Answers
The trigonometric identities can be proved as follows;
32. Using the substitution, tan(x) = sin(x)/cos(x), and cot(x) = cos(x)/sin(x), we get;
cot(x) - tan(x) = (1 - 2·sin²(x))/(sin(x)·cos(x)) = sec(x)·(csc(x) - 2·sin(x))
36. tan(θ)/(1 + sec(θ)) = sin(θ)/(1 + cos(θ)) = (1 - cos(θ))/(sin(θ)) = -cot(θ) + csc(θ)
What are trigonometric identities?Trigonometric identities are equations involving trigonometric functions which are valid for the values of the input variable.
as follows;
cot(x) - tan(x) = sec(x)·(csc(x) - 2·sin(x))
The left hand side of the equation can be expressed using sin(x) and cos(x) as follows;
cot(x) = cos(x)/sin(x)
tan(x) = sin(x)/cos(x)
Therefore;
cot(x) - tan(x) = cos(x)/sin(x) - sin(x)/cos(x) = (cos²(x) - sin²(x))/(sin(x)·cos(x))
cos²(x) = 1 - sin²(x), therefore
(cos²(x) - sin²(x))/(sin(x)·cos(x)) = (1 - sin²(x) - sin²(x))/(sin(x)·cos(x))
(1 - sin²(x) - sin²(x))/(sin(x)·cos(x)) = (1 - 2·sin²(x))/(sin(x)·cos(x))
(1 - 2·sin²(x))/(sin(x)·cos(x)) = csc(x)·sec(x)·(1 - 2·sin²(x))
csc(x)·sec(x)·(1 - 2·sin²(x)) = sec(x)·(csc(x) - 2·sin(x))
Therefore;
cot(x) - tan(x) = csc(x)·sec(x)·(1 - 2·sin²(x)) = sec(x)·(csc(x) - 2·sin(x))
cot(x) - tan(x) = sec(x)·(csc(x) - 2·sin(x))36. tan(θ)/(1 + sec(θ)) = -cot(θ) + csc(θ)
The left hand side can be manipulated as follows;
tan(θ) = sin(θ)/cos(θ)
sec(θ) = 1/cos(θ)
Therefore; tan(θ)/(1 + sec(θ)) = (sin(θ)/cos(θ))/(1 + (1/cos(θ)))
(sin(θ)/cos(θ))/(1 + (1/cos(θ))) = (sin(θ)/cos(θ))/((cos(θ) + 1)/(cos(θ)))
((sin(θ)/cos(θ))×cos(θ))/((cos(θ) + 1)/(cos(θ)) × cos(θ))) = sin(θ)/(cos(θ) + 1)
tan(θ)/(1 + sec(θ)) = sin(θ)/(cos(θ) + 1)
sin(θ)/(cos(θ) + 1) = sin(θ)/(1 + cos(θ)) × ((1 - cos(θ))/(1 - cos(θ)))
sin(θ)/(1 + cos(θ)) × ((1 - cos(θ))/(1 - cos(θ))) = ((sin(θ)·(1 - cos(θ))/(1 - cos²(θ)))
((sin(θ)·(1 - cos(θ))/(1 - cos²(θ))) = ((sin(θ)·(1 - cos(θ))/(sin²(θ)) = (1 - cos(θ))/(sin(θ))
sin(θ)/(cos(θ) + 1) = (1 - cos(θ))/(sin(θ)) = csc(θ) - cot(θ) = -cot(θ) + csc(θ)
tan(θ)/(1 + sec(θ)) = sin(θ)/(cos(θ) + 1) = -cot(θ) + csc(θ)
Therefore;
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Related Questions
B) The area of a triangle = ½ the base the
height
What is the area of an equilateral triangle where
every side measures 10 cm?
Answers
The area of the equilateral triangle with side length 10 cm is 25 cm².
What is triangle ?
Triangle can be defined in which it consists of three sides , three angles and sum of three angles is always 180 degrees.
To find the area of an equilateral triangle, we need to find the height first. The height of an equilateral triangle is equal to the length of the perpendicular line from the center of the triangle to one of its sides.
To find the height, we can divide the equilateral triangle into two 30-60-90 triangles. The height is equal to half of the hypotenuse of one of these triangles, which can be calculated as:
10 cm / 2 = 5 cm
Now that we have the height, we can find the area of the equilateral triangle as follows:
Area = ½ * base * height
Area = ½ * 10 cm * 5 cm
Area = 25 cm²
Hence, The area of the equilateral triangle with side length 10 cm is 25 cm².
