Answers
Hello there. To solve this question, we'll have to remember some properties about inverse trigonometric functions.
Given the expression:
[tex]\sin (\sin ^{-1}(x)-\cos ^{-1}(x))[/tex]We want to express it as an algebraic function of x.
For this, imagine the following triangle:
Now, we find the missing leg of the triangle applying Pythagoras theorem:
[tex]\begin{gathered} 1^2=x^2+?^2 \\ 1-x^2=?^2 \\ ?^{}=\sqrt[]{1-x^2} \end{gathered}[/tex]Now, finding the sine and cosine of the angles alpha and beta, we get:
First, remember the sine of an angle is equal to the ratio between the opposite side to the angle and the hypotenuse of the triangle. The cosine of the same angle is equal to the ratio between the adjacent side to the angle and the hypotenuse of the triangle.
Therefore, we have:
[tex]\begin{gathered} \sin (\alpha)=\frac{x}{1}=x \\ \cos (\beta)=\frac{x}{1}=x \\ \sin (\beta)=\frac{\sqrt[]{1-x^2}}{1}=\sqrt[]{1-x^2} \\ \cos (\alpha)=\frac{\sqrt[]{1-x^2}}{1}=\sqrt[]{1-x^2} \end{gathered}[/tex]This means that:
[tex]\begin{gathered} \alpha=\sin ^{-1}(x) \\ \beta=\cos ^{-1}(x) \end{gathered}[/tex]Now, the expression we had earlier turns into:
[tex]\sin (\alpha-\beta)[/tex]For this, we'll use the angle sum formula:
[tex]\sin (u-v)=\sin (u)\cos (v)-\sin (v)\cos (u)[/tex]Which gives us:
[tex]\sin (\alpha)\cos (\beta)-\sin (\beta)\cos (\alpha)_{}[/tex]Plugging the results we got earlier, this is simply:
[tex]x\cdot x-\sqrt[]{1-x^2}\cdot\sqrt[]{1-x^2}[/tex]As x > 0, because we're using it as a triangle side (but it could be negative considering inverse sine and cosine as functions), we get:
[tex]\begin{gathered} x^2-(1-x^2) \\ x^2-1+x^2 \\ 2x^2-1 \end{gathered}[/tex]
Related Questions
erica has 32 pairs of shoes. 5/8 of erica's shoes are high heels how many of her shoes are high heels?
Answers
Given Data:
Total number of shoes Erica has is: t=32
The portion of high heel shoes is: r=5/8
The expression to calculate the total number of high heel shoes is,
[tex]\begin{gathered} \text{Total Number of high h}eel\text{ shoes=r}\times t \\ =\frac{5}{8}\times32 \\ =5\times4 \\ =20 \end{gathered}[/tex]Thus, Erica has 20 high heel shoes.
Got cut off while saying thanks to last tutorial t bg at helped me.
Answers
We are given the following equation.
[tex]A=p+prt[/tex]we are asked to find an equation for "t". To do that, we are going to solve for "t", first by subtracting "p" on both sides, like this:
[tex]\begin{gathered} A-p=p-p+prt \\ A-p=prt \end{gathered}[/tex]Now we will divide both sides of the equation by "pr"
[tex]\begin{gathered} \frac{A-p}{pr}=\frac{prt}{pr} \\ \frac{A-p}{pr}=t \end{gathered}[/tex]A thus we found a relationship for "t".
If 14 people want to share 24 oranges equally, how many oranges will each person get?
Answers
Given data:
The total number of oranges is: 24
The total number of people is: 14
The expression to calculate the number of oranges a person has is,
[tex]\begin{gathered} \frac{Total\text{ number of oranges}}{\text{Total number of people}}=\frac{24}{14} \\ =1+\frac{5}{7} \\ =1.714\text{ oranges.} \end{gathered}[/tex]Thus, each person gets 1 orange and 5/7 of an orange.
find the volume of a hemisphere when the diameter is 24 cm. Leave answer in terms of Pi. I had the answer of 1152 which is not correct.
Answers
Explanation
the volume of a hemisphere is given by:
[tex]\text{Volume}_{hemisphere}=\frac{2}{3}\cdot\pi\cdot r^3[/tex]where r is the radius
then
[tex]\begin{gathered} Diameter=2\text{radius} \\ \frac{\text{Diameter}}{2}=r \\ \frac{24\text{ cm}}{2}=r \\ r=12\text{ cm} \end{gathered}[/tex]now, replace.
