Solve each equation for θ with 0 ≤ θ <2 π.
tanθ=tan²θ
Answers
To solve the equation tanθ = tan²θ, we can use trigonometric identities to simplify it.
First, let's rewrite tan²θ as (sinθ/cosθ)². Next, we can multiply both sides of the equation by cos²θ to eliminate the denominators.
This gives us sin²θ = sinθ. Now, we can rearrange the equation to get
sin²θ - sinθ = 0. Factoring out sinθ, we have sinθ
(sinθ - 1) = 0.
To find the values of θ that satisfy this equation, we set each factor equal to zero.
For sinθ = 0,
we have θ = 0, π, and 2π.
For sinθ - 1 = 0, we have
sinθ = 1, which occurs at θ = π/2.
Therefore, the solutions for θ are.
θ = 0, π/2, π, and 2π.
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To solve the equation tanθ = tan²θ, we can use the trigonometric identity and algebraic manipulation to find the values of θ that satisfy the equation within the given range of 0 ≤ θ < 2π.
To solve the equation tanθ = tan²θ, we can use the trigonometric identity for tangent: tan²θ = (sinθ/cosθ)².
First, let's rewrite the equation using the identity: tanθ = (sinθ/cosθ)².
Next, we can simplify the equation by multiplying both sides by cos²θ: tanθ * cos²θ = sin²θ.
Using the identity for sine squared: sin²θ = 1 - cos²θ, we can substitute it in the equation: tanθ * cos²θ = 1 - cos²θ.
Now, let's isolate cos²θ by moving it to one side of the equation: tanθ * cos²θ + cos²θ = 1.
Factoring out cos²θ: cos²θ * (tanθ + 1) = 1.
Dividing both sides by (tanθ + 1): cos²θ = 1 / (tanθ + 1).
Now, we can take the square root of both sides to solve for cosθ: cosθ = ±√(1 / (tanθ + 1)).
Finally, since we're solving for θ, we need to find the values of θ that satisfy 0 ≤ θ < 2π. We can substitute the values of cosθ into the equation cosθ = ±√(1 / (tanθ + 1)) and find the corresponding values of θ within the given range.
Remember to check for any restrictions or limitations on the solutions, such as asymptotes or undefined values.
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Related Questions
The manager where you work with has made a contest to encourage you to learn more about a new company you are working with for business. Every time you get a correct answer to the managers question, you get to draw a $1, $5, or $10 bill from a bag. There are 10 bills in the bag, 5 are $1 bills, 4 are $5 bills, and 1 is a $10 bill.
If you are the first to draw, what is your expected value in rewards?
Possible answers:
A: $4.00
B: $3.50
C: $3.00
D: $4.50
Answers
Answer:
$3.50
Step-by-step explanation:
add the dollar amount of the bills together, 5x1 + 4x5 + 10x1= 35
then divide that by the total number of bills to get your answer of $3.50
Answer:
$3.50
Step-by-step explanation:
a book has 136 pages. each page has the same number of words, and each page has no more than 100 words on it. the number of words in the book is congruent to 184, modulo 203. how many words are on each page?
Answers
The words are on each page is 73.
Given:
A book has 136 pages. each page has the same number of words, and each page has no more than 100 words on it.
The number of words in the book is congruent to 184, modulo 203.
136n mod 203 =184,
solve for n
Using CRT + MMI
n =203m + 73, where m =0, 1, 2, 3....
Therefore the words are on each page is 73.
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On Tuesday, the temperature in Fairfield, Montana fell from 51℉ at noon to −℉ at midnight.What was the average temperature change per hour?
Answers
= -4.333 deg F
Negative sign shows decrease in temperature
6.
The length of a rectangle is 3 in more than its width. The perimeter of the rectangle is 30 in.
a. Define a variable for width
b. Write an expression for length in terms of the width.
c. Write an equation to find the width of the rectangle.
d. Solve your equation.
e. What is the length of the rectangle?
Answers
Answer:
The answer is in the attached file below
Step-by-step explanation:
In the ardmore hotel, 20 percent of the customers pay by american express credit card. (a) of the next 10 customers, what is the probability that none pay by american express? (b) at least two? (c) fewer than three? (d) what is the expected number who pay by american express? (e) find the standard deviation.
Answers
a) The probability that none pay by American express is 0.107.
b) The probability of atleast two is 0.8350
c) The probability of fewer than three is 0.1650
d) The expected number who pay by American Express is 4.
e) The Standard deviation is 1.788
Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.
