To prove this Pythagorean Identity, we have to know that:
cot = 1 / tan
and we also know that:
tan = sin / cos
Therefore,
cot = 1 / tan = cos / sin
So we can write the given equation in the form of:
1 + (cos^2 θ / sin^2 θ) = csc^2 θ
Expanding the left hand side of the equation:
(sin^2 θ / sin^2 θ) + (cos^2 θ / sin^2 θ) = csc^2 θ
(sin^2 θ + cos^2 θ) / sin^2 θ = csc^2 θ
We know that given a unit circle, sin^2 θ + cos^2 θ = 1. So:
1 / sin^2 θ = csc^2 θ
The equation above is already true basing on the trigonometric identities. Therefore:
csc^2 θ = csc^2 θ
The sine function is an odd function
Answer:
The given statement is true.
Step-by-step explanation:
We have been given a statement and we are supposed to determine whether our given statement is even or odd.
Statement: The sine function is an odd function.
We know that sine is symmetric about origin and [tex]\text{sin}(-x)=-\text{sin}(x)[/tex]. Therefore, sine is an odd function and our given statement is true.
The population of a local species of beetle can be found using an infinite geometric series a1=960 and the common ratio is .25 write the sum in sigma notation and calculate the sum of possible that will be the upper limit of this population
Sigma 960 (1/4)^i-1 ; the sum is 1280
Signa 960 (1/4)^-1 the sun is divergent
Sigma 960 (1/4)^i sum is 1280
Sigma (1/4)^i sum is divergent
I=1 for them all
The answer is:
Sigma 960 (1/4)^i-1 ; the sum is 1280
Step-by-step explanation:We are given the first term of the geometric sequence as:
[tex]a_1=960[/tex]
Also, the common ratio of the terms in geometric sequence is: [tex]\dfrac{1}{4}[/tex]
We know that if the series is a geometric series than the sum of the terms is given by:
[tex]a+ar+ar^2+.....\\\\=a(1+r+r^2+....)\\\\=\sum ar^{n-1}[/tex]
where a is the first term of the series and r is the common difference.
Here a=960
and r=1/4
Hence,
The sum of the series is:
[tex]=\sum 960(\dfrac{1}{4})^{i-1}[/tex]
Now we know that the sum of the infinite geometric series is given by:
[tex]S=\dfrac{a}{1-r}[/tex]
where S is the sum of the series.
Hence, here the sum of the series is calculated by:
[tex]S=\dfrac{960}{1-\dfrac{1}{4}}\\\\\\S=\dfrac{960}{\dfrac{3}{4}}\\\\\\S=\dfrac{960\times 4}{3}\\\\\\S=1280[/tex]
Hence, the sum is: 1280
The value of m in the following system of equations is:
m-2n=8
n=m-2
A) m= -4
B) m= -6
C) m= 2
D) None of the choices are correct
The correct option is A. [tex]m=-4[/tex]. By substituting the second equation into the first one in the system of equations, the value of m is -4.
The value of m in the system of equations can be found by substitution or elimination. Starting with the given equations:
[tex]m - 2n = 8 (1)[/tex]
[tex]n = m - 2 (2)[/tex]
Using the second equation (2) to substitute for n in the first equation (1):
[tex]m - 2(m - 2) = 8[/tex]
[tex]m - 2m + 4 = 8[/tex]
[tex]-m + 4 = 8[/tex]
[tex]-m = 8 - 4[/tex]
[tex]-m = 4[/tex]
[tex]m = -4[/tex]
This means that the value of m is -4, which corresponds to option A.
True or false some drugs like Tylenol or available over-the-counter because they are safe in any dose
A function f satisfies g(7) = 13 and g(1) = 6. a function f satisfies h(13) = 2 and h(6) = 7. what is the value of g(h(6))
What is the slope of a line that is perpendicular to the line x=-3?
A(-3
B(0
C(1/3
D(underfined
The slope of a line perpendicular to x = -3 is 0. Option B is correct answer.
What is the slopes of perpendicular lines?Perpendicular line slopes are the negative reciprocals of one another. In other words, if one line has a slope of m, a line perpendicular to it has a slope of -1/m. The definition of perpendicular lines, which states that the angles created by the lines are 90 degrees, may be used to demonstrate this relationship. The product of the slopes of perpendicular lines must be -1 in order to meet the requirement that the tangent of a 90 degree angle be specified.
Given the line x = -3.
Any line perpendicular to this line will have slopes of negative reciprocals of each other.
The slope of a horizontal line is 0, since the line does not change in the y-direction.
Hence, the slope of a line perpendicular to x = -3 is 0.
