Use an identity to write the expression as a single trigonometric function or as a single number.
Answers
We can use the following trigonometric identity to solve the exercise:
[tex]\frac{2\tan\theta}{1-\tan^2\theta}=\tan 2\theta[/tex]In this case, θ = 15°. Then, we have:
[tex]\begin{gathered} \frac{2\tan15\degree}{1-\tan^215\degree}=\tan 2(15\degree) \\ \frac{2\tan15\degree}{1-\tan^215\degree}=\tan 30\degree \end{gathered}[/tex]We can use a notable angles table to find the value of tan 30°. Finally, we have:
[tex]\frac{2\tan15\degree}{1-\tan^215\degree}=\frac{\sqrt[]{3}}{3}[/tex]Related Questions
11. Given the information, write the appropriate equation for: A vertical line going through the point (-3,-1). Equation:
Answers
A vertical line has the form:
x = constant
Given that the point (-3, -1) must be included, then the equation is:
x = -3
2. Calculate the rate of change for the following linear functions.
Answers
Formula to find the slope, m, or change in function of a graph is given below as,
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]For the first linear graph,
[tex]\begin{gathered} \text{Where x = 0, y = 0 }_{} \\ (x_1,y_1)=(0,0) \\ \text{Where }x\text{ = -2, y = 2} \\ (x_2,y_2)=(-2,2) \end{gathered}[/tex]Substituting the coordinates into the equation,
[tex]m=\frac{2-0}{-2-0}=\frac{2}{-2}=-1[/tex]For the second linear graph,
[tex]\begin{gathered} \text{Where x = 0, y = 2} \\ (x_1,y_1)=(0,2) \\ \text{Where x = 2, y = 2} \\ (x_2,y_2)=(2,2) \end{gathered}[/tex]Substituting the coordinates into the equation,
[tex]m=\frac{2-2}{2-0}=\frac{0}{2}=0[/tex]For the third linear graph,
[tex]\begin{gathered} \text{Where x = -4, y = 0} \\ (x_1,y_1)=(-4,0) \\ \text{Where x = -4, y = 2} \\ (x_{2,}y_2)=(-4,2) \end{gathered}[/tex]Substituting the coordinates into the equation,
[tex]m=\frac{2-0}{-4-(-4)}=\frac{2}{-4+4}=\frac{2}{0}=\infty[/tex]Hence, the slope of the first graph is -1.
Slope of the second graph is 0 since it doesn't rise vertically.
Slope of the third graph is infinite because it is a vertical line that neither move to the left or right.
Find the total surface area of the rectangular prismA) 72cm squaredB) 48cm squaredC) 24cm squaredD) 96cm squared
Answers
Given data:
The given figure of the rectangular prism.
The expression for the area of the prism is,
[tex]\begin{gathered} A=(2\text{ cm+3 cm+2 cm+3 cm)}(6\text{ cm)+2}(3\text{ cm)(2 cm)} \\ =72cm^2 \end{gathered}[/tex]Thus, option (A) is correct.
please help with question #10. the teacher said to solve it using systems of equations. and we are required to show all of our work. I hope you can figure this out, thanks!!
Answers
ANSWER
x = 20 and y = 25
EXPLANATION
We are given three angles
We have that:
From the figure, we see that
This means that they are equal to one another.
That is:
=> 3x + 2 = 4x - 18
Collect like terms:
4x - 3x = 2 + 18
x = 20
We also see that
Adjacent angles of a parallelogram add up to 180 degrees.
This means that:
5y - 7 + 4x - 18 = 180
Recall that x = 20:
5y - 7 + 4(20) - 18 = 180
5y - 7 + 80 - 18 = 180
5y + 55 = 180
5y = 180 - 55 = 125
=> y = 125 / 5
y = 25
Therefore, x = 20 and y = 25.