Probability(Event) = Favorable Outcomes/Total Outcomes = x/n
In the Ardmore Hotel, 20 percent of the customers pay by American Express credit card.
Here,
Mean = μ = np = 20*0.2 = 4
Standard deviation = σ = √(npq) = √(20×0.2×0.8) = √(3.2) = 1.7889
Let A be the event in which they pay by American express.
∴ P(A) = 0.2
∴ The probability that they do not pay by American Express = 1 - P(A)= 0.8.
(a) P(none in 10) = 0.8^10 = 0.10737
(b) At least two:
P(none pay) + P(one pays) + P(at least two pay) = 1
P(at least two) = 1 - [P(none) + P(one pay)
= 1 - [0.10737 + 20C1(0.2)^1*(0.8)^19] = 1 - [0.10737+0.05765] = 0.8350
(c) Fewer than three = P(one) + P(two) = 1- 0.8350 = 0.1650
(d) The expected number who pay by American Express is:
Mean = np = 20*0.2 = 4
(e) Standard deviation:
σ = √(npq) = √(4* 0.8) = √(3.2) = √(1.6*2) = 1.788
Thus,
a) The probability that none pay by American express is 0.107.
b) The probability of atleast two is 0.8350
c) The probability of fewer than three is 0.1650
d) The expected number who pay by American Express is 4.
e) The Standard deviation is 1.788
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If f'(c) < 0 then f(x) is decreasing and the graph of f(x) is concave down when x = c. True False Question 4 (1 point). A local extreme point of a polynomial function f(x) can only occur when f'(x) = 0. True False Question 5 (1 point) If f'(x) > 0 when x < c and f'(x) < 0 when x > c, then f(x) has a maximum value when x = C. True False
Answers
Question 3: True
Question 4: False
Question 5: True
If the derivative of a function, f'(x), is positive for values of x less than c and negative for values of x greater than c, then it indicates a change in the slope of the function. This change from positive slope to negative slope suggests that the function has a maximum value at x = c.
This is because the function is increasing before x = c and decreasing after x = c, indicating a peak or maximum at x = c.
Question 3: If f'(c) < 0 then f(x) is decreasing and the graph of f(x) is concave down when x = c.
True
When the derivative of a function, f'(x), is negative at a point c, it indicates that the function is decreasing at that point. Additionally, if the second derivative, f''(x), exists and is negative at x = c, it implies that the graph of f(x) is concave down at that point.
Question 4: A local extreme point of a polynomial function f(x) can only occur when f'(x) = 0.
False
A local extreme point of a polynomial function can occur when f'(x) = 0, but it is not the only condition. A local extreme point can also occur when f'(x) does not exist (such as at a sharp corner or cusp) or when f'(x) is undefined. Therefore, f'(x) being equal to zero is not the sole requirement for a local extreme point to exist.
Question 5: If f'(x) > 0 when x < c and f'(x) < 0 when x > c, then f(x) has a maximum value when x = c.
True
If the derivative of a function, f'(x), is positive for values of x less than c and negative for values of x greater than c, then it indicates a change in the slope of the function. This change from positive slope to negative slope suggests that the function has a maximum value at x = c. This is because the function is increasing before x = c and decreasing after x = c, indicating a peak or maximum at x = c.
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Which point in the table does not lie on the same line aa the other three points?
Answers
Answer:
If three or more points do not lie on the same straight line, then they are said to be non-collinear points. If any point of all the points is not on the same line, then as a group they are non-collinear points.
Q. Find missing angle
Answers
Answer:
c =√a2 + b2=√152 + 122=√369=19.20937
∠α =arcsin(a)c= arcsin(15)19.209372712299=arcsin(0.78086880944303) = 51.34° = 51
answer is C
Answer:
Tan x = opp/ adj
Tan x = 15 / 12
Tan x = 1.25
Angle = 51.34
C. 51 degrees
HOPE ITS HELPS..!!
Easy question! TOTALLY not a lot of points!
Find the scale factor for the figure below.
Answers
2:6= 1:3 or 1/3
or
Large to small is;
6:2 = 3:1
good luck!
Too easy XD I don’t remember how to do that T-T
Hhhhhhhhelllllllllllp In physics, you are using the formula E = mc2, where c is positive and represents the speed of light. Which equations are equivalent to
this?
Answers
The equivalent equation of E = mc² is m = E/c², and c = √(E/m) thus options (D) and (E) are correct.