[tex]\begin{gathered} \text{Volume}_{hemisphere}=\frac{2}{3}\cdot\pi\cdot r^3 \\ \text{Volume}_{hemisphere}=\frac{2}{3}\cdot\pi\cdot(12\operatorname{cm})^3 \\ \text{Volume}_{hemisphere}=\frac{2}{3}\cdot\pi\cdot1728cm^3 \\ \text{Volume}_{hemisphere}=1152\text{ }\pi cm^3 \end{gathered}[/tex]so, the answer is
[tex]\text{Volume}_{hemisphere}=1152\text{ }\pi cm^3[/tex]I hope this helps you
Find the unit price. If necessary, round your answer to the nearest cent. You would enter an answer like $0.49/pound, the value (like 0.49) in the first box and the appropriate unit (like pound) in the second boX $8.39 for 12 kg
Answers
For a determined product you can buy 12Kg for $8.39. To determine how much 1Kg of said product costs, you can apply cross multiplication to calculate it:
If 12 Kg cost $8.39
1 Kg costs $x:
[tex]\begin{gathered} \frac{8.39}{12}=\frac{x}{1} \\ \frac{8.39}{12}=x \\ x=0.699\cong0.70 \end{gathered}[/tex]The cost is $0.70/Kg
Find the reference angle for a rotation of 334º.
Answers
We are asked to find the reference angle for a standard angle of 334°
Recall that the reference angle is the angle that is measured with respect to the x-axis.
First, we need to find out in which quadrant the given standard angle lies.
334° lie in the 4th quadrant (180° to 360°)
So, the reference angle can be found as
[tex]\begin{gathered} \theta^{\prime}=360\degree-\theta \\ \theta^{\prime}=360\degree-334\degree \\ \theta^{\prime}=26\degree \end{gathered}[/tex]Therefore, the reference angle is 26°
16. The area of the rectangle is x^2+14x+24.Find the length and width (the factors),then determine the perimeter of the rectangle.Length:Width:Perimeter:
Answers
Given a rectangle of width W and length L, the area is calculated by:
A = W.L
The perimeter of the rectangle is:
P = 2(W + L)
We are given the area of a rectangle with the equation:
[tex]A=x^2+14x+24[/tex]To find the dimensions of the rectangle, we must factor the polynomial.
It's easily done by inspection: find two numbers that added result in 14 and multiplied result in 24. Those numbers are 12 and 2, thus:
[tex]x^2+14x+24=(x+2)(x+12)[/tex]The factors are the width and the length of the rectangle (the width smaller than the length), thus:
Width = x+2
Length = x + 12
Now calculate the perimeter:
P = 2( x + 2 + x + 12)
P = 2( 2x + 14)
Perimeter = 4x + 28
I need to know the answer and how to solve
Answers
We have by theorem that supplementary angles add up to 180° [The sum of both angles gives as result a straight line; then, the following is true:
[tex]151+x=180\Rightarrow x=29[/tex]So, the value of x is 29°.
Rewrite the following equation in slope-intercept form.6x+y=12Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
According to the given data we have the following equation:
6x+y=12
To rewrite the following equation in slope-intercept form we would have to make the following steps:
6x+y=12
Move 6x to the other side, by doing this it would change its sign
So,
y=-6x + 12
So, equation in slope-intercept form would be y=-6x + 12
someone help me please this is the last question and can't get it wrong
Answers
We can use the fact that the sum of angles in a triangle is 180degrees and the sine rule to find the missing angle and sides. This will give us a single unique triangle.
So the right answer is 1 triangle
Distance travelled = rate (or speed) * time.If Alan drives 44 miles at a constant speed of 44 mph. How long will it take? (Be sure to includeunits.)
Answers
We are given that Alan at a constant speed of 44 miles per hour. We are asked to determine the time to travel a distance of 44 miles. To do that we will use the following formula:
[tex]d=vt[/tex]Where:
[tex]\begin{gathered} d=\text{ distance} \\ v=\text{ velocity} \\ t=\text{ time} \end{gathered}[/tex]Now, since we are asked to determine the time we will divide both sides by the velocity:
[tex]\frac{d}{v}=t[/tex]Now we substitute the values:
[tex]\frac{44miles}{44\frac{miles}{hour}}=t[/tex]Now we solve the operations:
[tex]1\text{hour}=t[/tex]Therefore, the time to travel 44 miles is 1 hour.
add and subtract scientific notation 1) 5.1 x 106 + 2.3 x 106 =
Answers
Solucionamos de la siguiente manera:
[tex]5.1\cdot10^3+2.3\cdot10^6=(5100)+(2300000)=2305100[/tex]Como ya se sabe, la notación 10^3 representa miles y la notación 10^6 representa millones, por lo que al sumarles obtendremos notación de 10^6 (generalmente); aunque en nuestro caso al llevarle nuevamente a notación científica obtendremos lo siguiente:
[tex]2.3051\cdot10^2[/tex][Otra parte de la notación científica es que esta indica cuantos valores a derecha o izquierda se "mueve" el punto.]
