Can someone give me the answer to the following 4 trig questions using sum and difference formulas
Answers
1. Therefore, the exact value of cos(105°) is (√2 - √6)/4.
2. Therefore, the exact value of sin(75°) is (√6 + √2)/4.
3. Therefore, the exact value of tan(105°) is (√3 + 1)/(3 - √3).
4. Therefore, the exact value of cos(105°)cos(15°) is (2 + 2√3)/8.
What is trigonometry?A branch of mathematics known as trigonometry is concerned with the study of triangles and the correlations between their sides and angles.
It entails computing different triangle measurements and attributes using trigonometric functions including sine, cosine, and tangent.
Trigonometry is an essential component of calculus and other advanced mathematical studies, and it has applications in many disciplines including engineering, physics, navigation, astronomy, and architecture.
1. We can use the formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B) with A = 60° and B = 45°. Then:
cos(105°) = cos(60° + 45°)
= cos(60°)cos(45°) - sin(60°)sin(45°)
= (1/2)(√2/2) - (√3/2)(√2/2)
= (√2 - √6)/4
Therefore, the exact value of cos(105°) is (√2 - √6)/4.
2. We can use the formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B) with A = 45° and B = 30°. Then:
sin(75°) = sin(45° + 30°)
= sin(45°)cos(30°) + cos(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4
Therefore, the exact value of sin(75°) is (√6 + √2)/4.
3. We can use the formula tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)) with A = 60° and B = 45°. Then:
tan(105°) = tan(60° + 45°)
= (tan(60°) + tan(45°))/(1 - tan(60°)tan(45°))
= (√3 + 1)/(3 - √3)
Therefore, the exact value of tan(105°) is (√3 + 1)/(3 - √3).
4. We can use the formula cos(A + B) = cos(A)cos(B) - sin(A)sin(B) with A = 60° and B = -45°. Then:
cos(105°)cos(15°) = cos(60° - 45°)cos(15°)
= [cos(60°)cos(-45°) - sin(60°)sin(-45°)]cos(15°)
= [(1/2)(√2/2) - (√3/2)(-√2/2)]cos(15°)
= (√2 + √6)/4 * cos(15°)
= (√2 + √6)/4 * (√6 + √2)/4
= (2 + 2√3)/8
Therefore, the exact value of cos(105°)cos(15°) is (2 + 2√3)/8.
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Related Questions
(6-x):(7-x)::(18-x):(22-x)
Answers
Using the property of proportionality for the equation (6-x) : (7-x) :: (18-x) : (22-x), then, x = ⁶/₇.
How did we get the value of x?Solving this problem by using the property of proportionality, which states that the ratio of the first two terms in a proportion is equal to the ratio of the last two terms. In this case:
(6-x) : (7-x) :: (18-x) : (22-x)
Solving for x, cross-multiply the ratios:
(6-x) x (22-x) = (7-x) x (18-x)
Expanding both sides:
132 - 28x + x² = 126 - 25x + x²
Simplifying, we get:
7x = 6
Therefore, x = ⁶/₇.
Thus, the solution to the proportion is:
(6-x) : (7-x) :: (18-x) : (22-x)
or
⁶/₇ : ¹/₇ :: ¹²/₇ : ¹⁶/₇
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Amanda’s new job has a 50% employer match on the first 4% of her salary contributed to her 401(k). How much was deposited in Amanda’s account if she makes $61,000 a year and she took advantage of the entire contribution match?
Answers
Answer:
$2,440 + $1,220 = $3,660
Step-by-step explanation:
If Amanda makes $61,000 a year, and she contributes 4% of her salary to her 401(k), that would be:
$61,000 x 0.04 = $2,440
Since her employer has a 50% match on the first 4% of her salary, they would contribute:
$2,440 x 0.5 = $1,220
What does |23| mean?
Answers
|23| = 23
|-23| = 23
Your parents are purchasing a mobile home for $89,000. The sales tax is 4.2%, they make a $3,000 down payment, and they have an average credit score. How much is the PRINCIPAL BALANCE after applying their first month’s payment of $925.67?