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Wendy and Roger camped out at a state park, and then hiked back to their cars, which were
parked on opposite ends of the park. Wendy headed due north at 3 miles per hour, and Roger
hiked due south at 6 miles per hour. In how long were the two friends 2 miles apart?
If necessary, round your answer to the nearest minute.
Answers
In 30 minutes the two friends 2 miles apart.
What is the speed?The speed formula can be defined as the rate at which an object covers some distance. Speed can be measured as the distance travelled by a body in a given period of time. The SI unit of speed is m/s.
Given that, Wendy headed due north at 3 miles per hour, and Roger hiked due south at 6 miles per hour.
We know that, Time= Distance/Speed
Now, T₁= 2/3
T₂= 2/6 =1/3
T=T₁-T₂
T=2/3+1/3
T=1/3
T=0.33 hours
= 30 minutes
Therefore, in 30 minutes the two friends 2 miles apart.
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Rounded to 3 decimal places, what is the value of r for this data set?
Answers
The correlation coefficient r of the data given is 0.938
What is the correlation coefficient?The correlation coefficient is a statistical measure of the strength of a linear relationship between two variables.
Given that a set of data showing the values of variables x and y, we are asked to find the correlation coefficient r of the data given,
We know that,
The correlation coefficient r is a number between and calculated to represent the linear dependence of two variables or sets of data.
Using an Excel tool :- (CORREL function) = CORREL(B3:B10,C3:C10)
We get, r = 0.938
Hence, the correlation coefficient r of the data given is 0.938
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Marianna finds an annuity that pays 8% annual interest, compounded quarterly. She invests in this annuity and contributes $10,000 each quarter for 6 years. How much money will be in her annuity after 6 years? Enter your answer rounded to the nearest hundred dollars.
Answers
Answer:
Marianna will have $19,700 in her annuity after 6 years.
Step-by-step explanation:
To find the amount of money in the annuity after 6 years, we need to calculate the compound interest earned over that time period.
First, let's find the interest rate per quarter: 8% annual interest divided by 4 quarters per year = 2% interest per quarter.
Next, we'll calculate the number of quarters that the money is invested: 6 years x 4 quarters per year = 24 quarters.
Now we can use the formula for compound interest to find the total amount of money in the annuity after 6 years:
A = P * (1 + r/n)^(nt)
Where:
A = final amount
P = initial investment ($10,000 in this case)
r = interest rate per quarter (2% or 0.02)
n = number of times the interest is compounded in a year (4)
t = number of years invested (6)
Plugging in the values:
A = $10,000 * (1 + 0.02)^(24)
A = $10,000 * 1.96634438
A = $19,663.44
Rounding to the nearest hundred dollars: $19,700
So, Marianna will have $19,700 in her annuity after 6 years.
Use the data display to answer 11 and 12.
11. Analyze and Persevere What does the mean
describe in the data set?
12. Analyze and Persevere What does the range
describe in the data set?
Answers
Answer:
563
Step-by-step explanation:
In a certain college, 4 instructors are single out of 24 instructors employed. What is the ratio of single to married instructors?
A. 1 to 6
B. 1 to 5
C. 24 to 4
D. 20 to 24
Answers
Answer:
(B) "1 to 5."
Step-by-step explanation:
The ratio of single to married instructors is 1 to 5. Since 4 instructors are single out of 24 total instructors, the ratio of single instructors to married instructors is 4 to (24 - 4) = 4 to 20. Simplifying the ratio to its simplest form, we get 4:20 = 1:5, which is the answer choice (B) "1 to 5."
Composition function
Answers
The composite function is given below with examples.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
Consider two functions f(x) and g(x).
Now,
f o g = f(g(x)) is called a composite function.
f(x) = 2x + 1
g(x) = x + 2
So,
(f o g)(x)
= f(g(x))
= f(x + 1)
= 2 (x + 1) + 1
= 2x + 2 + 1
= 2x + 3
Thus,
The composite function is given above with examples.
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please help me with math i’ll give you brainlist
Answers
Answer:we can’t help if it is barely readable
Step-by-step explanation:
Analyzing Functions Using an Equation and a Table
The function f(x)=x+is used to complete this
table.
-1
0
2
f(x)
1
312
2
512
2
Which statements are true of the given function?
Check all that apply.
머글)=-2
Of(0) = 2/1/2
Of(1) = -1
Of(2)=1
f(4) = 글
Answers
Answer:
2 . of(0)=2/1/2 this is answer number 2
The solution is, option 2 neither linear nor exponential.