Which of the following expressions represents the simplified version of the expression below
Answers
we have the expression
[tex](5x^3y^2-3xy+2)+(2x^3y^2-3x^2y^2+4xy-7)[/tex]step 1
Combine like terms
so
[tex](5x^3y^2+2x^3y^2)+(-3xy+4xy)+(2-7))-3x^2y^2[/tex][tex](7x^3y^2)+(xy)+(-5)-3x^2y^2[/tex]therefore
the answer is the second optionleast to greatest [tex]\pi[/tex][tex] \frac{13}{4} [/tex]22/2[tex] \sqrt{12} [/tex][tex] - 2[/tex]3.07[tex] - 3.27[/tex]
Answers
We have this number and we have to sort them from least to greatest.
We start by expressing them in decimals in order to compare them easily.
Take into account some of them are irrational, so they will be expressed approximately by a decimal (for example, pi).
Then, we have:
[tex]\begin{gathered} \pi\approx3.14 \\ \frac{13}{4}=3.25 \\ \frac{22}{2}=11 \\ \sqrt{12}\approx3.46 \\ -2 \\ 3.07 \\ -3.27 \end{gathered}[/tex]The least will be the negative numbers of this group, so we start with the negative value with the most absolute value: -3.27.
Then, we continue with -2.
Then, we start with the positive values: 3.07, pi, 13/4, sqrt(12) and 22/2.
Then, we can write them in order as:
[tex]\begin{gathered} -3.27 \\ -2 \\ 3.07 \\ \pi \\ \frac{13}{4} \\ \sqrt{12} \\ \frac{22}{2} \end{gathered}[/tex]7 - 6u = 5u + 29 solve for u
Answers
Answer:
u = -2
Explanation:
Given the expression;
7 - 6u = 5u + 29
We are to find the value of u;
Collect like terms;
-6u - 5u = 29 - 7
-11u = 22
Divide both sides by -11;
-11u/-11 = 22/-11
u = -2
Hence the value of u is -2
The solution of u in the equation is,
⇒ u = -2
Given that;
the expression is,
⇒ 7 - 6u = 5u + 29
Now, We have to find the value of u;
⇒ 7 - 6u = 5u + 29
Combine like terms;
⇒ -6u - 5u = 29 - 7
⇒ -11u = 22
Divide both sides by -11;
⇒ -11u/-11 = 22/-11
⇒ u = -2
Therefore, the value of u is -2.
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Name:Date:6. The table shows the postage charges for sending letters to Country A.How much does it cost to send a letter weighing 80 ounces to Country A?First 30 Ounces60For Every Additional 10 Ounces35
Answers
The first 30 ounces costs 60¢.
For every additional 10 ounces the cost increases in 35¢.
From the given 80 ounces, the first 30 costs 60¢ and the other 50 ounces, by considering it is 5 times 10 ounces, cost 5*(35¢) = 175¢.
Then, the total cost is:
total = 60¢ + 175¢ = 235¢
The ratio of 8th graders in a chess club to the total number of members is 0.45.What is the decimal in fraction form?Enter the correct answers in the boxes.
Answers
The ratio of 8th graders in a chess club to the total number of members is 0.454545...
This is called a repeating decimal.
Let us convert this decimal into a fraction.
[tex]x=0.4545\ldots[/tex]Step 1:
Multiply by 100
[tex]\begin{gathered} 100\times x=100\times0.4545\ldots \\ 100x=45.45\ldots \end{gathered}[/tex]Step 2:
Subtract x
[tex]\begin{gathered} 100x-x=45.45\ldots-0.4545 \\ 99x=45 \end{gathered}[/tex]Step 3:
Finally, divide by 99
[tex]\begin{gathered} \frac{99x}{99}=\frac{45}{99} \\ x=\frac{45}{99} \end{gathered}[/tex]Therefore, the fraction form is
[tex]x=\frac{45}{99}[/tex]Which of the following completes the statement below?MZBAC = {(MBC + mED00חוBERXRmZEADA0DBDC
Answers
Answer:
m(arc)ED
Explanation:
In the diagram:
[tex]m\angle\text{BAC}=\frac{1}{2}(m\widehat{BC}+m\widehat{ED})[/tex]The first option is correct.
