The proving of the trigonometric function [tex]\frac{1+\cos A}{\sin A}+\frac{\sin A}{1+\cos A} = 2 cosec A[/tex] that LHS = RHS is shown.
What are trigonometric functions?Trigonometric functions are real functions in mathematics that relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many others. In trigonometry, there are six functions of an angle that are commonly used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their names and abbreviations (csc).So, [tex]\frac{1+\cos A}{\sin A}+\frac{\sin A}{1+\cos A} = 2 cosec A[/tex]:
LHS = [tex]\frac{1+\cos \mathrm{A}}{\sin \mathrm{A}}+\frac{\sin \mathrm{A}}{1+\cos \mathrm{A}}[/tex]
[tex]=\frac{(1+\cos \mathrm{A})(1+\cos \mathrm{A})+\sin \mathrm{A} \sin \mathrm{A}}{\sin \mathrm{A}(1+\cos \mathrm{A})}\\\begin{aligned}&=\frac{(1+\cos \mathrm{A})^2+\sin ^2 \mathrm{~A}}{(1+\cos \mathrm{A}) \sin \mathrm{A}} \\&=\frac{1+\cos ^2 \mathrm{~A}+2 \cos \mathrm{A}+\sin ^2 \mathrm{~A}}{(1+\cos \mathrm{A}) \sin \mathrm{A}}\end{aligned}\\[/tex]
[tex]\begin{aligned}&=\frac{1+2 \cos A+1}{(1+\cos A) \sin A} \\&=\frac{2+2 \cos A}{(1+\cos A) \sin A}\end{aligned}\\\begin{aligned}&=\frac{2(1+\cos A)}{(1+\cos A) \sin A} \\&=2 \frac{1}{\sin A}\end{aligned}\\=2cosecA[/tex]
= RHS
Hence Proved.
Therefore, the proving of the trignometric function [tex]\frac{1+\cos A}{\sin A}+\frac{\sin A}{1+\cos A} = 2 cosec A[/tex] is shown.
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The correct question is given below:
Prove that: [tex]\frac{1+\cos A}{\sin A}+\frac{\sin A}{1+\cos A} = 2 cosec A[/tex]
A television programme begins at 6:10pm the programme ends at 7:05 how long will the television last
Answer:
55 minutes
Step-by-step explanation:
→ Work out long it will take to get to 7:00
7:00 - 6:10 = 50 minutes
→ Work out how much extra time after 7:00
7:05 - 7:00 = 5 minutes
→ Add the 2 answers together
50 + 5 = 55 minutes
Solve each system by elimination.
x + y = 12 , x - y = 2
By elimination, the solution of the system of equations, x + y = 12 and x - y = 2, is (7 , 5).
A system of equations is a set of two or more equations which includes common variables. To solve system of equations, we must find the value of the unknown variables used in the equations that must satisfy all the equations.
There are three methods that can be used to solve system of equations.
1. Elimination
2. Substitution
3. Graphing
Using the elimination method, given two equations in x and y, a variable should be eliminated by adding/subtracting the two equations.
Adding the two equations will eliminate the variable y.
x + y = 12 (equation 1)
x - y = 2 (equation 2)
2x + 0y = 14
x = 7
Substitute the value of x to any of the two equation and solve for y.
x + y = 12 (equation 1)
7 + y = 12
y = 12 - 7
y = 5
Hence, the solution of the system of equations is (7 , 5).
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Step up an inequality for this situation below. Define your variable, but do not solve.
Mr. Bass charges his music students $25 per lesson plus a recital fee of $50. How many lessons can a student take if they pay no more than $175?
The inequality that models the situation is:
$50 + $25*x < $175
Where x is the number of lessons that the student can take.
How to write the inequality?
We know that Mr. Bass charges $25 per class plus a fixed fee of $50.
Then if a student takes x classes, the cost is given by the expression:
$50 + $25*x
And we know that this needs to be smaller than $175, then we can write the inequality:
$50 + $25*x < $175
Now we can solve this for x:
$25*x < $175 - $50
$25*x < $125
x < $125/$25
x < 5
So the student needs to take less than 5 classes.