Secured Unsecured
Credit APR (%) APR (%)
Excellent 4.75 5.50
Good 5.00 5.90
Average 5.85 6.75
Fair 6.40 7.25
Poor 7.50 8.40
A. $89,249.80
B. $88,245.05
C. $88,912.03
D. $89,021.98
Answers
The initial loan amount is the purchase price of the mobile home minus the down payment and the sales tax.
Initial loan amount = $89,000 - $3,000 - (4.2% * $89,000) = $81,738
Next, we need to determine the interest rate based on your parents' credit score. Since the problem states that they have an "average" credit score, we will use the average interest rates for secured and unsecured credit with an APR of 5.85% and 6.75%, respectively. Since this is a mobile home purchase, we will assume that the loan is secured and use an interest rate of 5.85%.
Using these values, we can calculate the principal balance after the first month's payment using the formula for the present value of an annuity:
PV = C * [(1 - (1 + r)^(-n)) / r]
where PV is the present value, C is the monthly payment, r is the monthly interest rate (5.85% / 12 = 0.4875%), and n is the number of months (30 years or 360 months).
Substituting the given values, we get:
PV = $925.67 * [(1 - (1 + 0.004875)^(-360)) / 0.004875]
PV = $81,276.05
Therefore, the principal balance after applying the first month's payment is $81,276.05.
Answer: B. $88,245.05 is not correct, the correct answer is $81,276.05
You are making scarves for presents. Each scarf needs 5/6 yd of fabric. How many yards of fabric do you need for 6 scarves?
Answers
There are 8 blue marbles, 6 red marbles, 2 green marbles and 4 black marbles in a box. What is the probability that the marble is not green?
A) 92
B) 10
C) 85
D)20
Answers
We have 20 marbles in the bag. The probability of a marble that is not green is 20/20.
So the probability of the randomly that the marble is not green is 20/20 or just 20.
The entire graph of the function g is shown in the figure below.
Write the domain and range of g as intervals or unions of intervals.
domain=
range =
Answers
For the given function 'g':
domain = (-4, -2) ∪ (1, 3]
range = (-5, 4]
If the function is defined as f(x) = y, then the values of x for which the value of function exists, the set of that values is said to be the domain of the function f.
If the function is defined as f(x) = y, then the set of all possible values of the function that is the set of all possible values of y is said to be the range of the function f.
Here given the graph of 'g'.
From the graph we can see that for -4 < x < -2 and 1 < x [tex]\le[/tex] 3 the function has graph.
So the domain = (-4, -2) ∪ (1, 3]
And it is clear that the value of y lies between -5 < y [tex]\le[/tex] 4.
So the range = (-5, 4].
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The expression 5(2) gives the number of leaves on a plant as a function
of the number of weeks since it was planted.
What does 2 represent in this expression?
Choose 1 answer:
B
The plant was planted 2 weeks ago.
There were initially 2 leaves on the plant.
The number of leaves is multiplied by 2 each week.
Answers
C: the number of leaves is multiplied by 2 each week
Step-by-step explanation
The proper expression ought to be 5(2) ^ t =5(2)^{t}
where t speak to the number weeks after it planted.
On the off chance that we put to the time(t) we are going know how numerous clears out at first are. The calculation will be:
5(2)^{t}= 5(2)^{0} =5
This appear that coefficient 5 decide how much the takes off at first are.
Let see how the number going for distinctive esteem of t. On the off chance that its 1 week after plant, the leaf will ended up:
5(2)^{t}= 5(2)^{1} = 10
At that point the number of takes off 2 weeks after planted will be:
5(2)^{t}= 5(2)^{2}= 20
The number keeps twofold each week since coefficient 2 speaks to the proportion of the development of the work and the proportion decides how much takes off duplicated each week.
So, the reply will be C.