Completing the square can be use to find the minimum value of the function represented by the equation y=x^2+4x+7. Where is the minimum value of the function located?
Answers
Given:
[tex]y=x^2+4x+7[/tex]Find: the manimum value of the function.
Explanation:
[tex]\begin{gathered} y=x^2+4x+7 \\ y^{^{\prime}}=2x+4 \\ y^{^{\prime}}=0 \\ 2x+4=0 \\ x=-2 \end{gathered}[/tex]ar x=-2,
the function will be
[tex]\begin{gathered} y=x^2+4x+7 \\ y(-2)=(-2)^2+4(-2)+7 \\ =4-8+7 \\ =11-8 \\ =3 \end{gathered}[/tex]Which two European nations took control of the regions in the Middle East that had been controlled by the Ottoman Empire?d
Answers
The two European nations took control of the regions in the Middle East that had been controlled by the Ottoman Empire are Britain and France.
The Ottoman Empire had long held the top position among Islamic nations in terms of geography, culture, and theology. Due to the division of the Ottoman Empire after the war, Western powers like Britain and France came to dominate the Middle East, and the modern Arab world and the Republic of Turkey were established.
The Turkish National Movement initiated opposition to these forces, but it did not extend to the other post-Ottoman states until the time of quick decolonization following World War II.
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each student in a class received some textbooks one third of these were english books which expresión shows how manys english books each student received
Answers
Let x be the total number of books each student recieved, since one third of them is an english book we have that the expression is:
[tex]\frac{1}{3}x[/tex]The peak of Mt. Whitney in California is 14,494 feet above sea level. Write this number as an integer.
Answers
The peak of the mountain is 14, 494 ft above sea level therefore the number can be represented as follows as an integer
[tex]+14,494\text{ feet}[/tex]The mountain is plus 14,494 ab
Find the inverse function of f.f(x) = 7 - 9x3f^-1(x) =
Answers
ANSWER
[tex]f^{-1}(x)=\sqrt[3]{\frac{7-x}{9}}[/tex]EXPLANATION
We want to find the inverse function of the given function:
[tex]f(x)=7-9x^3[/tex]To do this, we have to make x the subject of the formula.
Let f(x) be y:
[tex]\begin{gathered} y=7-9x^3 \\ 9x^3=7-y \\ x^3=\frac{7-y}{9} \\ \Rightarrow x=\sqrt[3]{\frac{7-y}{9}} \end{gathered}[/tex]Now, replace x with f^(-1)(x) and y with x:
[tex]f^{-1}(x)=\sqrt[3]{\frac{7-x}{9}}[/tex]That is the inverse function of f.
After a 30% discount, the price of a t-shirt was $14. What was the price BEFORE the discount?
Answers
Price after discount = $14
discount = 30%
x (1-30/100) = 14
x (1-0.3) = 14
x 0.7 = 14
Solve for x (original price )
x = 14/0.7
x = 20
Good morning I could really use some help solving this problem!
Answers
If the new line passes through the point (-6, 2), let's use x = -6 and y = 2 in the equation and then solve it for 'a':
[tex]\begin{gathered} y=\frac{1}{2}x+a\\ \\ 2=\frac{1}{2}(-6)+a\\ \\ 2=-3+a\\ \\ a=2+3\\ \\ a=5 \end{gathered}[/tex]Therefore the value of 'a' is 5.
Please answer all. How do you determine the dose-specific response of a drug given f(x)?
Answers
__________________
Just replace in value of x in the function
f(0.1) = 100* (0.1)^2 / ((0.1)^2 + 0.17) = (1)/(0.18) = 5.56
f(0.1) = 5.56
f(0.8) = 100* (0.8)^2 / ((0.8)^2 + 0.17) = (64)/(0.81)= 79.01
f(0.8) = 79.01
_______________________
derivative of a quotient
Q(x)= f(x)/g(x)
Q'(x)= (f'(x)*g(x) - f(x)*g'(x)) / (g(x) ^2)
[tex]Q^{\prime}\mleft(x\mright)=\frac{f^{\prime}\mleft(x\mright)\cdot g\mleft(x\mright)-f\mleft(x\mright)\cdot g^{\prime}\mleft(x\mright)}{g\mleft(x\mright)^2}[/tex]_____________________________
If f(x) = 100* (x^2) / (x^2 + 0.17),
f'(x)= 100 * ( 2x* (x^2 + 0.17) - 2x*x^2 ) / (x^2 + 0.17)^2
f'(x)= 100 * ( 2x^3 + 2x* 0.17 - 2x^3 ) / (x^2 + 0.17)^2
f'(x)= 100 * ( 2x* 0.17 ) / (x^2 + 0.17)^2
f'(x)= 34*x/ (x^2 + 0.17)^2
_________________________
Just replace in value of x in the function
f'(0.1)= 34*(0.1)/ (0.1^2 + 0.17)^2
f'(0.1)= 3.4/ (0.18)^2
f'(0.1)= 104.9
f'(0.8)= 34*(0.8)/ (0.8^2 + 0.17)^2
f'(0.8)= 27.2/ (0.81)^2
f'(0.8)= 41.46
Find the volume of the triangular pyramid to the nearest whole number.HighlightA)181 in 3B)361 in 3722 inD)1,082 in 3what's the answer
Answers
the volume of a pyramid is
[tex]V=\frac{1}{3}Ah[/tex]where A is the area of the basis and h is the height of the pyramid.