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Determine whether each sequence is geometric. If so, identify the common ratio and find the next two terms. 1, 1/2, 1/4,1/8, . . .
Common ratio= 1/2, Next two term= 1/16, 1/32
This is a geometric sequence since there is a common ratio between each term.
i.e.,
the common ratio (r) = [tex]\frac{1/2}{1}[/tex] = [tex]\frac{1/4}{1/2}[/tex]
= 1/2
We can write the formula for general term of the series as
[tex]a_{n}[/tex] = a[tex](\frac{1}{2})^{n-1}[/tex]
If the further sequence does continue in the same way then the whole infinite sequence indicated is a geometric series.
So we have
[tex]a_{2}[/tex] =(1) [tex](\frac{1}{2} )^{2-1}[/tex]= 1/2
[tex]a_{3}[/tex] = 1× [tex](\frac{1}{2} )^{3-1}[/tex]= 1/4
[tex]a_{4}[/tex] = 1× [tex](\frac{1}{2} )^{4-1}[/tex]= 1/8
[tex]a_{5}[/tex] = 1× [tex](\frac{1}{2} )^{5-1}[/tex]= 1/16
[tex]a_{6}[/tex] = 1× [tex](\frac{1}{2} )^{6-1}[/tex]= 1/32
......
Therefore the sequence is in the same series, and we get the next two terms of the sequence which are 1/16, 1/32
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Which floating-point literal correctly represents the scientific notation value: 2.3 x 10^7?
The floating-point literal that correctly represents the scientific notation value: 2.3 x 10^7 will be A. 2.3e^7.
What is scientific notation?It should be noted that scientific notation simply means. way that's used to express numbers that are either too large or too small to be written in decimal form.
Therefore, it should be noted that the floating-point literal that correctly represents the scientific notation value: 2.3 x 10^7 will be:
= 2.3 × 20^7
= 2.3 × e^7
= 2.3e7
The correct option is A.
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Which floating-point literal correctly represents the scientific notation value: 2.3 x 10^7?
a. 2.3e^7
b. 2.3 × 10e^7
c. 2.3e × 10^7
d. 2.3 × e^7
Simplify. *
-2√- 45 - 3√125
A.-13i√5
B.-17i√5
C. -17√5
D.-21i√5
According to the question, the given expression can be simplified with the help of factors.
Given expression = [tex]-2\sqrt{-45}-3\sqrt{125}[/tex]
Do the factors of 45 as well as 125 and take the common things out from the square root:
Required expression = [tex]-2(3)\sqrt{-5}-3(5)\sqrt{-5}= -21i\sqrt{5}[/tex]
In the theory of complex number, [tex]i^{2} = -1[/tex] is already defined.
Hence, the answer for the given expression is [tex]-21i\sqrt{5}[/tex].
What is simplification?
Simplification is the process to make the expression as simple for the calculation. That means, to calculate the solution of a complex expression.
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54 yards to mm bridge method
Evaluate A² for A = -3.
-6
9
6
-9
Answer:
b) 9
Step-by-step explanation:
Given that,
→ A = -3
Then the value of A² will be,
→ (-3)² = 9
Hence, option (b) is correct.
Solve each inequality. Check your solution. (x/3) + 5 ≥ 1/6
The inequality given by (x/3) + 5 ≥ 1/6 has the solution x ≥ -29/2 or [-29/2 , +∞).
Inequality refers to the relationship between two non-equal expressions. It can be denoted by > for greater than, < for less than, >/= for greater than and equal to, and </= for less than and equal to.
To solve an inequality is to find the values of x such that when we substitute that number for x we have a true statement.
Given the inequality (x/3) + 5 ≥ 1/6, subtract 5 from both sides.
(x/3) + 5 - 5 ≥ 1/6 - 5
x/3 ≥ -29/6
Multiply both sides by 3.