For which of the following geometric series can the infinite sum be determined? a1 = 9, r = –0.3 a1 = 5, r = –3 a1 = 0.4, r = 2 a1 = –0.4, r = –6
Answers
A geometric sequence is given by the general form:
[tex]a_n=a\cdot r^{n-1}[/tex]Where a is the first term of the sequence and r is known as the common ratio. If the common ratio is smaller than one then the sum of all the elements is equal to:
[tex]\sum ^{\infty}_{n\mathop=1}a\cdot r^{n-1}^{}=\frac{a}{1-r}[/tex]So as I stated before this expression can only be used if r<1. From the four options given by the exercise the only one with r<1 us the first one. Then the answer is the first option.
f(x)=(3x-1)(2x+3)Find the zero property
Answers
Given: f(x)=(3x-1)(2x+3)
Find: zero perperty of the given function
Explanation: zero propert allow us to factor equation and solve them .
here we have already a factor is given, we will put them equal to 0 for to find the value of x.
(3x-1)(2x+3)=0
3x-1=0 and 2x+3=0
on solving we get,
[tex]x=\frac{1}{3},-\frac{3}{2}[/tex]Final answer: the value of x is
[tex]\frac{1}{3},-\frac{3}{2}[/tex]Let 100,92, 84, 76, 68, ... be a sequence. Which of the following are true about thissequence? (Hint: there is definitely more than one answer)a = -Sn + 108100(-8)-1The common difference is -8The common ratinis
Answers
The arithmetic sequence is given by:
[tex]a_n=-8n+108_{}[/tex]Where the common difference is -8.
Therefore, the correct choices are:
[tex]a_n=-8n+108[/tex]and the common difference is -8
if there are 11 books on a shelf 4 are new the rest are old what is the ratio of the new books to the old books.and what is the ratio of the old books to the new ones.
Answers
Recall that the ratio of two quantities x and y is the fraction
[tex]\frac{x}{y},[/tex]or x:y, which indicates how many times y is the quantity x.
Now, if there are 4 new books, then there are 11-4=7 old books.
Therefore, the ratio of new books to the old books is:
[tex]\frac{4}{7}\text{.}[/tex]The ratio of old books to new books is:
[tex]\frac{7}{4}\text{.}[/tex]Answer:
New to old: 4/7.
Old to new: 7/4.
5. In this figure, triangle GHJ is similar to triangle PQR P Q 00 R G Based on this information, which ratio represents tan G? Sin G? Cos G?
Answers
In the given problem,
[tex]\begin{gathered} \Delta GHJ\approx\Delta PQR \\ \frac{GH}{PQ}=\frac{HJ}{QR}=\frac{GJ}{PR} \\ \angle G=\angle P \\ \angle H=\angle Q \\ \angle J=\angle R \end{gathered}[/tex]Thus value of tanG, sinG and cosG can be determined as,
[tex]\begin{gathered} \tan G=\tan P=\frac{QR}{PR}=\frac{15}{8} \\ \sin G=\sin P=\frac{QR}{QP}=\frac{15}{17} \\ \cos G=\cos P=\frac{PR}{QP}=\frac{8}{17} \end{gathered}[/tex]Thus, the above expression gives the requried value of tanG, sinG and cosG.
[tex]x + y = 4 \: and \: 2x + y = 6[/tex]Sketch and estimate the solution.
Answers
Do you have a picture of your question?
It's because I am not sure about the equations
ok
x + y = 4 2x + y = 6
x y x y
-2 6 -2 10
0 4 0 6
2 2 2 2
Graph the points
The solution is where both lines intercept (2, 2)
What is the balance if you invest $2000 at 3.5% for 4 years? Help please:(
Answers
I = P x R x T / 100
WhereP = Principal
R= rate and T = time
$2000 was invested which is the principal
R = 3.5%, t = 4 years
I = 2000 x 3.5 x 4 / 100
I = 28, 000 / 100
I = $280
The interest after 4 years is $280
The balance inside the account is
B = I + P
Balance = 2000 + 280
Balance = $2280
The answer is $2,280
if mice are allowed to reproduce without any restrictions of population where shares a special growth which table shows an example that might models in mice population over time
Answers
When reproducing without any restrictions, the population will grow exponentially.
The table that represents an exponential growth is the one in option 2.