What is the equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.
In another word, the equation must be constrained with some constraints.
As per the given,
E = mc²
m = E/c²
And,
c² = E/m
c = √(E/m)
Hence "The equivalent equation of E = mc² is m = E/c², and c = √(E/m)".
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Multiply and simplify if possible - radical functions. Thank you!
Answers
We want to simplify the following expression
\(\sqrt{7x}(\sqrt{x}+7\sqrt{7})\)First, we can apply the distributive rule to rewrite this product
\(\sqrt{7x}(\sqrt{x}+7\sqrt{7})=(\sqrt{7x})(\sqrt{x})+(\sqrt{7x})(7\sqrt{7})\)Then, using the following property
\(\sqrt{x}\cdot\sqrt{y}=\sqrt{x\cdot y}\)we can rewrite our expression as
\((\sqrt{7x})(\sqrt{x})+(\sqrt{7x})(7\sqrt{7})=\sqrt{7x\cdot x}+7\sqrt{7x\cdot7}\)We can remove the squared terms out of the root
\(\sqrt{7x\cdot x}+7\sqrt{7x\cdot7}=x\sqrt{7}+7\cdot7\sqrt{x}=x\sqrt{7}+49\sqrt{x}\)and this is our answer.
\(\sqrt{7x}(\sqrt{x}+7\sqrt{7})=x\sqrt{7}+49\sqrt{x}\)given x=-10.
Which equation is true?
5m + 5= -40
\(\frac{m}{10}\) + 4.7 = 3.7
\(\frac{m}{2}\) + 21 = 15
-49 + 3m = 72
Answers
Using mathematical operations to evaluate the equations, the only equation that fits is m/10 + 4.7 = 3.7
Evaluating an EquationAn expression is an algebraic statement that is made up of constant, variables etc. An equation on the other hand is two equal expression often times, there's are unknown variables which the values makes them equal to one another.
In this problem, we have to plug the value of m = 10 in all equations and see which fits, or we can simply just solve for m
a) 5m + 5 = -40
5m = -40 - 5
5m = -45
m = -9
This does not fit
b) m/10 + 4.7 = 3.7
multiply through by 10
m + 47 = 37
m = 37 - 47
m = -10
The equation is true.
c) m / 2 + 21 = 15
collect like terms
m / 2 = 15 - 21
m / 2 = -6
m = -12
This is false
d)
-49 + 3m = 72
3m = 72 + 49
3m = 121
m = 40.3
This is not true
Only m /10 + 4.7 = 3.7 is true for m = -10
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in the given figure find x if ABIICD
Answers
Answer:
i hope it will help uh....
Question 291 ptHow many real solutions does the quadratic equation below have?y = 2? + 5x + 100 1 real solutionNo real solutions2 real solutionsInfinite number of real solutionsNexDrevious
Answers
The quadratic equation is:
\(y=2x^2+5x+10\)To find if the number of solutions, we use the discriminant of the equation. But first, we compare the given equation with the general quadratic equation:
\(y=ax^2+bx+c\)By comparison, we find the values of a, b, and c:
\(\begin{gathered} a=2 \\ b=5 \\ c=10 \end{gathered}\)Now, as we said previously, we have to use the discriminant to find the number of solutions. The discriminant is defined as follows:
\(D=b^2-4ac\)• If the value of D results to be equal to 0, there will be 1 real solution.
• If the value of D results to be greater than 0, there will be 2 real solutions.
• And if the value of D results to be less than 0, there will be no real solutions.
We substitute a, b and c into the discriminant formula:
\(D=5^2-4(2)(10)\)Solving the operations:
\(\begin{gathered} D=25-4(2)(10) \\ D=25-80 \\ D=-55 \end{gathered}\)As we can see, the value of D is less than 0 (D<0) which indicates that there will be no real solutions for this quadratic equation.
Answer: No real solutions
Rewrite the following in the form log(c)
log(6)−log(3)
Answers
Explanation:
The rule to use here is log(x)-log(y) = log(x/y)
So,
log(6)-log(3) = log(6/3) = log(2)
Answer:
log(2)
Step-by-step explanation:
As we have,
log(a)−log(b)=log(a/b)
Replace 'a' as 6 and 'b' as 3.