3(x/3) ≥ 3(-29/6)
x ≥ -29/2
To check, let x = -29/2
x = -29/2 ≥ -29/2
Substitute this value to the inequality.
(x/3) + 5 ≥ 1/6
(-29/2)/3 + 5 ≥ 1/6
1/6 ≥ 1/6
let x = -14,
x = -14 ≥ -29/2
Substitute this value to the inequality.
(x/3) + 5 ≥ 1/6
-14/3 + 5 ≥ 1/6
1/3 ≥ 1/6
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7x+ 6y=2
- 28x- 24y = -8
Answer:
x=2/7 - 6y/7
Step-by- explanation:
Stan owns 30 shares of Wal-Mart stock worth $48 a share and 50 shares of another company’s stock worth $61 a share. What is Stan’s net worth assuming he has no liabilities?
Answer: $4490
Step-by-step explanation:
30×48=1440
1440+(50×61)
URGENT
AN arithmetic sequence is given below
23,19,15,11
Write an explicit formula for the nth term an
Answer:
11, 15, 19, 23 = An+4
23,19,15,11=An-4
State whether the sentence is true or false. If false, replace the underlined word or number to make a true sentence.
Line t is called the transversal for lines p and q .
The statement "Line t is called the transversal for lines p and q" is true
Given,
The statement "Line t is called the transversal for lines p and q"
In the figure we can see that line t cut two parallel line p and q
So the line t is transversal for lines p and q
So the given statement "Line t is called the transversal for lines p and q" is true
Hence, the The statement "Line t is called the transversal for lines p and q" is true
The complete question is:
State whether the sentence is true or false. If false, replace the underlined word or number to make a true sentence.
Line t is called the transversal for lines p and q .
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The opposite of the sum of -7 and y
Please help!!
Answer:
7 - y
Step-by-step explanation:
The sum of -7 and y: -7 + y
The opposite of the sum: -(-7 + y) = 7 - y
Find the quotient of the quantity 32 times ten raised to the negative twenty second power end quantity divided by the quantity 8 times ten raised to the negative twenty second power end quantity.. write the final answer in scientific notation. 4 x 10−44 4 x 1044 4 x 1022 4 x 100
The final answer in scientific notation is 4 × 10⁰.
Firstly writing both dividend and divisor in scientific notation.
Dividend = 32 × 10⁻²²
Divisor = 8 × 10⁻²²
Now performing division to find the quotient.
Quotient = Dividend ÷ Divisor
Quotient = 32 × 10⁻²² ÷ 8 × 10⁻²²
As per the known rule of exponents, aˣ ÷ aⁿ = aˣ⁻ⁿ
Quotient = (32 ÷ 8) × 10⁻²² ⁻ ⁽⁻²²⁾
Dividing the base numbers
Quotient = 4 × 10⁻²²⁺²²
Performing subtraction on exponents
Quotient = 4 × 10⁰
Also, as per the known fact, a⁰ = 1. So, the final answer is 4 and in scientific notation the final answer is 4 × 10⁰.
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Given two terms of each arithmetic sequence, find a₁ and d . a₄=8 and a₇=20
The two terms of each arithmetic sequence [tex]a_{4}[/tex] = 8 [tex]a_{7}[/tex] = 20 are d = 4 and a = -4.
What is sequence?[tex]a_{4}[/tex] = 8
8 = a +3d.............1
[tex]a_{7}[/tex] = 20
20 = a + 6d...........2
equating equation 1 and 2
8 = a +3d
20 = a + 6d
d = 12/3
d = 4...............3
from equation 3
8 = a + 12
a = -4
An ordered group of numbers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6.
Arithmetic sequences are sequences containing these patterns. The distance between succeeding terms in an arithmetic series is always the same. The difference between consecutive words is always two, hence the sequence 3, 5, 7, 9... is arithmetic.
An explicit formula that states a = d (n - 1) + c, where d is the common difference between succeeding words, and c = a1, can be used to establish an arithmetic sequence.