Answer: Option 2.
Which description compares the domains of Function A and Function B correctly?
Answers
Given,
The expression of the function A is,
[tex]f(x)=\sqrt[\placeholder{⬚}]{(-x)}[/tex]The graph of the function B is,
Required
The domain of the function A and function B.
For the function A, if the value of x is positive real number or the number greater than 0, then the value of y will be imaginary number. Which can be neglected.
Hence, the domain of the function A is set of real numbers less than or equal to 0.
For function B, it is clearly seen from the graph that the function have the value of x as the set of all real number less than or equal to 0.
Hence, the domain of the both function is set of real numbers less than or equal to 0.
please help. there is also another picture that goes with this one
Answers
We see that A, B, C, and E are straight lines, but D is a parabola. The graph of a quadratic equation is a parabola, so que equation of D is y = 0.1x² (the only quadratic function of the set).
Among the straight lines, there is only one of them with a negative slope, and this is line E. The only equation that has a negative slope is y = 9 - 0.5x, so this is the equation of the line E.
Now, we see that C passes through the origin, so the y-intercept must be 0. The only linear equation that has a y-intercept equal to 0 is y = x, so this is the equation of the line C.
Additionally, B and C are parallel, so they must have the same slope. Since B has a slope of 1, the equation of B must be y = x + 2, which has a slope 1.
Finally, the equation of A is y = 2x + 2 (the only one remaining).
I need to solving 9-49x²=0
Answers
1) Let's start out by rewriting that into the standard form then solve for x this incomplete quadratic equation:
[tex]\begin{gathered} -49x^2+9=0 \\ -49x^2+9-9=0-9 \\ -49x^2=-9 \\ \frac{-49x^2}{-49}=\frac{-9}{-49} \\ x^2=\frac{9}{49} \\ x=\sqrt{\frac{9}{49}},\:x=-\sqrt{\frac{9}{49}} \\ x=\frac{3}{7},\:x=-\frac{3}{7} \end{gathered}[/tex]Note that there are two roots.
HelppppWhich choices are equivalent to the expression below (see the picture please)
Answers
Step 1
Given;
[tex](2^3)^4[/tex]Required; To find the choices that are equivalent to the expression.
Step 2
Find the options that are equivalent to the result of the expression given when simplified.
[tex]\begin{gathered} (2^3)^4=4096\text{ } \\ Opt\imaginaryI on\text{A}(\text{2})^{3(4)}=4096\text{ }\imaginaryI\text{s correct} \end{gathered}[/tex][tex]Option\text{ B 2}^{12}=4096\text{ is correct}[/tex]Answer; Option A and Option B
A construction crew is lengthening a road. Let I be the total length of the road (in miles). Let D be the number of days the crew has worked. Suppose that L = 3D + 200 gives L as a function of D. The crew can work for at most 60 days.Identify the correct description of the values in both the domain and range of the function. Then, for each, choose the most appropriate set of values.Onumber of days the crew has worked Olength of the road (in miles)?Onumber of days the crew has worked Olength of the road (in miles)?
Answers
The domain of a function is the set of all values for which the function is defined. In this case, we can see that the function depends of the variable D, which represents the number of days the crew has worked.
Since the crew can work for at most 60 days and the time can not be negative, the domain of the function is the set of all real numbers from 0 to 60.
On the other hand, the range of a function is the set of all values that the function takes. In this case, we can see that the dependent variable is L, which represents the length of the road. Then, we have:
[tex]\begin{gathered} L=3D+200 \\ \text{ If }D=0 \\ L=3(0)+200 \\ L=0+200 \\ L=200 \\ \\ \text{ If }D=60 \\ L=3\left(60\right)+200 \\ L=180+200 \\ L=380 \end{gathered}[/tex]Thus, the range of the given function is the set of all real numbers from 200 to 380.
