Answer:
The solution of the equation is Ф = 0 or Ф = 3π/2
Step-by-step explanation:
* Lets revise some facts in trigonometry
- The identity sin² Ф + cos² Ф = 1
- By subtracting sin² Ф from both sides then cos² Ф = sin² Ф - 1
- In the rectangular plane the point (x , y) represents (cos Ф , sin Ф)
where x = cox Ф and y = sin Ф
- The point (1 , 0) lies on the positive part of x-axis means cos Ф = 1
and sin Ф = 0, then Ф = 0 or 2π
- The point (-1 , 0) lies on the negative part of x-axis means cos Ф = -1
and sin Ф = 0, then Ф = π
- The point (0 , 1) lies on the positive part of y-axis means cos Ф = 0
and sin Ф = 1, then Ф = π/2
- The point (0 , -1) lies on the negative part of y-axis means cos Ф = 0
and sin Ф = -1, then Ф = 3π/2
* Lets solve the problem
∵ sin Ф + 1 = cos² Ф
- To solve we must change cos² Ф to sin² Ф
∵ cos² Ф = sin² Ф - 1
- substitute cos² Ф in the equation by 1 - sin² Ф
∴ sin Ф + 1 = 1 - sin² Ф ⇒ add sin² Ф to both sides
∴ sin² Ф + sin Ф + 1 = 1 ⇒ subtract 1 from both sides
∴ sin² Ф + sin Ф = 0
- Take sin Ф as a common factor from both terms
∴ sin Ф (sin Ф + 1) = 0
- Equate each factor by 0
∴ sin Ф = 0 OR sin Ф + 1 = 0
- Remember 0 ≤ Ф < π
∵ sin Ф = 0 ⇒ from the information above
∴ Ф = 0
∵ sin Ф + 1 = 0 ⇒ subtract 1 from both sides
∴ sin Ф = -1
- From the information above
∴ Ф = 3π/2
* The solution of the equation is Ф = 0 or Ф = 3π/2
guys help me to solve these questions please
Answer:
Step-by-step explanation:
The first one I would factor the difference of cubes on top and then cancel a common factor this would end up giving you sin^2(A/2)+sin(A/2)cos(A/2)+cos^2(A/2)
(the 1st+3rd term here actually equals 1)
now you have
1+sin(A/2)cos(A/2)
1+1/2*2sin(A/2)cos(A/2) (this step was just to help show how I'm going to use the next identity)
1+1/2sin(A) (double angle identity for sine)
hint for the next one: I would multiply top and bottom by bottom's conjugate first
Each side length of a triangle is 4 cm. What type of triangle is it?
acute
right
equilateral
scalene
Which are the solutions of the quadratic equation?
x2 = 7x + 4
Step-by-step explanation:
[tex]x {}^{2} = 7x + 4[/tex]
It can also be written as:-
[tex]x {}^{2} - 7x - 4 = 0[/tex]
Now, here a=1, b= -7 and c = -4
Now by quadratic formula,
[tex]d = b {}^{2} - 4ac[/tex]
Calculate d which is
[tex] \sqrt{65} [/tex]
Now, the roots will be:-
[tex]x = ( - b + \sqrt{d} ) \div 2a[/tex]
and hence it will become:-
[tex]x = (7 + \sqrt{65} ) \div 2[/tex]
and
[tex] x = (7 - \sqrt{65} ) \div 2[/tex]
Answer: third option.
Step-by-step explanation:
Move the [tex]7x[/tex] and [tex]4[/tex] to the left side of the equation:
[tex]x^2 = 7x+ 4[/tex]
[tex]x^2 -7x- 4=0[/tex]
Now you need to apply the Quadratic formula:
[tex]x=\frac{-b\±\sqrt{b^2-4ac} }{2a}[/tex]
In this case you can identify that:
[tex]a=1\\b=-7\\c=-4[/tex]
Then you can substitute values into the Quadratic formula and get the following solutions:
[tex]x=\frac{-(-7)\±\sqrt{(-7)^2-4(1)(-4)} }{2*1}\\\\x_1=\frac{7-\sqrt{65} }{2}\\\\x_2=\frac{7+\sqrt{65} }{2}[/tex]
Do 9, 12, 14 form a right triangle
Answer:
No
Step-by-step explanation:
To determine if the 3 sides form a right triangle.
Use the converse theorem of Pythagoras.
If the square of the longest side is equal to the sum of the squares of the other two sides then the triangle is right,
longest side = 14 and 14 ² = 196
9² + 12² = 81 + 144 = 225
Since 196 ≠ 225 the triangle is not right
Select the correct answer.