What is function?Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable.
here, we have,
The given function is
x : -5 -3 -1 1 3
f(x) :8192 512 32 2 18
when we plot this function in the excel we get to know that it a regression as shown in the figure attached
because value of lies between (0,1)
which shows regression
therefore it is neither linear nor exponential.
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What are the transformations of y= -4(x-5)^3+6 from the parent function y=x^3.
Answers
A parent function is the most fundamental function that still satisfies the criteria for a specific type of function. When considering the linear functions that make up a family of functions, for instance, the parent function would be y = x. The simplest function is this linear one.
What is linear function?An uninterrupted line represents the graph of a linear function. The following form represents a linear function. a + bx = y = f (x) (x).Two variables—one independent and one dependent—make up a linear function. Both X and Y are considered independent and dependent variables.The words "linear function" in mathematics are used to describe two different but related ideas: A polynomial function of degree 0 or 1 with a graph that is a straight line is known as a linear function in calculus and fields related to it.A straight line can be visualised on the coordinate plane using a linear function. A linear function is demonstrated by the equation y = 3x - 2 because it depicts a straight line in the coordinate plane.To learn more about linear function refer to:
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Gabrielle's age is three times Mikhail's age. The sum of their ages is 64. What is Mikhail's age?
Answers
Answer:
16 years old
Step-by-step explanation:
Let's call Mikhail's age "x". Gabrielle's age is 3 times that, so her age is 3x. The sum of their ages is 64, so:
x + 3x = 64
4x = 64
x = 16
So, Mikhail's age is 16 years old.
porfavor me ayudarian es urgente
Answers
Answer: no exact answer
Step-by-step explanation:
The result of fraction multiplication are:
2/3 x 2/5 = 4/15
4/2 x 6/3 = 4
2/4 x 3/6 = 1/4
2/3 x 6/3 = 4/3
Given that, simplify the following:
2/3 x 2/5
4/2 x 6/3
2/4 x 3/6
2/3 x 6/3
For the expression 2/3 x 2/5:
To multiply fractions, we multiply the numerators together and the denominators together.
So, (2 x 2)/(3 x 5) = 4/15.
For the expression 4/2 x 6/3:
Again, we multiply the numerators together and the denominators together.
Thus, (4 x 6)/(2 x 3) = 24/6 = 4.
For the expression 2/4 x 3/6:
Following the same process, we multiply the numerators and denominators.
Therefore, (2 x 3)/(4 x 6) = 6/24 = 1/4.
Lastly, for the expression 2/3 x 6/3:
Similarly, we multiply the numerators and denominators together.
Hence, (2 x 6)/(3 x 3) = 12/9 = 4/3.
Therefore, the result of fraction multiplication are:
2/3 x 2/5 = 4/15
4/2 x 6/3 = 4
2/4 x 3/6 = 1/4
2/3 x 6/3 = 4/3
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Quadratic Expressions:Question 1
Which is a factor of x²+ 7x + 10?
Select one:
a x-5
bx-1
cx+5
dx+1
Answers
Answer:
Option C. (x + 5)
Step-by-step explanation:
This quadratic equation can either be solved using the quadratic formula or be factorised using The Double Bubble Method.
Double Bubble Method:
After listing the different pairs of factors 10, the pair (that would add together to give 7) are 2 and 5. These two factors will be used to rewrite +7x:
[tex]x^{2} + 2x + 5x + 10[/tex]
Factorization by grouping (i.e factorising each separate group sharing a common factor) will then be applied:
[tex]= x(x + 2) + 5(x + 2)[/tex]
The common factor, which is the (x + 2) expression, will be extracted:
[tex]= (x + 2)(x + 5)[/tex]
∴ [tex]x^{2} + 7x + 10[/tex] can be factorised to [tex](x + 2)(x + 5)[/tex]
It is observed that a factor of the given quadratic equation is x + 5
∴Option C
If x = -2, what is y = ?
Answers
Answer:
y = 2
Step-by-step explanation:
If x = -2, what is y = ?
