section at x = 0, can be obtained by plugging in x = 0, into the equation as follows;
y²/8 - 0²/8 - z² = 1
y²/8 - z² = 1
The equation is the equation of an hyperbola in the yz plane
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A large restaurant chain is curious what proportion of their customers in a given day are new customers. They are thinking of taking a sample of either n=50 or n=100 customers and building a one-sample z interval for a proportion using the data from the sample. Assuming the sample proportion is the same in each sample, what is true about the margins of error from these two samples?
Answers
Answer:
none of the above
a triangle has one side length of 26 and another side length of 18. select all of the possible lengths of the third side.
Answers
Since the longest side cannot be longer than the total of the other two sides, the third side's potential lengths range from 8 to 44.
A triangle's third side must be longer than the difference between the other two sides' lengths and shorter than the total length of the other two sides. The two side lengths in this situation are 18 and 26. The third side must be greater than 8 because the difference between them is 8. The third side must be less than 44 because the total length of the first two sides equals 44. Therefore, the third side length might be any side length between 8 and 44.Since the longest side cannot be longer than the total of the other two sides, the third side's potential lengths range from 8 to 44.
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The complete question is
a triangle has one side length of 26 and another side length of 18. select all of the possible lengths of the third side potential lengths range from 8 to 44?
132 is what percent of 880
Answers
Answer:
Step-by-step explanation:
132 is what percent of 880? = 15.
Solve for x in simplest form.
Answers
Answer:
\(x=-\frac{4}{5}\)
Step-by-step explanation:
\(6=\frac{3}{4}(10x+16)\\\mathrm{or,\ }6\times 4=3(10x+16)\\\\\mathrm{or,\ }24=30x+48\\\\\mathrm{or,\ }-24=30x\\\\\mathrm{\therefore}\ x=-\frac{4}{5}\)
Select the correct answer from each drop-down menu. The inequality 5m − 7 > 16 holds true for all numbers _ than _ in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Answers
The values of {m} that is greater than 4.6 represent the solution of the given inequality.
An inequality is used to compare two or more expressions or numbers.
For example -
2x > 4y + 3
x + y > 3
x - y < 6
The given inequality is -
5m - 7 > 16
Adding 7 on both sides, we get -
5m - 7 + 7 > 16 + 7
5m > 23
m > 23/5
m > 4.6
Therefore, the values of {m} that is greater than 4.6 represent the solution of the given inequality.
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Find the distance between the two points in simplest radical form.
(2,-5) and (7,7)
Answers
Answer:
13
Step-by-step explanation:
use distance formula \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\d= \sqrt{(7-2)^2+(7-(-5))^2} =13\)
In July, Lee Realty sold 10 homes at the following prices: $140,000; $166,000; $80,000; $98,000; $185,000; $150,000; $108,000; $114,000; $142,000; and $250,000. Calculate the mean and median.
Answers
Median:the number in the middle
80,000 98,000 108,000 114,000 140,000 142,000 150,000 166,000 185,000 250,000
They are 10 numbers so you’ll have 2 numbers left
140,000+142,000=282,000
Then 282,000 divided it by 2
Which gives you 141,000
So the median is:141,000
80,000+98,000+108,000+114,000 140,000+142,000+150,000+166,000+185,000+250,000=1,293,000
1,293,000 divided by 10 which equals 129,300
So the mean is=129,300
The mean is 143000 and Median is 141000 for data $140,000; $166,000; $80,000; $98,000; $185,000; $150,000; $108,000; $114,000; $142,000; and $250,000.
What is Statistics?A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data
The mean is give by sum of n numbers to the total number of observations
Mean=Sum of observations/ Number of observations
Given,
10 homes at the following prices: $140,000; $166,000; $80,000; $98,000; $185,000; $150,000; $108,000; $114,000; $142,000; and $250,000.
Sum of observations=$140,000+$166,000+$80,000+$98,000+ $185,000+$150,000+ $108,000+$114,000+$142,000+ $250,000=1433000
n=10
Mean=1433000/10=143000
So mean is 143000
Now let us find the median, Median is the middle most number.
First we have to arrange the observation in ascending order.
$80,000, $98,000, $108,000, $114,000, $140,000, $142,000, $150,000, $166,000, $185,000, $250,000
Now Median= ($140,000+$142,000)/2
=282000/2=141000
Hence Mean is 143000 and Median is 141000.