What is the domain of the function f(x) = x^2 + 3x + 5?
A.
all whole numbers
B.
all positive real numbers
all integers
D.
all real numbers
Answer:
All real numbers
Step-by-step explanation:
f(x) is a polynomial and is well defined for all real values of x
Domain is x ∈ R
The domain of the function[tex]$$f(x)=x^{2}+3 x+5$$[/tex] is (D). All real numbers
Domain of functionThe domain of a function exists as the set of all possible inputs for the function.A function with a fraction with a variable in the denominator. To discover the domain of this kind of function, set the bottom equal to zero and exclude the x value you find when you solve the equation. A function with a variable inside a radical sign.It is provided that,
[tex]$$f(x)=x^{2}+3 x+5$$[/tex]
This exists as the equation of a vertical parabola opens upward.
The vertex exists at a minimum
utilizing a graphing tool
The vertex is the point (-1.5,2.75)
The range is the interval -------> [2.75.∞)
[tex]y \geq 2.75[/tex] ------->All real numbers greater than or equal to 2.75
The domain is the interval -------> (-∞,∞) -----> All real numbers
Hence, The domain of the function[tex]$$f(x)=x^{2}+3 x+5$$[/tex] is (D). All real numbers.
To learn more about the Domain of function refer to:
https://brainly.com/question/13856645
#SPJ2
I need help fast !!!
Answer:
D
Step-by-step explanation:
By definition:
rational number = fraction or decimals that are non-whole numbers
negative = well..... negative numbers have a minus (-) in front of them.
Looking at the options, only D has a fraction that has a (-).
The surface area of square pyramid is 1,089 in^2.
If the dimensions are multiplied by 1/3, what will
be the new surface area?
The new surface area of a square pyramid when its dimensions are multiplied by 1/3 is calculated by squaring the scale factor (1/3) and multiplying it with the original surface area. In this case, the new surface area is 121 in².
Explanation:When the dimensions of a square pyramid are multiplied by 1/3, each linear dimension of the pyramid becomes one-third of its original size. Since surface area is a two-dimensional measure (length × width), when you scale down each dimension by a factor of 1/3, the new surface area will be the square of that scale factor times the original area. The surface area is proportional to the square of the scaling factor because area is calculated using two dimensions.
The original surface area is 1,089 in². To find the new surface area, we use the scale factor squared: (1/3) ² = 1/9. We then multiply the original surface area by this factor:
New Surface Area = Original Surface Area × (Scale Factor)²
New Surface Area = 1,089 in² × 1/9
New Surface Area = 1,089 in² / 9
New Surface Area = 121 in²
Therefore, the new surface area of the square pyramid when the dimensions are multiplied by 1/3 will be 121 in².
What is the explicit rule for the arithmetic
sequence on which the series is based?
Answer:
12,6
Step-by-step explanation:
The explicit rule for the arithmetic sequence on which the series is based is an = 12 + (n - 1) * 6
How to determine the explicit rule of a sequence?Series are sum of sequences. The nth term of an arithmetic series is expressed as:
an = a + (n - 1)d
where
a is the first term = 12
n is the number of terms
d is the common difference = 18 - 12 = 24 - 18 = 6
Substitute
an = 12 + (n - 1) * 6
Hence the explicit rule for the arithmetic sequence on which the series is based is an = 12 + (n - 1) * 6
Learn more on explicit rule here: https://brainly.com/question/2059043
Which table represents a direct variation function?
a.
Input (x)23456
Output (y)7891011
b.
Input (x)246810
Output (y)-3-5-6-7-8
c.
Input (x)-5-4-3-2-1
Output (y)108642
d.
Input (x)-21012
Output (y)-3-3-3-3-3
is it a?
Answer:
Option C is correct.
Step-by-step explanation:
A direct variation function is
y/x = k
i.e. we can say that the ratio of y and x is equal to a constant value k.
We will check for each Option given.
Option A
7/2 = 7/2
8/3 = 8/3
9/4 = 9/4
10/5 = 2
11/6 = 11/6
Option D is incorrect as y/x ≠ k as ratio of y/x for each value of table doesn't equal to constant
Option B
-3/2 = -3/2
-5/4 = -5/4
-6/6 = -1
-7/8 = -7/8
-8/10 = -4/5
Option B is incorrect as y/x ≠ k as ratio of y/x for each value of table doesn't equal to constant
Option C
10/-5 = -2
8/-4 = -2
6/-3 = -2
4/-2 = -2
2/-1 = -2
Option C is correct as y/x = k as ratio of y/x for each value in table c is equal to constant value -2
Option D
-3/-2 = 3/2
-3/1 = -3
-3/0 = 0
-3/1 = -3
-3/2 = -3/2
Option D is incorrect as y/x ≠ k as ratio of y/x for each value of table doesn't equal to constant .