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Help me
In solving this question
Answer:
She could have planted 250 seeds on Wednesday
Evaluate each expression for x=5 .
(5/3) (3x-6)-(6-4 x)
The value of the given expression (5/3)(3x - 6) - (6 - 4x) at x = 5 is 29.
According to the given question.
We have an expression (5/3)(3x - 6) - (6 - 4x).
So, the value of the given expression (5/3)(3x - 6) - (6 - 4x) at x = 5 is given by
(5/3)(3x - 6) - (6 - 4x)
= 5x - 10 - (6 - 4x) (by distributive rule)
= 5x - 10 - 6 + 4x
= 9x - 16
= 9(5) - 16 (at x = 5)
= 29
Hence, the value of the given expression (5/3)(3x - 6) - (6 - 4x) at x = 5 is 29.
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WHAT are the angle relationships formed when a third line
intersects two parallel lines?
the angles are the same as their pair
the pairs are
1 and 5
2 and 6
4 and 8
3 and 7
express the function f(x)=1+|2x-5| in piecewise form without using absolute values
The function f(x)=1+|2x-5| in piecewise form without using absolute values will be f₁ = 4, f₂ = 2 and a = 5/2.
Given that,
f(x) = 1 + |2x - 5|
And,
f(x) = {f₁ x < a
{f₂ x ≥ a
Now,
We have,
f(x) = 1 + |2x - 5|
And,
By definition of absolute value,
|x| = {x x ≥ 0
{-x x < 0
And,
Given the absolute value,
|2x - 5|
Solve for x,
i.e.
2x - 5 = 0
We get,
x = 5/2 = 2.5
i.e.
a = 5/2
i.e.
For all values greater than x = 5/2, 2x - 5 is positive,
And,
For all values less than x = 5/2, 2x - 5 is negative.
So,
|x| = {x x ≥ 5/2
{-x x < 5/2
i.e.
|2x - 5| = {2x - 5 x ≥ 5/2
{-(2x - 5) x < 5/2
Now,
f(1) = 1 + |2(1) - 5|
We get,
f₁ = 4
And,
f(2) = 1 + |2(2) - 5|
We get,
f₂ = 2
Hence we can say that the function f(x)=1+|2x-5| in piecewise form without using absolute values will be f₁ = 4, f₂ = 2 and a = 5/2.
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What is the unit price of a 24- ounce box of cereal that costs $3.60
Answer:
$0.15 per ounce
Step-by-step explanation:
Take the total price and divide by the ounces to get the price per ounce. 3.60/24 equals 0.15
Jai's family took a road trip to Mount Rushmore. Jai fell asleep after they had
travelled 259 miles. If the total length of the trip was 740 miles, what
percentage of the total trip had they travelled when Jai fell asleep?
The total percentage they travelled when Jai fell asleep is 35%.
In mathematics, a percentage is a ratio or value that is expressed as a fraction of 100. The Latin term "per centum," which means "by a hundred," is the source of the English word "percent." The sign "%" stands for percentage. We apply the percentage method to determine the portion of a whole expressed in terms of 100. Use the percentage formula to convert a number to a fraction of 100.
The total length of trip was 740 miles.
Jai travelled before they fell asleep = 259 miles
Therefore the percentage of total trip travelled before Jai fell asleep =
= [tex]\frac{259}{740}\times100[/tex]
= 35 %
So by 35% of the total trip, they travelled when Jai fell asleep.
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Stuart’s bank account had a balance of $-51.41 last month. This month he won a cash prize of $77.63 for winning a competition. How much money does Stuart have at the end of this month , if there are no other transactions?
Answer:
$26.22
Step-by-step explanation:
Stuart's bank account balance (before) : $ - 51.41
Profit/Loss : + $ 77.63
Therefore:
- $ 5 1 . 4 1
+
$ 7 7 . 6 3
_____________
$ 2 6 . 2 2
Therefore, Stuart has $26.22 in his bank account at the end of the month assuming there are no other other transactions.