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You make both rocking chairs and bar stools. You charge $100 for every rocking chair and $40 for every bar stool. Which expression represents your income if you sell r rocking chairs and b bar stools?
Answer:
The answer is [tex]Income=100*r+40*b[/tex]
Step-by-step explanation:
In this case we have to construct an expression that describe the income of selling the rocking chairs ($100 each - r) and the bar stool ($40 each - b), so the total income will depend on the addition of this two variables multiplied by its individual price, this is:
[tex]Income=100*r+40*b[/tex]
The answer is [tex]Income=100*r+40*b[/tex]
Help please. I need the answer fast.
Can anyone help!? Points are high!
total was 49.10
mileage would be 49.10 = 2.10x +5.00
total miles =
49.10 = 2.10 +5.00
44.10 = 2.10x
x = 44.10/2.10 = 21 miles
Does Verona have enough information to determine the areas of both triangles?
A cube has side lengths of 8 inches. What is the volume of the cube
The population of snakes at a wildlife park over the course of six months is modeled by the equation y = x5 − 5x4 + 2x3 − 10x2 + 100x + 5, where x is the number of months. If the park officials count 505 snakes, how many months have passed since the initial count was taken?
P(A∩B)=1150,P(A|B)=1120,P(B)=?
For the given probabilities P(A∩B) = 1150 and P(A|B) = 1120, using Bayes' theorem for conditional probability, the probability of event B, P(B), is found to be approximately 1.027.
Explanation:The topic of this question is probability, specifically concurrent events in probability theory. Here, we use Bayes' theorem for conditional probability which states that P(A|B) = P(A ∩ B) / P(B).
Given that P(A∩B) = 1150 and P(A|B) = 1120, we can plug these values into the formula to find P(B). Hence, P(B) = P(A ∩ B) / P(A|B), substituting the given numbers, we get P(B) = 1150 / 1120 = 1.027 approximately.
So P(B), the probability of event B occurring, we would expect to be approximately 1.027 (or 102.7% assuming the probabilities were in decimal form to start with).
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Alright, let's solve the problem with these given values.
The formula for calculating the probability of B in the context of A and B both happening is:
`P(A and B) / P(A|B)`.
In this case, the probability of A and B occurring is 1150, while the likelihood of A given that B has occurred is 1120. We can substitute these values into the formula:
So, P(B) = P(A and B) / P(A|B)
Substitute the given into the formula:
P(B) = 1150 / 1120 = 1.0267857142857142
Therefore, the Probability of event B (P(B)) is approximately 1.027 when rounded to three decimal places.
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Please help!!! Geometry.
Look at the triangle. What is the value of sin x°?
If there are 190 contestants in a competition, how many ways can the first, second, and their place winner be chosen
Assume that you randomly select 6 cards from a deck of 52. what is the probability that all of the cards selected are hearts?
Which of the relationship below are function
A functions B and D are valid functions because they meet the criteria of each x value having only one corresponding y value.
Function B: You mentioned that B is a function because each x value has only one corresponding y value. In mathematical terms, a function is defined as a relation between a set of inputs (x values) and a set of outputs (y values), where each input (x) maps to exactly one output (y). So, if B satisfies this definition, it is indeed a function.
Function D: Similar to B, if D also adheres to the definition of a function, where each x value corresponds to only one y value, then it is a function as well.
Function A: You mentioned that for A, the x value of 1 has 2 y values. In mathematical terms, a function cannot have one input (x) mapping to multiple outputs (y). If this is the case for A, it does not meet the definition of a function.
Function C: You mentioned that for C, the x value of 7 has 2 y values, 10 and 3. Again, if one input (x) is associated with multiple outputs (y) in C, it does not qualify as a function.
In summary, functions B and D are valid functions because they meet the criteria of each x value having only one corresponding y value. Functions A and C do not meet this criterion and are not functions in the mathematical sense.
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In triangle ABC, m∠A=55 triangle ABC, m∠A=55°, c=11c=11, and m∠B=19m∠B=19°. Find the perimeter of the triangle
What are the two equations that need to be solved to find the solutions for the absolute value equation |2x-3|=x+6?
2x – 3 = x + 6 and 2x – 3 = –x – 6
2x – 3 = x + 6 and 2x + 3 = –x + 6
2x – 3 = x + 6 and 2x – 3 = x – 6
2x – 3 = x + 6 and 2x + 3 = x + 6
The equations to solve for [tex]\( |2x-3|=x+6 \)[/tex] are [tex]\( 2x - 3 = x + 6 \)[/tex] and [tex]\( 2x - 3 = -(x + 6) \).[/tex]
Let's solve the absolute value equation |2x-3|=x+6 step by step.