A quality expert can test 18 units in 32 minutes. If there are 400 units to be tested, about how long will it take to test them?
Answers
It will take about 711.12 minutes or approximately 11.85 hours to test all 400 units.
To find out how long it will take to test 400 units, we need to use the information given in the problem. We know that the quality expert can test 18 units in 32 minutes. We can use this information to find out how long it will take to test one unit.
First, we need to find out how many units can be tested in one minute:
18 units / 32 minutes = 0.5625 units per minute
This means that the quality expert can test 0.5625 units per minute. To find out how long it will take to test one unit, we can divide 1 by 0.5625:
1 / 0.5625 = 1.7778 minutes per unit
So it takes about 1.7778 minutes to test one unit. To find out how long it will take to test 400 units, we can multiply this value by 400:
1.7778 minutes per unit * 400 units = 711.12 minutes
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what is the mode of 14, 17, 21, 28, 40
Answers
Answer:
24
Step-by-step explanation:
14 + 17 + 21 + 28 +40 / 5
120 / 5
24
A survey asked students whether they have any siblings and pets. The survey
data are shown in the relative frequency table.
Pets
No pets
Total
Siblings
0.3
0.45
0.75
OA. 30%
OB. About 67%
O C. 75%
D. 40%
Given that a student has a sibling, what is the likelihood that he or she also
has a pet?
Answers
The likelihood that he or she has a pet given that a student does not have a sibling is 60%. Therefore, the correct option is option D.
Let A denote the event that a student does not have a sibling.
Let B denote the event that a student has a pet.
Then A∩B will denote the event that the student does not have a sibling but he has a pet.
Let P denote the probability of an event in this question.
Then, we have to find the conditional probability of event B which means the likelihood of event B occurring based on the occurrence of event A.
P(B|A)
We know that:
[tex]P(B|A) = \frac{P(A\cap B)}{P(A)}[/tex]
Also from the table, we have:
P(A)=0.25
and P(A∩B)=0.15
Hence,
[tex]P(B|A) = \frac{0.15}{0.25}[/tex]
[tex]P(B|A) = \frac{3}{5}[/tex]
[tex]P(B|A) = 0.6[/tex]
which in percentage is:
0.6 * 100 = 60 %
Therefore, the likelihood that he or she has a pet too is 60% and the correct option is option D.
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Suppose that the function h is defined as follows.
Answers
For the given function h(x) the points can be plotted on,
(-1,-1)(0,0)(1,1)(2,2)The given function h(x) is having four conditions and each condition can be clearly discussed below:
Given that for the first condition, the x-axis is ranging from -1 [tex]\leq[/tex] x < 0, the y-axis point is present exactly at -1. Similarly, for the second condition, the x-axis is ranging from 0 [tex]\leq[/tex] x < 1, and the y-axis point is present at 0. For the third condition, the x-axis is ranging from 1 [tex]\leq[/tex] x < 2, and the y-axis point is at 1.Finally, for the fourth condition, the x-axis is ranging from 2 [tex]\leq[/tex] x < 3, and the y-axis point is at 2.From, the above analysis we plotted the graph and attaching below.
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The numbers of polio cases in the world are shown in the table for various years.
Year Number of polio cases (thousands)
1988 350
1992 138
1996 33
2000 4
2005 3.2
2007 1.3
PREDICT THE NUMBER OF POLIO CASES IN 2014
hint: The function F gives the number of thousands of polio cases
Answers
The value of t is given as 2.3 years.