Answer14.1 figur 13. Opposite sides of a parallelogram are parallel Prove that opposite angles of a parallelogram are congruent. Nam Glven: ABCD is a parallelogram Prove: ZA A ZC, 2BA 2D Statements Reasons 1. AB II CD 2. m2B+ m2cm 180 mzA+ m2D - 180 2 3. BC || AD 3. 4. m2A + m2B = 180 m2C + m2D - 180 4. M m21 5. MZA+ m2B = m2B + m2c 5. 6. 6. MzA - mac m22 m23 7. ZANZC 7. m24 8. m2B + m2 m2C+ m2D 8. 9. m2B = m2D 9. 10. 2B SZD 10
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Required: To prove that opposite angles of a parallelogram are congruent.
[tex]\text{Prove: }\angle A\cong\angle C,\text{ }\angle B\cong\angle D[/tex]1. Given ( opposite sides of a parallelogram are parallel)
2. Adjacent angles of a parallelogram are supplementary.
3. Given (opposite of a parallelogram are parallel)
4. Adjacent angles of a parallelogram are supplementary.
5. Congruency property
6. Transitive property
7. Congruency
8. Transitive property
9. Transitive property
10. Congruency
How many cans of corn does he buy?Enter the correct answer.Alex stocks up for winter. He buys 24cans of vegetables. He pays 80 cents percan for tomatoes and 40 cents per canfor corn, for a total cost of $12.8.OOBDONEClear all3
Answers
From the question, the two vegetables involved are tomatoes and corn
We have to generate some sets of simultaneous linear equations from the information given in the question.
Let the number of cans of tomatoes be represented by x
and the number of cans of corn be represented by y
Alex bought 24 cans of vegetables. This means;
x + y = 24 ..................................................... equation 1
He paid 80cents ($0.80) for 1 can of tomato.
This means for x cans of tomatoes, he would pay $0.80x
He paid 40cents ($0.40) for 1 can of corn.
This means for y cans of corn, he would pay $0.40x
Alex paid a total cost of $12.8. This means;
0.8x + 0.4y = 12.8 ........................................... equation 2
Hence, we have derived two equations out from the question and we solve them simultaneously
x + y = 24 ........................................................(equation 1)
0.8x + 0.4y = 12.8..........................................(equation 2)
Using substitution method, if x + y = 24
x = 24 - y
Substituting x = 24 - y in equation 2, we have;
0.8(24-y) + 0.4y = 12.8
19.2 - 0.8y + 0.4y = 12.8
Collecting like terms,
19.2 - 12.8 = 0.8y - 0.4y
6.4 = 0.4 y
[tex]\begin{gathered} y=\frac{6.4}{0.4} \\ y=16 \end{gathered}[/tex]Since x + y = 24
x + 16 = 24
x = 24 - 16
x = 8
Therefore, x = number of tomato can = 8
y = number of corn can = 16
Hence, the cans of corn he bought are 16cans.
Hello, does anyone know how to solve this? I keep getting it wrong on my homework, thanks!Find the first term of the sequence given by the following~n= 1, 2, 3...an = 5(3)^n-1 I attached an image to this question.
Answers
Given the nth term of the required sequence expressed according to the equation:
[tex]a_n=5(3)^{n-1}[/tex]You need to get the first four terms of the sequence as shown:
For the first term, when n = 1
[tex]\begin{gathered} a_1=5(3)^{1-1} \\ a_1=5(3)^0 \\ a_1=5(1) \\ a_1=5 \end{gathered}[/tex]For the second term, when n = 2
[tex]\begin{gathered} a_2=5(3)^{2-1} \\ a_2=5(3)^1 \\ a_2=5(3)_{} \\ a_2=15 \end{gathered}[/tex]For the third term, when n = 3
[tex]\begin{gathered} a_3=5(3)^{3-1} \\ a_3=5(3)^2 \\ a_3=5(9) \\ a_3=45 \end{gathered}[/tex]For the fourth term, when n = 4
[tex]\begin{gathered} a_4=5(3)^{4-1} \\ a_4=5(3)^3 \\ a_4=5(27) \\ a_4=135 \end{gathered}[/tex]Therefore the first four terms of the sequence will be 5, 15, 45 and 135
Matthew thought he could make 19free throws, but he only made 13.What was his percent error?