[tex]A=\frac{12.3\text{ in }\times10\text{ in}}{2}=61.5in^2[/tex]Then the volume is
[tex]\begin{gathered} V=\frac{1}{3}(61.5in^2)(17.6in) \\ =360.8in^3 \end{gathered}[/tex]Then rounding the number, the answer is B).
22% 38% 1/4 21/10 three 5/10 and 3/8 from least to greatest value
Answers
The ascending order for the given values will be 22%<1/4<3/8<38%<21/10<3 5/10 as the definition of ascending order will be "When numbers are arranged in ascending order, they are done so from smallest to largest."
What is ascending order?When numbers are arranged in ascending order, they are done so from smallest to largest. Ascending order, also known as increasing order of importance, is the exact opposite of descending order. The order of the items is lowest to highest value. The smallest value is placed first in the order, and the biggest value is placed last. As a result, if you were to put the numbers from the previous section in ascending order, they would be as follows: 11, 20, 49, 80, 56, and so on. The first number is always the smallest, in this case 11. The last number is always the largest, in this case 80.
Here,
22%=0.22
38%=0.38
1/4=0.25
21/10=2.1
3 5/10=3.5
3/8=0.375
22%<1/4<3/8<38%<21/10<3 5/10
According to the definition of ascending order, the given values will be in ascending order as follows: 22%<1/4<3/8<38%<21/10<3 5/10 "Numbers are arranged from smallest to largest when they are arranged in ascending order."
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Find the general equation of the circle having a diameter with endpoints at A(-2,3) and B(4,5).
Answers
The equation of a circle whose center is (a, b) and which has a radius r is given as:
(x - a)² + (y - b)² = r²
For the circle with endpoints at A(-2,3) and B(4,5).
The center (a, b) of the circle is calculated as:
[tex]\begin{gathered} a=\frac{-2+4}{2} \\ a=\frac{2}{2} \\ a=1 \\ b=\frac{3+5}{2} \\ b=\frac{8}{2} \\ b=4 \end{gathered}[/tex]The center, (a, b) = (1, 4)
The diameter(D) of the circle is the distance between the endpoints A(-2,3) and B(4,5).
[tex]\begin{gathered} D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ D=\sqrt[]{(4-(-2))^2+(5-3)^2} \\ D=\sqrt[]{(4+2)^2+2^2} \\ D=\sqrt[]{6^2+2^2} \\ D=\sqrt[]{36+4} \\ D=\sqrt[]{40} \end{gathered}[/tex]The diameter of the circle = √40
Radius = Diameter / 2
r = d/2
r = √40 / 2
Substituting a = 1, b = 4, and r = √40 / 2 into the general equation of a circle:
[tex]\begin{gathered} (x-1)^2+(y-4)^2=(\frac{\sqrt[]{40}}{2})^2 \\ (x-1)^2+(y-4)^2=\frac{40}{4} \\ (x-1)^2+(y-4)^2=\text{ 10} \end{gathered}[/tex]The general equation of the circle is:
[tex](x-1)^2+(y-4)^2=\text{ 10}[/tex]given expression :4x-2y=11find the missing coordinate in ordered pair (-3,?)
Answers
Equations
We are given the equation:
4x - 2y = 11
And it's required to find the missing coordinate in the ordered pair (-3, ?).
The first coordinate is x=-3, thus we need to calculate the y-coordinate by solving the equation for y, using the value of x.
Substituting:
4(-3) - 2y = 11
Operating:
-12 - 2y = 11
Adding 12:
-2y = 23
Dividing by -2:
y = 23/(-2)
y = -23/2
The ordered pair is:
[tex](-3,-\frac{23}{2})[/tex]