When x = -2 the line crosses the y axis at 2, so y = 2
Which of the following expressions is equivalent to the logarithmic expression below? log4 7/x^3
Answers
Answer:
D
Step-by-step explanation:
using the rules of logarithms
• log a - log b = log ([tex]\frac{a}{b}[/tex] )
• log [tex]x^{n}[/tex] = nlog x
then
[tex]log_{4}[/tex] [tex]\frac{7}{x^3}[/tex]
= [tex]log_{4}[/tex] 7 - [tex]log_{4}[/tex] x³
= [tex]log_{4}[/tex] 7 - 3[tex]log_{4}[/tex] x
NEEDED TODAY THANK YOU
Answers
Step by step
First I will find the total area
A = L x W
A = 11 x 7
A = 77
Now to find the area of the missing space and subtract that from the total area of 77
L = 11 - 4 - 2 = 5
W = 7 - 3 = 4
A = 5 x 4
A = 20 area of cut out missing space
77 - 20 = 57 units squared
How far did Mai run in her first 30 minutes
Answers
Given the costs associated with insurance, why do companies provide insurance plans to employees?
Answers
Company should provide insurance policies to employees because the employees are assets to them.
What is meant by Insurance Policies?An insurance policy is a contract between the insurance company and the person getting the insurance.
Employees are assets of a company.
Companies should look after by insuring this asset like any other asset.
Although it is not the only factor for an employee to stick to a company, it is also one of the contributing factor for the employees to stick to a company longer.
It is a very good benefit for the employee since the premium is paid by the employer for the employee.
Hence companies should provide insurance policies to employees.
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How much time will it take to earn $760.00 in interest if you deposit $3,200.00 into a simple interest account earning 4.5% annum Simple interest
Answers
Using the formula of simple interest, the time it will take to accumulate to $3200 at a rate of 4.5% is 71.35 years
What is Simple InterestSimple interest is a method of calculating the interest charge on a loan or deposit. It is based on the principle of earning interest only on the initial amount (principal) that was borrowed or deposited.
The formula for simple interest is:
I = P * r * t
where:
I is the interest amount
P is the principal amount
r is the interest rate (expressed as a decimal)
t is the time period (in years)
Substituting the values into the formula;
A = P(1 + rt)
3200 = 760(1 + 0.045(t))
3200 = 760 + 34.2t
34.2t = 3200 - 760
34.2t = 2440
t = 2440 / 34.2
t = 72.35 years
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The graph shows Felix's English scores versus the number of hours he studies:
What will Felix's English scores most likely be if he spends 7 hours a week studying English?
a 64 points
b 72 points
c 84 points
d 92 points
Answers
Felix's English score would be around 72 points if he spends 7 hours a week studying English.
the answer is option b, 72 points.
What is a graph of an equation?
A graph of an equation is a visual representation of the relationship between the variables in that equation. In mathematics, an equation typically involves one or more variables, and the graph of that equation shows how the value of one variable changes as the value of another variable change.
Look at the horizontal axis of the graph to find the point that corresponds to 7 hours of studying per week. In this case, it is approximately halfway between 6 and 8, so we can estimate it to be around 7.
From that point, draw a vertical line up to the graph to find the corresponding English score.
Follow the line until it intersects with the blue line representing Felix's English scores.
From that intersection point, draw a horizontal line over to the y-axis to read the corresponding score.
Based on this analysis, we can estimate that Felix's English score would be around 72 points if he spends 7 hours a week studying English.
Therefore, Felix's English score would be around 72 points if he spends 7 hours a week studying English.
the answer is option b, 72 points.
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100 points!!!!!
Using the information for West Coast Entertainment Inc. presented in the table below, determine the following:
1. The number of shares bought and sold on this trading day.
2. Amount of income per share paid to a stockholder in the last 12 months.
3. Percentage of change in the stock's price since the beginning of the year.
Answers
Percentage of change in the stock's price since the beginning of the year is: 12.32%
What is stock's price?The price at which a stock's share is currently trading on the market is referred to as the stock price. Every publicly listed company issues its shares at a price, which is an estimation of their value that, ideally, corresponds to the worth of the firm itself.
The market forces of supply and demand govern a company's share price once it becomes public and its shares begin trading on a stock exchange. Price increases will occur if there is a large demand for its shares. A stock's price might be reduced by sellers if the company's prospects for future growth appear uncertain.
Given that,
Closing price today = $21.20
Closing price on the previous trading day= $28.83
Difference between closing price today and closing price on the previous trading day= $21.20 - $28.83
= $7.63
Percentage of change in the stock price since the beginning of the year
= Year-to-date % change
= 12.32%
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Solve the following polynomial equation. y3+8=0
Answers
I need the answer NO LINKS URGENT NO LINKS URGENTS
Answers
Answer: See the attached image below for the proof table.
Explanation:
We start with what we're given. Simply repeat the given information word for word. This is statement 1 of the proof.