Answers
EXPLANATION
Since he only made 13 from 19 throws, the percent error is as shown as follows:
[tex]Percent\text{ error=}\frac{\parallel measured-real\parallel}{real}*100[/tex]Plugging in the numbers into the expression:
[tex]Percent\text{ error=}\frac{\parallel19-13\parallel}{13}*100[/tex]Subtracting numbers:
[tex]Percent\text{ error=}\frac{6}{13}*100[/tex]Multiplying terms:
[tex]Percent\text{ error=46.15\%}[/tex]In conclusion, the percent error was 46.15%
Simplify this expression. (3c^5)^-6
Answers
We need to simplify the expression,
[tex](3c^5)^{-6}[/tex]We are going to use the following exponent rule,
[tex]\begin{gathered} (a^xb^y)^z \\ =a^{xz}b^{yz} \end{gathered}[/tex]Let's simplify the expression with the rule shown above,
[tex]\begin{gathered} (3c^5)^{-6} \\ =3^{-6}(c^5)^{-6} \\ =3^{-6}c^{-30} \end{gathered}[/tex]We like to keep all exponents positive so we will use the following rule,
[tex]a^{-x}=\frac{1}{a^x}[/tex]So, the simplified form becomes:
[tex]\begin{gathered} \frac{1}{3^6}\cdot\frac{1}{c^{30}} \\ =\frac{1}{729c^{30}} \end{gathered}[/tex]Answer[tex]\frac{1}{729c^{30}}[/tex]Perform the indicated operation. Assume all variables are positive.[tex]3 \sqrt[5]{x} + 9 \sqrt[5]{x} [/tex]
Answers
write the radicals as a power
[tex]3\cdot x^{\frac{1}{5}}+9\cdot x^{\frac{1}{5}}[/tex]factor the common term
[tex]\begin{gathered} x^{\frac{1}{5}}\cdot(3+9) \\ 12\cdot x^{\frac{1}{5}} \\ \text{write as a radical} \\ 12\sqrt[5]{x} \end{gathered}[/tex]Consider the following statements. Select all that are always true.- The sum of a rational number and a rational number is rational. - The sum of a rational number and an irrational number is irrational. - The sum of an irrational number and an irrational number is irrational. - The product of a rational number and a rational number is rational. - The product of a rational number and an irrational number is irrational. - The product of an irrational number and an irrational number is irrational.which Venn diagram correctly represents the relationship between rational numbers and irrational numbers
Answers
It is important to know that the sum of two rational numbers is rational. Similarly, the sum between a rational and an irrational is irrational, but not always. Similarly, the sum of two irrational numbers is sometimes irrational, not always.
But, the product between two rational numbers is always rational. However, the product between a rational number and an irrational is not always irrational because the number zero would be a counterexample.
At last, the product of two irrational numbers is sometimes irrational.
The following image shows the diagram
As you can observe, rational and irrational numbers don't have common elements, so they don't intersect.
Hence, the true statements are
• The sum of two rational numbers is rational.
,• The sum of a rational number and an irrational number is irrational.
,• The product between two rational numbers is always rational.
estion 7 (1 point) What is the mean of this set of data: 1.2.2.2.3.4.4.5.6.7.7.8.9. 9. 10? Round your answer to the nearest tenth. 2.7 5.1 5.3 5.5
Answers
We are asked to find the mean of a set of data. Let's remember that the mean is the sum of all the terms in a set of data divided by the number of terms. The given set of data is:
[tex]1,2,2,2,3,4,4,5,6,7,7,8,9,9,10[/tex]We sum all the terms and divide them by 15, like this:
[tex]M=\frac{1+2+2+2+3+4+4+5+6+7+7+8+9+9+10}{15}[/tex]Solving the sum:
[tex]M=\frac{79}{15}[/tex]The mean is then:
[tex]M=5.3[/tex]10. If P (n, 4 ) = 17 160 , then n what is the value of n? A. 9 B. 11 C. 13D. 14
Answers
This is a permutation problem.