How to solvewe get the table for function as
t f(t)
8 350
12 138
16 36
20 4
25 3.2
27 1.3
So it is better to model the data by using an exponential model
using a graphing calculator we get our model as
f(t)=a(b)^t
[tex]f(t)=\boldsymbol{3648.6915(0.7380)^t}[/tex]
we have an exponential decay, so we get a rate of decrease as
b=1-r
r=1-b=1-0.738=0.262=26.2\\%
So the number of polio cases is decreasing by 26.2% per year
The number of polio cases in 2014 is
t=2014-1980=34
f(34)=3648.6915(0.7380)^{34}=0.1991389
this is in thousands so we get the number of cases in 2014 approximately as
[tex]0.1191389*1000 \approx \boldsymbol{119} \textup{ cases}[/tex]
for 1 case of polio :
f(t)=0.001
3648.6915(0.738)^t=0.001
we get
[tex]t=\frac{\ln (\frac{0.001}{3648.6915})}{\ln(0.738)}\approx 50[/tex]
so we get year as
1980+50={2030},
to find half life
[tex]\frac{a}{2}=a(b)^t[/tex]
[tex]b^t=\frac{1}{2}[/tex]
[tex]0.738^t=\frac{1}{2}[/tex]
[tex]t=\frac{\ln (1/2)}{\ln (0.738)}\approx\boldsymbol{2.3} \textup{ years}[/tex]
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What is the domain of g(x)=log(7x−10)?
Answers
The domain of g(x) = log (7x - 10) is all real numbers greater than 10/7.
Set-builder notation: x > 10/7. Interval notation: (10/7,∞)
What is the domain of the function?Given the function in the question:
g(x) = log( 7x - 10 )
For the function g(x) = log (7x - 10), the domain consists of all the possible values of x that can be plugged into the function without resulting in an undefined expression.
Here, the logarithm (in this case, (7x - 10) must be greater than zero, since the logarithm of zero or a negative number is undefined in the real numbers.
So we set the argument of the logarithm to be greater than zero, and solve for x:
7x - 10 > 0
Solve for x
7x - 10 + 10 > 0 + 10
7x > 10
Dividing both sides by 7:
7x/7 > 10/7
x > 10/7
Therefore, the domain of g(x) = log (7x - 10) is x > 10/7
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Match the following:
1. Counts as Demand
2. Does Not Count as Demand
3. Demand Increases
4. Demand Decreases
What happens to the demand for
goods and services if the population in
town decreases?
Sue buys a new computer.
Adam likes the new cell phone, but
cannot afford it.
What happens to the demand for
jeans if the price of a popular brand of
jeans decreases?
Answers
2. Does Not Count as Demand - Sue buying a new computer does not affect the overall demand for computers, it only affects the demand for that specific brand or model of computer.
3. Demand Increases - if the price of a popular brand of jeans decreases, the demand for jeans will increase as more people will be willing and able to buy them.
4. Does Not Count as Demand - Adam liking a new cell phone but not being able to afford it does not affect the overall demand for cell phones, as he is not able to purchase one.
Someone help I have no idea how to do this, it would help if I could get a step by step
Answers
Answer: x=3
Step-by-step explanation:
shaded part is 5/6 of the full circle, so just make an equation.
5/6 pi r^2(formula to find area)=15/2 pi(area given)
cancel out pi and simplify
r^2=9
r=+-3 but cant be negative so r=3
please help for this question
Answers
The dilation of the object is given by the coordinates:
(-6,-8)
(-2, 0)
(-2, -8)
A dilation is a function f from a metric space M into itself that fulfills the identity d=rd for all locations x, y in M, where d is the distance between x and y and r is some positive real integer.
For the first dilation
Original points are
(2,1)
(4,5)
(4,1)
Multiply by scale factor
(2,1) x 2 = (4,2)
(4,5) x 2 = (8, 10)
(4,1) x 2 = (8, 2)
This given us the coordinates of triangle B).
For the second dilation
(2,1)
(4,5)
(4,1)
Adjust for the center of dilation which is (5,5)
(2,1) less (5,5) = (-3, -4)
(4,5) less (5,5) = (-1, 0)
(4,1) less (5,5) =(-1, -4)
Multiply the New original point by scale factor
(-3, -4) x 2 = (-6,-8)
(-1, 0) x 2 = (-2, 0)
(-1, -4) x 2 = (-2, -8)
Thus, the new coordinates of the dilated triangle C are:
(-6,-8)
(-2, 0)
(-2, -8).