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Write the equation of the parabola that has the same shape as f(x)=-3 x² but with vertex (5,9) in the form f(x)=a(x-h)²+k.
f(x)=
Answers
The equation of the parabola from the vertex is f(x) = -3(x - 5)² + 9
How to determine the equation of the parabola?The equation of the function is given as
f(x) = 3x²
Also, we have
Vertex = (5, 9)
The form of the equation is given as
f(x) = a(x - h)² + k
Where
Vertex = (h, k)
This means that
Vertex = (h, k) = (5, 9)
Substitute (h, k) = (5, 9) in the equation f(x) = a(x - h)² + k
So, we have
f(x) = a(x - 5)² + 9
This also means that
f(x) = -3(x - 5)² + 9
The -3 is gotten from f(x) = -3x²
Hence, the equation of the parabola is f(x) = -3(x - 5)² + 9
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Without multiplying, how can you tell which product will be greater, 3 x 4 or 6 x 2?
Answers
Answer:
we can compare 4 x 3 and 2 x 6, equal to 12. So, we can see that both products are similar and neither is greater.
Step-by-step explanation:
We can use the commutative property of multiplication to see that 3 x 4 is the same as 4 x 3 and 6 x 2 is the same as 2 x 6. When multiplying two numbers, the order of the numbers does not affect the product. Therefore, we can compare 4 x 3 and 2 x 6, equal to 12. So, we can see that both products are similar and neither is greater.
NEED MORE EXPLANATION?
The commutative property of multiplication states that when multiplying two numbers, the order of the numbers does not affect the product. This means that 3 x 4 is the same as 4 x 3, and 6 x 2 is the same as 2 x 6. We can use this property to compare 4 x 3 and 2 x 6, equal to 12. Therefore, we can see that both products are identical and neither is greater.
HELP ME raaaaaaaa NOWWWWWWWWWW
Answers
Answer:
C: EAF Is the correct answer
Given that f(x)=x2−9x and g(x)=x+1, find
Answers
The value of the functions are,
f + g = x² - 8x + 1
f - g = x² - 10x - 1
fg = x³-8x²-9x
What is a function?The expression that established the relationship between the dependent variable and independent variable is referred to as a function. In the function as the value of the independent variable varies the value of the dependent variable also varies.
Given functions are f(x) = x² − 9x and g(x) = x + 1. The values will be calculated as,
f + g = x² − 9x + x + 1
f + g = x² − 8x + 1
f - g = x² − 9x - x - 1
f - g = x² − 10x - 1
f x g = (x² − 9x) x (x + 1)
f x g = (x³ +x ² - 9x² -9x)
f x g = x³ - 8x² - 9x
Therefore, the value of the functions are, f + g = x² - 8x + 1, f - g = x² - 10x - 1, and fg = x³-8x²-9x.
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PQ = 7x + 13, QR = 10x-2, PR = x + 27 what is the range of x-values
Answers
The range of x-values is 1 < x < 21.
What is triangle Inequality?According to triangle inequality theorem, for any given triangle, the sum of two sides of a triangle is always greater than the third side.
Given:
We have three sides of Triangle as
PQ = 7x + 13, QR = 10x-2, PR = x + 27.
So, PQ + QR > PR
(7x + 13) + (10x - 2) > x + 27, which simplifies to x > 1
and, PQ + PR > QR
(7x + 13) + (x + 27) > 10x - 2, which simplifies to x < 21
and, QR + PR > PQ
(10x - 2) + (x + 27) > 7x + 13, which simplifies to x > -3
So, the Range of sides is 1 < x < 21.
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Solve this equation: –3(x – 14) + 9x = 6x + 42
Answers
Answer:
0
Step-by-step explanation:
Step 1 - Expand bracket:
\(-3(x - 14) + 9x = 6x + 42\\-3(x) + -3(-14) + 9x = 6x + 42\\-3x + 42 + 9x = 6x + 42\)
Step 2 - Combine like terms:
\(-3x + 42 + 9x = 6x + 42\\42 + 6x = 6x + 42\)
As both sides are the same, the equation equals 0.
\(42 + 6x = 6x + 42\\42 - 42 = 6x + 42 - 4x\\6x = 6x\\6x - 6x = 6x - 6x \\0 = 0\\\)
Hope this helps!
Which of the following relations is a function?
A{(3,-1), (2, 3), (3, 4), (1,7)}
B{(1, 2), (2, 3), (3, 4), (4, 5)}.
C{(3, 0), (4, -3), (6, 7), (4,4)}
D{(1, 2), (1, 3), (2, 8), (3, 9)}
Answers
Answer:
B
Step-by-step explanation:
A is not a function because the same x value is repeated twice with different y values. The same goes for C and D so the answer is C.