SO, Option C is correct.
Answer: OPTION C
Step-by-step explanation:
The function of direct variation has this form:
[tex]y=kx[/tex]
Where k is the constant of variation.
Let's check if there is a constant of variation on the Options "a" and "b":
On Option A:
[tex]\frac{7}{2}=3.5\\\\\frac{8}{2}=4[/tex]
On Option B:
[tex]\frac{-3}{2}=-1.5\\\\\frac{-5}{4}=-1.25[/tex]
There is no constant of variation, then these tables do no represent a direct variation.
On the table shown in Option "d" you can observe that "y" does not change when "x" changes. Then it does not represent a direct variation.
Since on the table shown in Option "c":
[tex]k=-2[/tex]
This table represents a direct variation.
There are 20 students in the high school glee club. They need to pick four students to serve as
the student representatives for an upcoming trip. How many ways are there to choose the
representatives?
Answer:
4845 ways
Step-by-step explanation:
To solve this problem we use the formula of combinations
[tex]nCr=\frac{n!}{r!(n-r)!}[/tex]
Where n is the total number of students that can be chosen and you choose r from them.
So
[tex]n = 20\\r = 4[/tex]
Note that in this case the order in which the 4 students are chosen does not matter.
Then
[tex]20C4=\frac{20!}{4!(20-4)!}[/tex]
[tex]20C4=\frac{20!}{4!*16!}=4845\ ways[/tex]
Answer:
4845
Step-by-step explanation:
Total number of students : 20
Students to be picked : 4
Does order matter? : No
Use combination formula (see attached)
POINTS! APLENTY!
A cone with a base radius of 8 cm fits inside a sphere of radius 10 cm. Find the perpendicular height of the cone.
Please Explain!
Thank you very much for your help!
A stock has a 25% probability of increasing by $10 and a 75% probability of decreaing by $5. What is the stocks expected increase or decrease?
Please help me!!!!
Step-by-step explanation:
The expected value is the sum of each probability times the value.
E = 0.25 (10) + 0.75 (-5)
E = -1.25
The stock is expected to decrease by $1.25.
Please help me I AM STUCK, and please explain how you get the answer.
Answer:
x= -10
Step-by-step explanation:
(3)8+ [tex]\frac{\sqrt{2x+29} }{3}[/tex]= 9(3) (multiply both sides by 3)
24+ [tex]\sqrt{2x+29}[/tex]= 27 (move the 24 to the right)
[tex]\sqrt{2x+29}[/tex]=27-24 (subtract)
([tex]\sqrt{2x+29}[/tex])²= 3² (square both sides)
2x+29=9 (move the 29 to the other side)
2x=9-29 (subtract)
2x= -20 (divide both sides by 2)
x= -10
hope this helps and good luck! :)
An automobile dealer has 10 Fords, 7 Buick’s, and 5 Plymouth’s in her used-car lot. If a person purchases a used car, find the probability that it is a Ford or Buick.
Answer:
The probability is [tex]0.7727[/tex] or [tex]77.27\%[/tex]
Step-by-step explanation:
we know that
The probability of an event is the ratio of the size of the event space to the size of the sample space.
The size of the sample space is the total number of possible outcomes
The event space is the number of outcomes in the event you are interested in.
so
Let
x------> size of the event space
y-----> size of the sample space
so
[tex]P=\frac{x}{y}[/tex]
In this problem we have
[tex]x=10+7=17[/tex]
[tex]y=10+7+5=22[/tex]
substitute
[tex]P=\frac{17}{22}=0.7727[/tex]
Convert to percentage
[tex]0.7727*100=77.27\%[/tex]
Consider the following sequence 24,18,..
Assuming the sequence is geometric,
write the next three terms.
24, 18,
b.
Write an explicit equation
representing the sequence above.