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Mrs Smith uses 5 lemons to make 2/3 gallon lemonade. How many gallons of lemonade can Mrs Smith make with 15 lemons?
Answers
5 lemons / (2/3 gallon) = 15 lemons / x gallons
To solve for x, we can cross-multiply:
5 lemons * x gallons = (2/3 gallon) * 15 lemons
5x = 10
x = 2
Therefore, Mrs. Smith can make 2 gallons of lemonade with 15 lemons.
Explanation: Since Mrs. Smith uses 5 lemons for 2/3 of a gallon, you can multiply 2/3 by 3. (You will end up with the product 2. This then translates into 2 gallons.) Hope that helped!
PLEASE SOMEONE HELP ME THIS IS SO HARD. ILL GIVE 60 POINTS TO WHOEVER HELPS ME. PLEASE I GIVE BRAINLIEST
Answers
Answer:
B
Step-by-step explanation:
Vertex form is: y=a(x-h)²+k
Vertex would be (h,k)
Based on this, the only equation in vertex form is B
Answer:
Step-by-step explanation:
B is the easiest way to see vertex because that is the vertex form:
y=a(x-h)²+k where (h,k) is the vertex
Vertex (2, 27)
Two coins are tossed. Assume that each event is equally likely to occur.
a) Use the counting principle to determine the number of sample points in the sample space.
b) Construct a tree diagram and list the sample space.
c) Determine the probability that no tails are tossed.
d) Determine the probability that exactly one tail is tossed.
e) Determine the probability that two tails are tossed.
f) Determine the probability that at least one tail is tossed.
Question content area bottom
Part 1
a) There is/are --------- sample point(s) in the sample space.
Answers
a. There are 4 sample points in the sample space.
What are the probability values for the toss?a) There are 4 sample points in the sample space.
b) The tree diagram and sample space are:
HH (two heads)
HT (one head, one tail)
TH (one head, one tail)
TT (two tails)
c) The probability that no tails are tossed (two heads) is 1/4 or 0.25.
d) The probability that exactly one tail is tossed (HT or TH) is 1/2 or 0.5.
e) The probability that two tails are tossed is 1/4 or 0.25.
f) The probability that at least one tail is tossed (HT, TH, or TT) is 3/4 or 0.75.
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A boy has 30 spellings to learn.
12 words included an 5, 16 a T and 18 an E.
10 words had an E and a T, 6 words had an Sand aT and 5 words had an E and an S.
4 words had all three.
Probability that?
A • the first two spellings in the test both included only one of the three letters?
C• the first two spellings in the test both had an E and at least one other letter
Answers
Answer:
A
Step-by-step explanation:
Someone help i really need help with this
Answers
The equation which can be used to find the amount the admission tickets cost would be 4x + 25 = 175. The cost of the admission tickets would therefore be $ 37.50.
How to find the equation ?The total spent by Ms. Boi was $ 175.00 so this is what the entire equation would be equal to. The cost of admission tickets will be denoted as x because they are the unknown figure.
There are 4 tickets purchased so the equation would be:
4 x Cost of tickets - Parking cost = Total spent
4 x - 25 = 175
The cost of each admission ticket is:
4 x - 25 = 175
x = 150 / 4
x = $ 37.50
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Given: mzA+m/B=m/B + mzC
Prove: m₂C=m/A
Write a paragraph proof to prove the statement.
Answers
We have proven that m∠C = m∠A.
We have,
We are given that m∠A + m∠B = m∠B + m∠C.
We want to prove that m∠C = m∠A.
Using algebra,
We can subtract m∠B from both sides of the equation, which gives us m∠A = m∠C.
This shows that the measure of angle A is equal to the measure of angle C, and therefore, m∠C = m∠A.
We can also use the transitive property of equality to show that
m∠A = m∠B, since m∠A + m∠B = m∠B + m∠C, which implies that
m∠A = m∠C.