We'll start by writing down the equation:
|2x - 3| = x + 6
Since the absolute value expression |2x - 3| can be either positive or negative, we'll split the equation into two cases:
Case 1: 2x - 3 = x + 6
Case 2: 2x - 3 = -(x + 6)
Let's solve Case 1:
2x - 3 = x + 6
Subtract x from both sides:
2x - x - 3 = 6
Simplify:
x - 3 = 6
Add 3 to both sides:
x = 9
Now, let's solve Case 2:
2x - 3 = -(x + 6)
Distribute the negative sign:
2x - 3 = -x - 6
Add x to both sides:
2x + x - 3 = -6
Simplify:
3x - 3 = -6
Add 3 to both sides:
3x = -3
Divide both sides by 3:
x = -1
So, the solutions to the equation |2x-3|=x+6 are x = 9 and x = -1.
What is the quotient and remainder when x2 + x − 12 is divided by x − 2?
What is the expected number of heads for Danielle's experiment? (Enter your answer as a decimal)
I’ve answered this before so here’s the complete given for
this problem.
Danielle conducts an experiment by tossing a fair
coin three times. She records the number of heads out of the three trials. The
probabilities are given in the table below (X = number of heads out of three
trials).
Given:
x 0 1 2 3
P(x)1/8 3/8 3/8 1/8
n = 3; p = 0.5; q = 0.5; P(x = 0) =
nCx p^x
x N(x) P(x) ΣP(x) 1- ΣP(x) x * P(x)
--- ----- --------- --------- ---------
---------
0 1 0.125 0.125 0.875 0
1 3 0.375 0.5 0.5 0.375
2 3 0.375 0.875 0.125 0.75
3 1 0.125 1 0 0.375
0+.375+.75+.375 = 1.5
So, in this problem, the expected number of heads for Danielle's experiment is 1.5.
What is 180 as product of prime factors and 180 as a Square root?
A person purchased a leather jacket that was marked down by 40%. if the sale price of the jacket was $101.79, what was its original price?
When flipping a coin three times, what is the probability of landing on heads all three times?
Scientific notation definition math is fun
Answer:
Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power, as 1385.62 written as [tex]1.3862*10^{4}[/tex].
Step-by-step explanation:
Ten children are to be divided into an a team and a b team of 5 each. the a team will play in one league and the b team in another. how many different divisions are possible?
Identify the factors of x2 − 4x − 12. (1 point)
(x + 4)(x − 3)
(x − 4)(x + 3)
(x − 2)(x + 6)
(x + 2)(x − 6)
Answer:
D. [tex](x-6)(x+2)[/tex]
Step-by-step explanation:
We have been given a polynomial [tex]x^2-4x-12[/tex]. We are asked to find the factors of our given polynomial.
We will use splitting the middle term format of factoring quadratic polynomial.
We need two numbers, whose product is [tex]-12[/tex] and whose sum is [tex]-4[/tex]. Such two numbers are [tex]-6\text{ and }2[/tex].
[tex]x^2-6x+2x-12[/tex]
[tex](x^2-6x)+(2x-12)[/tex]
Factor out GCF of each group:
[tex]x(x-6)+2(x-6)[/tex]
[tex](x-6)(x+2)[/tex]
Therefore, factors of our given expression are [tex](x-6)(x+2)[/tex] and option D is the correct choice.
(08.02)The coordinate grid shows the plot of four equations.
Which set of equations has (−1, 5) as its solution?
A and B
B and D
A and C
B and C
(05.01)
Which statement best explains if the graph correctly represents the proportional relationship y = −2x?
No, the points shown would not be part of y = −2x
No, proportions cannot be represented on a graph
Yes, all proportions can be shown on a graph of this line
Yes, the points shown on the line would be part of y = −2x
The best statement which is true for the graph y=-2x is that Yes, the points shown on the line would be a part of y=-2x.
What is line?A line is collection of various points on a surface which together joins two points on a surface. Equation of a line looks like y=a +b x where b is the slope of line.
How to make graph?We have been given an equation y=-2x and we have to find the graph of equation and for that we need to find points and for this we have to put the values of x which gives the values of y.
At x=0 ,y=0
At x=1 ,y=-2*1=-2
At x=2 ,y=-2*2=-4
At x=-1 ,y=-2*-1=2
Coordinates will be (0,0),(1,-2),(2,-4),(-1,2)
If we find these points on this line then will surely be able to find these points.
Hence the statement which is true is that Yes, the points shown on the line would be part of y = −2x.
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