Answer:
see explanation
Step-by-step explanation:
Since the sequence is geometric then the common ratio (r) is
r = [tex]\frac{18}{24}[/tex] = [tex]\frac{3}{4}[/tex] = 0.75
To find the next term in the sequence, multiply the previous term by r
The next 3 terms are
18 × 0.75 = 13.5
13.5 × 0.75 = 10.125
10.125 × 0.75 = 7.59375
The n th term equation of a geometric sequence is
[tex]a_{n}[/tex] = a[tex](r)^{n-1}[/tex]
where a is the first term, that is a = 24, hence
[tex]a_{n}[/tex] = 24[tex](0.75)^{n-1}[/tex]
1:
Tickets for the Battle of the Bands went on sale this week. On the first day of sales, 45 adult tickets and 20 student tickets were sold for a total of $875. On the last day of ticket sales, 25 adult tickets and 40 student tickets were sold for a total of $775. Let x represent the cost for each adult ticket and let y represent the cost for each student ticket. Write a system of equations to model this scenario and solve that system to determine the cost of each adult ticket.
First Day of Sales Equation:
Last Day of Sales Equation:
What is the cost of each adult ticket?
dollars.
Answer:
First day: 45x +20y = 875
Second day: 25x + 40y = 775
Adult ticket price: $15
Step-by-step explanation:
See paper attached. (:
The system of equations that model this scenario is:
45x + 20y = $875 equation 1
25x + 40y = $775 equation 2
The price of adults tickets is $15. The price of student's ticket is $10.
How many adults ticket were sold?In order to determine this value, multiplot equation 1 by 2 and subtract the resulting equation from equation 2.
90x + 40y = 1750 equation 3
975 = 65x
x = 15
How many students ticket were sold?Substitute for x in equation 1
45(15) + 20 = 875
675 + 20y = 875
y = (875 - 675) / 20
y = 10
To learn more about simultaneous equations, please check: https://brainly.com/question/25875552
Lines m and n are perpendicular if the slope of m is zero then the slope of n is
A.) zero
B.)undefined
C.)negative
Answer:
undefined
Step-by-step explanation:
A line with a slope of zero is a horizontal line parallel to the x- axis
Thus, a line perpendicular to it is a vertical line parallel to the y- axis, with an undefined slope.
Which equation is represented by the table ?
Answer:
The correct answer is B. b=3a + 2
Step-by-step explanation:
Plug in the values from column A into each equation and find which one works. In this case, the only option that works is B.
help me..it has to be rounded to the tenths
Answer:
10.2 cm²
Step-by-step explanation:
The area (A) of a regular hexagon is
A = [tex]\frac{1}{2}[/tex] × perimeter × apothem
Perimeter = 6 × 2 = 12 cm ( hexagon has 6 sides ), hence
A = 0.5 × 12 × 1.7 = 6 × 1.7 = 10.2 cm²
Answer:
10.2cm^2
Step-by-step explanation:
guys/girls help what is 30x1= and the times it by fish and eat it and tell me what u get
Answer:
you cant do that but the answer would be 0
Step-by-step explanation:
because if you eat something it would be gone in our tummy
I need to know how to do it and get the answer
Grace and her dad are planning to attend to State Fair. an adult ticket is $15 the price of an adult ticket is half the price of a student ticket plus $8. write an equation to determine how much grace will pay for the student ticket
Answer:
Equation:
[tex]15=\dfrac{1}{2}x+8[/tex]
Student ticket price = $14
Step-by-step explanation:
An adult ticket = $15
Let a student ticket = $x
An adult ticket is half the price of a student ticket plus $8, then an adult ticket
[tex]=\$\left(\dfrac{1}{2}x+8\right)[/tex]
Equate the prices for the adult ticket:
[tex]15=\dfrac{1}{2}x+8[/tex]
Subtract 8 from both sides:
[tex]15-8=\dfrac{1}{2}x+8-8\\ \\7=\dfrac{1}{2}x[/tex]
Multiply this equation by 2:
[tex]7\cdot 2=x\\ \\x=\$14[/tex]
Answer:
(1/2)x + 8 = 15
Step-by-step explanation:
The rectangle has rotational symmetry of order?
Answer:
A rectangle has rotational symmetry of order 2.
Step-by-step explanation:
The rotational symmetry involves rotation.
The rotational symmetry of order is the number of times a shape is mapped onto itself during a 360° rotation.
A rectangle has rotational symmetry of order 2.
When a rectangle is rotated on 180° or 360° it is mapped onto itself.
Answer:
2
180 & 360
Step-by-step explanation:
Write the expression in complete factored form 3c(p-6)+4(p-6)
Answer:
(p-6)(3c + 4)
Step-by-step explanation:
Note that (p-6) is a factor of both terms. Factoring out (p-6), we get:
(p-6)(3c + 4).
Write in Expanded form
(-3x)^4
Use parenthesis to indicate multiplication
A boy has as many sisters as brothers, but each sister has only half as many sisters as brothers.