The expression n permutation r is expressed as:
[tex]^nP_r=\frac{n!}{(n-r)!}[/tex]In like manner, n permutation 4 will be:
[tex]\begin{gathered} P(n,4)=17160 \\ \frac{n!}{(n-4)!}=17160 \end{gathered}[/tex]Evaluation the permutation operation above, we have:
[tex]\begin{gathered} \frac{n!}{(n-4)!}=17160 \\ \frac{n(n-1)(n-2)(n-3)(n-4)!}{(n-4)!}=17160 \\ (n-4)!\text{ cancels out (n-4)!, thus we have;} \\ n(n-1)(n-2)(n-3)=17160 \end{gathered}[/tex]Expanding the Left hand side of the equation; we have:
[tex]\begin{gathered} n^4-6n^3+11n^2-6n=17160 \\ n^4-6n^3+11n^2-6n-17160=0 \end{gathered}[/tex]By factorization, the equation becomes;
[tex]\begin{gathered} \mleft(n+10\mright)\mleft(n-13\mright)\mleft(n^2-3n+132\mright)=0 \\ (n^2-3n+132)\text{ is not factorizable and would also produce unreal roots, thus the value of n from the expression can't be correct} \\ n+10=0\text{ will produce n=-10, we can have a negative result for permutation problems} \\ \text{Thus, the correct answer is;} \\ n-13=0 \\ n=13 \end{gathered}[/tex]Hence, the value of n is 13, option C
Current information for the Healey Company follows:
Beginning raw materials inventory $ 14,200
Raw material purchases 50,000
Ending raw materials inventory 15,600
Beginning work in process inventory 21,400
Ending work in process inventory 27,000
Direct labor 37,800
Total factory overhead 29,000
All raw materials used were direct materials. Healey Company's total manufacturing costs for the year are:
Multiple Choice
$109,800.
$118,200.
$115,400.
$121,000.
$126,000.
Answers
Healey Company's total manufacturing cost for the year is $109800
Beginning raw materials inventory = $14200
Raw material purchases = $50000
Ending raw materials inventory = $15600
Beginning work in process inventory = $21400
Ending work in process inventory = $27000
Direct labor = 37800
Total factory overhead = 29000
Total manufacturing cost for the year = Beginning raw materials inventory + Raw material purchases - Ending raw materials inventory + Beginning work in process inventory - Ending work in process inventory + Direct labor + Total factory overhead
Substitute the values in the equation
= 14200+50000-15600+21400-27000+37800+29000
= $109800
Hence, Healey Company's total manufacturing cost for the year is $109800
Learn more about total manufacturing cost here
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Find the experimental probability of spinning green for this experiment.P (Green) = # of times green was spun/ # of trials5 Points: Setting up your problem5 Points: Correct answerSpinner Results3024211520Number of Times100GreenRedBlueColor
Answers
7/20
1) Since Probability is a quotient between the favorable outcomes and the total of trials, we can write out the following:
2) Total of trials:
21+15+24 =60
Probability of spinning green:
[tex]P(\text{green) =}\frac{21}{60}=\frac{7}{20}=0.35[/tex]Note that there was a simplification.
3) Hence, the experimental probability of this experiment displayed on the chart is 7/20 or 0.35
This geometry problem is confusing me.Find the area, and use 3.14 for the value of pi.
Answers
Explanation
In the image, we are given the diameter of the circle as 20 mi. The area of the circle will be calculated as;
[tex]Area=\pi(\frac{d}{2})^2=3.14\times(\frac{20}{2})^2=3.14\times10^2=3.14\times100=314mi^2[/tex]Answer
[tex]314mi^2[/tex]
Please help me write the equation for a cubic. Thanks!
Answers
Solution
For this case we have the following conditions:
Vertical reflection across the x axis
Horizontal compression of 1/3
Vertical shift down 4 units
The parent function is y= x³
A vertical reflection implies: y= - x³
The horizontal compression of 1/3 implies: y= - (3x)³
And the vertical shift down 4 units implies: y= -(3x)³ -4
Then the correct answer is the first one:
y= -(3x)³ -4