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What is the slope of the line that passes through the points (- 2, 6) and (4, - 2) ?
Answer:
-4/3
Step-by-step explanation:
[tex] \frac{6 - ( - 2)}{ - 2 - 4} = - \frac{8}{6} = - \frac{4}{3} [/tex]
Find two numbers with a sum of 65. Twice the lesser number is 10 more than greater number.
Answer:
a=25
b=40
Step-by-step explanation:
We can get two equations from these statements
1.a+b=65
2.a<b
therefore 2a=10+b
and 2a-10=b
substitute into original equation
a+(2a-10)=65
3a=75
a=25
b=65-25
=40
Add. 7/6 + 1/3
i've been asking a lot of fraction questions lol :D
Answer:
Hope this helps :)
All of these are the answers, the final answer is the simplified one, and the explanation is below.
9/6 (improper fraction)
1 3/6 (mixed number)
1 1/2 (simplified)
Step-by-step explanation:
1. Common denominators which is 6.
7/6 + 1/3
1/3 × 2/2 = 2/6
2. Add the numerators since they already have common denominators.
7/6 + 2/6 = 9/6
3. Change into mixed number and simplify if needed.
9/6 = 1 3/6 = 1 1/2
Engineers evaluate the efficiency of the memory and speed of a computer. What does evaluate mean in mathematics?
To evaluate anything is to determine its monetary value. Finding all of the values of a variable or collection of variables necessary to make a statement true is what it means to solve something.
What does evaluate mean in mathematics?It is to figure out or compute a formula, function, or relation's value.Middle school math involves evaluating solely arithmetic expressions.It is like to evaluate the equation with a specific value of x or y or any variable.As opposed to solution, which essentially involves going backwards and undoing all of the changes made to the input.Evaluating the value of an algebraic expression when a specified integer is used to replace a variable is known as evaluating the expression. We use the given number to replace the expression's variable before applying the order of operations to simplify the expression.To evaluate anything is to determine its monetary value. Finding all of the values of a variable or collection of variables necessary to make a statement true is what it means to solve something.To learn more about evaluate, refer to:
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Can someone please solve these? Identify the segment bisect of XY. Then find XY
The length of the line segment XY which was bisected at point M is;
1) 32
2) 6
How to solve division of line segments?
1) From the given line segments, it is clear that XM is congruent to YM since M bisects XY. Thus;
XM = YM and XM + YM = XY
We are given that;
XM = 3x + 1
YM = 8x - 24
Thus;
3x + 1 = 8x - 24
8x - 3x = 1 + 24
5x = 25
x = 25/5
x = 5
Thus;
XM = 3(5) + 1 = 16
YM = 8(5) - 24 = 16
XY = 16 + 16 = 32
2) From the given line segments, it is clear that XM is congruent to YM since M bisects XY. Thus;
XM = YM and XM + YM = XY
We are given that;
XM = 5x + 8
YM = 9x + 12
Thus;
5x + 8 = 9x + 12
9x - 5x = 8 - 12
4x = -4
x = -1
XM = 5(-1) + 8 = 3
YM = 9(-1) + 12 = 3
XY = 6
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Factor each expression. x²-2 x-15 .
Answer:(x+3)(x-5)
Step-by-step explanation:(x^2-5x)+(3x-15)
x(x-5)+3(x-5)
(x+3)(x-5)
PLEASE HELP ME
The frequency distribution shows the widths, in inches, of available cell phones at a technology store.
What is the mean of the frequency distribution? Round answers to tenths place, if answer is a whole number
than put a zero in the tenths place so it will be counted correct.
For given data set, The value of mean is 6.02.
What is mean?The sum of a set of numbers divided by the total number of numbers in the set is known as the arithmetic mean or arithmetic average in mathematics and statistics.
The formula to calculate mean is = sum of all observations / total number of observations
x = 4 5 6 7 8
f = 12 1 8 18 4 ∑f = 43
fx = 48 5 48 126 32 ∑fx = 259
The formula of mean = ∑fx / ∑f
= 6.02
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