Answer:
B.
Step-by-step explanation:
Well a relation is a set of points and a function is a relation where every x value corresponds to only 1 y value.
So lets see which x values in these relations have only 1 y value.
A. Well a isn’t a function because the number 3 which is a x value had two y values which are -1 and 4.
B. This relation is a function because there are no similar x values.
C. This is not a function because the x value 4 has two y values which are 4 and -3.
D. This is not a function because the number 1 has 2 and 3 as y values.
Yolanda scored 10 points in a basketball game. She could have scored with one‐point free throws, two‐point field goals, or three‐point field goals. In how many different ways could she have scored her 10 points?
Answers
she could have scored the 10 points in 302400 ways.
The given parameters are
n= total points =10
r1 =one-point free throw = 1
r2 = two-point field goals = 2
r3 = three-point field goals = 3
The number of ways (k) she could have scored the points is:
k=(n!)/(r1!×r2!×r3!)
The factorial function is a mathematical formula represented by an exclamation mark "!".
k= 10!/(1!×2!×3!)
k= 3628800/(1×2×6)
k= 302400
so.. she could have scored the 10 points in 302400 ways.
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2. Six months after John William becomes a shareholder, Peixotto Media announces
its first shareholder meeting. Because the co-presidents will be making some very
important announcements about decisions that will be voted upon by the
shareholders, John William attends. He finds that of the 20,000 shares that have
been offered so far, the shareholders who attend the meeting own 6,000 of them.
a. The co-presidents propose adding a new member to the company's board of
directors. The shareholders are allowed to vote on this matter. What percentage of
the total vote will John William control? (2 points)
b. The company's co-presidents announce that Peixotto Media will be issuing
another offering of stock, this time in the amount of 50,000 shares. Given his
preemptive rights as an existing shareholder, how many shares of this stock is John
William entitled to purchase before the offering is made to the public? (3 points)
Answers
John William is entitled to purchase 15,000 shares before the offering is made to the public.
a. To calculate the percentage of the total vote John William will control, we need to find the proportion of shares he owns out of the total shares represented at the meeting.
John William owns 6,000 shares out of the 20,000 shares offered so far. Therefore, the proportion of shares he owns is 6,000/20,000, which simplifies to 3/10 or 0.3.
To convert this proportion to a percentage, we multiply by 100:
Percentage of the total vote = 0.3 * 100 = 30%
Thus, John William will control 30% of the total vote at the. shareholder meeting.
b. Preemptive rights allow existing shareholders to purchase new shares before they are offered to the public. To determine the number of shares John William is entitled to purchase, we need to calculate the proportion of his current ownership.
John William owns 6,000 shares out of the total shares offered so far, which is 20,000. Therefore, his ownership proportion is 6,000/20,000, or 3/10.
If Peixotto Media plans to issue 50,000 new shares, John William's entitlement would be:
Entitlement = (3/10) * 50,000 = 15,000 shares.
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Lydia is organizing a cooking competition at her school. She ordered some basic supplies to share among the teams that are competing.
The teams will be bringing other ingredients as well.
Use the list at the right to answer the questions.
1. If 10 of the teams divide the olive oil equally, how
much will each team receive? Write an equation
to model your work.
738.4 grams, flour
8.25 liters, milk
5.4 liters, olive oil
87.6 grams, salt
36 eggs
Answers
The equation to model the given work is 5.4 liters / 10 teams = 0.54 liters per team, and 0.54 liters per team × 920 grams per liter = 497.28 grams per team.
The teams will share 5.4 liters of olive oil. To find out how much each team will receive, we can divide the total amount of olive oil by the number of teams:
5.4 liters ÷ 10 teams = 0.54 liters per team
However, it's usually more convenient to express this in grams for equation, so we need to convert liters to grams.
The conversion factor for olive oil is approximately 0.92 grams per milliliter (or 920 grams per liter). So, we can multiply 0.54 liters per team by 920 grams per liter to get:
0.54 liters per team × 920 grams per liter = 497.28 grams per team
Therefore, each team will receive 497.28 grams of olive oil as per the work.
Equation: 5.4 liters / 10 teams = 0.54 liters per team, and 0.54 liters per team × 920 grams per liter = 497.28 grams per team.
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The answer should be 3278 ft I just need help with how to get it and the work.
Answers
The Panther I traveled approximately 2413 miles before reaching a speed of 130 mph.
What is a trapezoid?A trapezoid is a quadrilateral having two parallel bases and one set of other sides called legs.