Then we get,
\(\hookrightarrow log(6)-log(3)\\\\\hookrightarrow log(\frac{6}{3} )\\\\\hookrightarrow log(2)\)
Pls help with algebra
Answers
Answer:
x=16
Step-by-step explanation:
Actually this is trig I believe.
cos60=8/x
x=8/cos60=16
I did this on a delta math the other day as well
write the equation of the line in fully simplified slope-intercept form
Answers
Answer:
y=-x+4
Step-by-step explanation:
First use slope formula using 2 points.
(0,3) and (1,2)
\(\frac{y2-y1}{x2-x1}\)
\(\frac{2-3}{1-0}=\frac{-1}{1} =-1\)
Slope is -1.
Now, use point-slope formula.
y-y1=m(x-x1)
Plug in the information that we have.
y-3=-1(x-0)
Use distributive property.
y-3=-x+1
Add 3 to both sides.
y=-x+4
It is now in slope-intercept form.
Hope this helps!
If not, I am sorry.
GEOLOGY Shan used a surveying tool to map a region of land for his science class. To determine the height of a vertical rock formation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters, and the angle was 36°. What is the height of the formation to the nearest meter? 36° 43 m
Answers
The formation's height is 31 meters to the closest meter.
What exactly is meant by height?In mathematics, height is defined as the vertical distance between an object's top and bottom. It is also referred to as 'altitude.' The term height in geometry refers to the measurement of an object along the y-axis in coordinate geometry.
Height, altitude, and elevation all refer to the vertical distance between the top and bottom of something or between a base and anything above it. Height is a standard bodily measurement that is commonly measured in feet (ft) + inches (in) in the United States and centimeters (cm) abroad. Because they are length measurements, the SI unit would be meters.
We can use the tangent function to calculate the formation's height, h.
tan ( 36°) = h/43
Here, we have to multiply both sides by 43 we get,
43tan (36°)= h
h=31m
Therefore, the height of the formation to the nearest meter is 31m.
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Frank reverses The older pairs show the height and ages of the same six tenants is age of function or high explain
Answers
Please do it with step by step explanation it will help me so much thanks!
Carolyn is searching online for tickets to a concert. Two weeks ago, the cost was $30. Now the cost is $39.
a. What is the percent increase?
b. What would be the percent increase if the ticket agent charges an additional $4.50 fee with the new ticket price?
Answers
Answer:
A) The percent increase is by 30%.
B) for the original price = 15%.
B2.0) for the existing price (39) = approximately 11.54%.
Step-by-step explanation:
A) First, we divide 30 by 100. The result is the amount that 1% would be(0.3). If you multiply it by 30(which will become 30%) it will be 9. And from 30-39 is 9.
B) First we divide 30 by 100. The result is the amount that 1% would be(0.39). then if you multiply it by 11.54(which will become 11.54%) it will be 4.50(approximately because it actually is 4.5006 and there is no amount that ends up in 4.50). But if you are asking what percentage from the original cost(30) it would be 15% of it(because 0.3 * 15 = 4.50).
The angle of elevation of a boy to the top of a tree is 30°. If the height of the tree is 25 feet, find the distance between the boy and the tree.
Answers
Answer:
43.3 ft
Explanation:
The diagram representing this problem is drawn and attached below:
The distance between the boy and the tree = x
Recall from trigonometrical ratios:
\(\tan \theta=\frac{\text{Opposite}}{\text{Adjacent}}\)Therefore:
\(\begin{gathered} \tan 30\degree=\frac{\text{2}5}{\text{x}} \\ \text{Cross multiply} \\ x\tan 30\degree=25 \\ x=\frac{25}{\tan 30\degree} \\ x=43.3\; ft \end{gathered}\)The distance between the boy and the tree is 43.3 ft (correct to 2 decimal places).
Chris and Alex are friends who have summer jobs as waiters. The distribution of Chris’s weekly tips is approximately normal with mean $225 and standard deviation $25. Alex works at a different restaurant; the distribution of his weekly tips is approximately normal with mean $240 and standard deviation $15. Assuming their weekly tips are independent, which is the probability that Chris will make more in tips than Alex in a randomly selected week? Show your work.
(A) 0.067
(B) 0.159
(C) 0.227
(D) 0.303
(E) 0.354
Answers
An amusement park is open from May through September. The table shows the attendance each month as a portion of the total attendance. How many times more guests visit the amusement park in the busiest month than in the least busy month?
Month May June July August September
Portion of Guests 0. 09 $\frac{8}{25}$
22% 0. 29 $\frac{2}{25}$
Answers
The busiest month has approximately 2.44 times more guests visiting the amusement park compared to the least busy month.