How many brothers and sisters are there in the family?
Answer:
There are four sisters and three brothers in the family.
Step-by-step explanation:
Each brother has only half as many brothers as sisters.
1 boy = 4 sisters and 3 brothers
1 girl = 3 sisters and four brothers
Be sure to count the sisters and brothers in total and in terms of their own number of siblings.
Which exponential function is represented by the graph?
Answer:
see explanation
Step-by-step explanation:
The exponential function is in the form
y = a [tex](b)^{x}[/tex]
Use points on the graph to find a and b
Using (0, 3), then
3 = a [tex](b)^{0}[/tex] ⇒ a = 3
Using (1, 6), then
6 = 3 [tex](b)^{1}[/tex] ⇒ b = 6 ÷ 3 = 2
The equation is y = 3 [tex](2)^{x}[/tex]
Answer:
y = [tex]3*2^{x}[/tex]
Step-by-step explanation:
The general form for an exponential function is :
y = [tex]ab^{x}[/tex]
We need to find out what a and b are using the values given in the graph.
we can see that (x=0, y = 3) and (x=1, y = 6) are points on the curve. Substitute these into the general equation
for (x=0, y = 3),
3 = [tex]ab^{0}[/tex]
3 = a (1) or a = 3
for (x=1, y = 6),
6 = [tex]ab^{1}[/tex]
6 = ab (substitute a=3 from previous calculation)
6 = 3b
b = 2
Hence the equation is:
y = [tex]3*2^{x}[/tex]
Edit reason: typo in the final answer
how many solutions are there?
Answer:
Infinitely many solutions
Step-by-step explanation:
There is a solution, but not one, there is an infinite amount of them.
Thats why if you were to graph this it'd just be a line continuously going up
Konichiwa~! My name is Zalgo and I am here to help you out on the marvelous day! The answer to your question is Answer Choice C;Infinitely Many Solutions.
I hope that this helps! :D
"Stay Brainly and stay proud!" - Zalgo
(By the way, do you think you could give me Brainliest? I'd really appreciate it! Have a nice day! :3)
Consider the system:
y = 3x + 5
y = ax + b
What values for a and b make the system
inconsistent? What values for a and b make the
system consistent and dependent? Explain
Answer:
An exponent value in a would make the system inconsistent because it will either gain or lose height over time.
Any real number without an exponent value will make the system consitent.
Any infintite/repeating number will make the system both consistent and inconsistent. it is consisent because it will stay at a constant rate but also inconsistent because it is repeating and will never end.
Answer:
When a = 3 and b ≠ 5, the system will be inconsistent because the lines will be parallel. When a = 3 and b = 5, the system will be consistent and dependent because they represent the same line.
Find x and y in 3^(2x-y)=1 and 16^x/4=8^(3x-y)
the solutions are (x = 2) and (y = 4).
To find the values of x and y in the given equations, let's solve them step by step:
1. (3^{2x-y} = 1)
For any nonzero number raised to the power of 0, the result is always 1. Therefore, we can rewrite the equation as:
[2x - y = 0]
Now, we have one equation in terms of (x) and (y).
2.[tex]\( \frac{16^x}{4} = 8^{3x-y} \)[/tex]
We'll simplify the left side first:
[tex]\[ \frac{16^x}{4} = \frac{(4^2)^x}{4} = \frac{4^{2x}}{4} = 4^{2x-1} \][/tex]
Now, the equation becomes:
[4^{2x-1} = 8^{3x-y}]
We know that (8 = 2^3), so we can rewrite the right side of the equation in terms of 2:
[4^{2x-1} = (2^3)^{3x-y} = 2^{9x-3y}]
Now, we have:
[4^{2x-1} = 2^{9x-3y}]
Since (4 = 2^2), we can rewrite the left side as:
[ (2^2)^{2x-1} = 2^{4x-2} ]
So, the equation becomes:
[2^{4x-2} = 2^{9x-3y}]
Since the bases are the same, the exponents must be equal:
[4x - 2 = 9x - 3y]
Now, we have two equations:
[2x - y = 0]
[4x - 2 = 9x - 3y]
From the first equation, we can solve for (y):
[y = 2x]
Substituting (y = 2x) into the second equation:
[4x - 2 = 9x - 3(2x)]
[4x - 2 = 9x - 6x]
[4x - 2 = 3x]
[x = 2]
Now, substituting (x = 2) into the first equation to find (y):
[2(2) - y = 0]
[4 - y = 0]
[y = 4]
So, the solutions are (x = 2) and (y = 4).