To estimate the distance traveled by the Panther I, we need to approximate the area under the velocity curve between 0 and 25.7 seconds. We can divide the area into trapezoids using the data given in the table.
The first trapezoid has a height of 30 mph and a base of 1.8 seconds. The second trapezoid has a height of 40 mph and a base of 1.3 seconds (the difference between 3.1 and 1.8).
Continuing in this way, we get the following table:
Speed Height
Range (mph) Base (s) Area (mph * s)
0-30 30 1.8 54
30-40 40 1.3 52
40-50 50 1.1 55
50-60 60 1.3 78
60-70 70 1.9 133
70-80 80 1.8 144
80-90 90 2.3 207
90-100 100 3.1 310
100-120 120 6.3 756
120-130 130 4.8 624
To find the total area, we add up the areas of all the trapezoids:
54 + 52 + 55 + 78 + 133 + 144 + 207 + 310 + 756 + 624 = 2413
Therefore, the Panther I traveled approximately 2413 miles before reaching a speed of 130 mph.
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Can someone PLEASE answer 5 and 6 for me?this assignment is due in a few minutes
Answers
Answer:
Yes they are equivalent because if you divide 18 by 6 it is 3 and if you do the same for 21 and 7 it also equals 3 which means they get paid 3 per hour
As for 6 they are not bec if we do the same from to 6 220/25=8.8 and 90/10=9 meaning they don’t have the same feet to inches
Step-by-step explanation:
I hope this helps and I’m not to late.
can anyone help please!?
Answers
9514 1404 393
Answer:
x = √66
Step-by-step explanation:
In a geometry like this, the right triangles are all similar. Then the length of the long side to the hypotenuse is ...
6/x = x/(6+5)
6·11 = x²
x = √66
_____
Additional comment
The relation g=√(ab) gives the geometric mean (g) of the two numbers 'a' and 'b'. In this geometry, there are three geometric mean relationships.
long side = geometric mean(long segment, hypotenuse) . . . x = √(6·11)
short side = geometric mean(short segment, hypotenuse) . . . y = √(5·11)
altitude = geometric mean(long segment, short segment) . . . BD = √(6·5)
A state lotto has a prize that pays $2,000 each week for 20 years.
Find the total value of the prize:
If the state can earn 5% interest on investments, how much money will they need to put into an
account now to cover the weekly prize payments?
Answers
Answer: state will need to put $686,539.98 into an account now to cover the weekly prize payments for the next 20 years, assuming a 5% interest rate.
Step-by-step explanation:
To find the total value of the prize, we can multiply the weekly prize amount by the number of weeks in 20 years:
Total value of prize = $2,000/week x 52 weeks/year x 20 years = $2,080,000
To determine how much money the state will need to put into an account now to cover the weekly prize payments, we can use the present value formula:
Present value = Future value / (1 + r)^n
where r is the interest rate and n is the number of compounding periods.
In this case, we want to find the present value of 20 years' worth of weekly payments, so n = 20 x 52 = 1,040 (assuming 52 weeks in a year).
The interest rate is 5%, so r = 0.05.
Plugging in the values, we get:
Present value = $2,080,000 / (1 + 0.05)^1,040
Present value = $686,539.98 (rounded to the nearest cent)
Therefore, the state will need to put $686,539.98 into an account now to cover the weekly prize payments for the next 20 years, assuming a 5% interest rate.
Simplify 3(2x +7x - 5 )
why are people so weird on this website
Answers
Answer:
I love algebra anyways
The ans is in the picture with the steps how i got it
(hope this helps can i plz have brainlist :D hehe)
Step-by-step explanation:
Sketch a graph of
What is the orientation of the graph?
Answers
It is impossible to sketch a graph or determine the orientation of the graph.
Without a specific function or data set provided, it is impossible to sketch a graph or determine the orientation of the graph. The orientation of a graph refers to the direction of the graph and can be described as either increasing or decreasing.
In order to sketch a graph, we must first have a set of data points or a function to plot. Once we have the necessary information, we can plot the points or graph the function and then determine the orientation of the graph.
To determine the orientation of a graph, we look at the direction of the graph as we move from left to right. If the graph is moving upward as we move from left to right, then the orientation is increasing.
If the graph is moving downward as we move from left to right, then the orientation is decreasing. If the graph is neither increasing nor decreasing, then the orientation is said to be undefined.
In summary, the orientation of a graph is determined by the direction of the graph as we move from left to right. Without a specific function or data set provided,
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