Therefore,
We have proven that m∠C = m∠A.
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McKenzie painted a yellow rectangle that measured 8 1/2 in. wide and 11 in. long for part of a mural she was working on. She also painted a
purple rectangle that was 1/4 the size of the yellow rectangle. What is the area of the purple rectangle that McKenzie painted?
Answers
Answer:
23.375 in^2
Step-by-step explanation:
To find the area of a rectangle or square, you need to multiple the length by the width.
The length: 8.5
Width: 11
8.5*11 = 93.5
Now we divide by 4
93.5/4
23.375
The area of a rectangular shaped rug is 81 square feet. If the rug is 9 ft long, what is ire perimeter?
Answers
The perimeter of the rug is 36ft
What is perimeter?Perimeter is the total length around the outside of a shape. The perimeter can be found by adding all the sides of the shape.
Perimeter of a rectangle is expressed as ;
P = 2(l+w)
The area of the rug is 81ft²
area = l× w
length = 9ft
width of the rug = 81/9 = 9ft
Therefore the perimeter of the rug will be
P = 2(l+w)
P= 2( 9+ 9)
P = 2 × 18
P = 36 ft
therefore the perimeter of the rectangular rug is 36ft
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e) Write a proof to show AABC ~ ACFD.
f) What is the longest cart that can pass through the second doorway?
Explain.
Some of the factory's products are long fragile rods that are carried through
the door by hand. The first door is 72 inches west of the start of the second
door. Assume the rod has zero width and BA = 13 in.
Answers
Answer:
Step-by-step explanation:
no
Solve the following system of equations.
Answers
The solutions for the system of equations " 3x²-2y²+5=0 and 2x²-y²+2=0" are :
First Solution : x = 1, y = 2
Second Solution : x = 1, y = -2
Third Solution : x = -1, y = 2
Fourth Solution : x = -1 , y = 2.
In order to solve the system of equations 3x²-2y²+5=0 and 2x²-y²+2=0, we use substitution method;
From the second equation, we solve for y² in terms of x²:
The second equation is : 2x² - y² + 2 = 0;
We get, y² = 2x² + 2
Substitute this expression for y² into the first equation:
We get,
3x² - 2(2x² + 2) + 5 = 0
3x² - 4x² - 4 + 5 = 0
-x² + 1 = 0
x² = 1
We get, x = ±1
Substitute these values of x into the equation for y²:
y² = 2x² + 2
For x = 1, y² = 4,
Which means the solution set is (1,2) and (1,-2),
For x = -1, y² = 4
the solution set is (-1,2) and (-1,-2).
Therefore, the solutions to the system of equations are (1, 2), (1,-2), (-1,2) and (-1, 2).
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Work out the largest integer value that x could take if
x+7<13
Answers
Answer:
The answer is going to 5
Step-by-step explanation:
I hope this helps you
Someone help i really need help with this
Answers
Step-by-step explanation:
step A is incorrect.
-40/5 does not equal to -40/-5.
-40/-5 is equal to 40/5 as we can simplify the negative 1 out.
Answer:
The correct answer is step A
Mrs Marcus combines her books and magazines then divides them among 10 different tables. If a represents the number of books and d represents the number of magazines Mrs. marcus has, write an expression to show how many books would be on each table.
Answers
An expression to show how many books would be on each table is a/10
To find out how many books would be on each table, we need to divide the total number of books by the number of tables.
First, let's express the total number of books in terms of a and d. Since we are not given any specific numbers for a and d, we can simply use variables to represent them. Therefore, the total number of books is given by a.
Next, we need to divide a by 10, the number of tables. This gives us the expression:
a/10
So, the above expression shows how many books would be on each table if Mrs. Marcus combines her books and magazines and then divides them among 10 different tables.
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Complete question is:
Mrs. Marcus combines her atlases and dictionaries and then divides them among 10 different tables. If a represents the number of atlases and d represents the number of dictionaries Mrs. Marcus has, write an expression to show how many books would be on each table.