Statement 2 mentions AB is parallel to CD. This is valid due to the definition of a parallelogram. The name means "two sets of opposite parallel sides. Statement 3 is a similar situation.
Statement 4 relies on statement 2. The red angles in the diagram are congruent alternate interior angles. They are only congruent because of AB parallel to CD.
Statement 5 relies on statement 3 for similar reasoning as the previous paragraph. Focus on the blue angles.
Statement 6 is hopefully self explanatory so there's not much for me to mention here.
Statement 7 has us connect statements 4, 5, and 6 to prove triangle ADB is congruent to triangle CBD. We use the ASA (angle side angle) theorem here.
Statements 8 and 9 finish up the proof. CPCTC stands for "Corresponding parts of congruent triangles are congruent". If two triangles are congruent, then their corresponding pieces must be congruent as well. It would be like saying two houses are identical if and only if their front doors are identical.
Find the equation of the lines given the two points.
a) (-8, 3) and (-6, 0)
b) (5, 4.5) and (5, -1)
2. A rental car costs $18 plus a fixed charge per mile driven. The total charge for 180 miles of use was $80. Write an equation for the cost, C (in dollars), in terms of the miles driven, x.
Solve the linear systems.
3. 5x -2y = 7
x = 2y-5
4. 4x + v=5
v = 2x - 10
Answers
The equation of the line given by the coordinates (-8, 3) and (-6, 0) is 2y = -3x -18.
What is slope of an equation?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Hence, it is possible to write the change in y coordinate with respect to the change in x coordinate as,
m = (y2 - y1) / (x2 - x1)
The coordinates of the points are:
(-8, 3) and (-6, 0)
The slope of the line is given as:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates we have:
m = (0 - 3)/ (-6 + 8)
m = -3/2
Using the point slope form:
y - y1 = m (x - x1)
Substituting the values of slope and coordinates we have:
y - 3 = -3/2(x + 8)
2(y - 3) = -3(x + 8)
2y - 6 = -3x -24
2y = -3x -18
Hence, the equation of the line given by the coordinates (-8, 3) and (-6, 0) is 2y = -3x -18.
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26 gallons of gas cost $80.08. What is the unit rate for 1 gallon of gas?
A. O$2.98 per gallon
B. O$3.08 per gallon
C. O$4.16 per gallon
D. O$308.00 per gallon
Answers
100 POINTS BRAINLIEST A business buys equipment with $300 cash and puts $425 on credit. What is the value of this asset?
A. $725
C. $425
B. $300
D. $125
Answers
Answer: A) $725
Step-by-step explanation:
The value of the asset that the business bought with $300 cash and $425 on credit is $725. The total amount that the business spent on the equipment is equal to the sum of the cash and the credit, which is $300 + $425 = $725.
The American flag has 50 stars, 7 red stripes, and 6 white stripes. Write a ratio to represent the ratio of stars to red stripes.
Answers
Answer:
50:7
Step-by-step explanation:
You just put ":" in between the numbers requested.
We will flip a balanced coin 3 times and for each toss, record whether we get a head or a tail. Write all possible outcomes of this experiment to find the probability that we get exactly 1 head.
Answers
The probability of getting exactly one head after flipping a coin three times is equal to ( 3 / 8 ) .
Possible outcomes of flipping a coin three times are :
= { HHH, THH, HTH, HHT, TTH, THT, HTT, TTT }
Possible outcomes of getting exactly one head are:
= { TTH, THT, HTT }
Total number of outcomes are equal to 8
Favourable number of outcomes are 3
Probability of getting exactly one head is equal to
= ( Number of favourable outcomes ) / ( total number of outcomes)
= 3 / 8
Therefore, the probability of getting exactly one head by flipping a coin three times is equal to ( 3 / 8 ).
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89392i÷5×5+788 what is the answer
Answers
Answer:
89392i + 788
Step-by-step explanation:
The divide by 5 and the multiply by 5 cancel out, and we cannot simplify 89392i + 788 any further, since we don't know what i is. Therefore the answer is 89392i + 788.
Hope that helps!
Find the length of CD.
E
2
D
A1 B
The length of CD is
5
C
unit(s).
Answers
The length of CD can be found to be 7. 21 units .
How to find CD ?The length of CD can be found by using the distance formula which is :
= √ [ ( x ₂ - x ₁ ) ² + ( y ₂ - y ₁ ) ² ]
Given the points, ( 3, - 2 ) and ( 7, - 8 ).
The length of CD is :
= √ [ ( 7 - 3 ) ² + ( - 8 - (- 2 ) ) ² ]
= 7. 21
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