To determine how many times more guests visit the amusement park in the busiest month compared to the least busy month, we need to compare the portions of guests in each month. The portions of guests are as follows:
May: 0.09
June: 8/25
July: 22%
August: 0.29
September: 2/25
To find the ratio between the busiest and least busy months, we divide the portion of guests in the busiest month by the portion in the least busy month.
The busiest month is July with a portion of 22%.
The least busy month is May with a portion of 0.09.
Ratio = (Portion of Guests in Busiest Month) / (Portion of Guests in Least Busy Month)
Ratio = 22% / 0.09
Converting 22% to a decimal: 22% = 0.22
Ratio = 0.22 / 0.09
Ratio ≈ 2.44
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What’s 45bc +abc+ 67(7-9)?
Answers
Answer:
A=67.6 ° L B=45.5 ° and AB=7.6 cm 7 i think thats your answer
Step-by-step explanation:
Answer:
hmmmmm ummmm 125bc
Step-by-step explanation:
WILL GIVE BRAINLIEST!
Consider the following.
a number increased by the total of the number and five
Answers
Answer:
n + (n+5)
Step-by-step explanation:
A number (N) increased (+) by the total of the number = (N+) and five (5)
Which graph has the greater slope?
Graph one
Or
Graph two
Answers
Answer:
Graph 1
Step-by-step explanation:
Graph one has a slope of 3/1 or 3, and Graph 2 has a slope of 1/3
Answer the following question by writing your response as a ratio in the form of a:b
Answers
ANSWER
\(2\colon3\)EXPLANATION
We want to write the ratio of hits to innings.
This means that we want to write the numbers in the form:
\(a\colon b\)There were 6 hits in 9 innings. Hence, the ratio of hits to innings (in simplest form) is:
\(\begin{gathered} 6\colon9 \\ \Rightarrow2\colon3 \end{gathered}\)That is the answer.
you can also add decimals by writing the problem vertically, lining up the decimal points to keep track of place values. The problem to the right it partially completed. Explain why there is a 1 above the tenths place
Answers
There is a 1 above the tenths place due to regrouping.
What is regrouping?Mathematicians use the addition with regrouping method to combine two or more integers of any size. It is used in conjunction with the column addition method, in which integers are added one column at a time and sums are arranged vertically.
Regrouping may also be referred to as "carrying over." The number gets carried over to the following place value column if you add up all the numbers in a column and the result is 10 or more. For instance, you would arrive to the sum of thirteen if the ones column had both a four and a nine.
The given expression is:
14.52 + 22.29 = 36.81
The expression on the right has 1 above the tenths place because of regrouping, This, is applied when the sum of the column is ten or more and the number is carried over.
Hence, there is a 1 above the tenths place due to regrouping.
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Solve for X
Y=5
3X+Y=2
Answers
Answer:
X=3/3=1
Hope it helps.
Have a good day Ahead
Can’t forget the negative!!
Calculate the Mean Absolute Percent Error (MAPE) from the following dataset for 6 days: Demand: 16, 19, 21, 21, 22, 17 and Forecast: 20, 20, 20, 20, 20, 20 respectively for each day.
a.
7%
b.
9%
c.
11%
d.
13%
)e.
15%
Answers
c. 11%. This means that the average forecasted demand deviates from the actual demand by approximately 11%.
The Mean Absolute Percent Error (MAPE) measures the accuracy of a forecast by calculating the percentage difference between the actual demand and the forecasted demand. To calculate the MAPE, you can follow these steps:
1. Calculate the absolute percent error (APE) for each day by subtracting the forecasted demand from the actual demand, taking the absolute value of the difference, and dividing it by the actual demand. Here are the APE values for each day:
Day 1: |(16-20)/16| = 20%
Day 2: |(19-20)/19| = 5.26%
Day 3: |(21-20)/21| = 4.76%
Day 4: |(21-20)/21| = 4.76%
Day 5: |(22-20)/22| = 9.09%
Day 6: |(17-20)/17| = 17.65%
2. Calculate the mean of the APE values by summing them up and dividing by the total number of days:
(20% + 5.26% + 4.76% + 4.76% + 9.09% + 17.65%)/6 ≈ 10.98%
3. Finally, multiply the mean APE by 100 to convert it into a percentage. This gives us the MAPE:
10.98% * 100 = 10.98%
Therefore, the MAPE for the given dataset is approximately 10.98%.
c. 11%. This means that the average forecasted demand deviates from the actual demand by approximately 11%.
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