Mathematics College

Write the given expression in terms of x and y only.
sin(sin−1(x) + cos−1(y))

Answers

Answer 1
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Write the expression below in terms of x and y only:

(I'm going to call it "E")

[tex]\mathsf{E=sin\!\left[sin^{-1}(x)+cos^{-1}(y)\right]\qquad\quad(i)}[/tex]


Let

[tex]\begin{array}{lcl} \mathsf{\alpha=sin^{-1}(x)}&\qquad&\mathsf{then~-\,\dfrac{\pi}{2}\le \alpha\le \dfrac{\pi}{2}}\\\\\\ \mathsf{\beta=cos^{-1}(x)}&\qquad&\mathsf{then~0\le \beta\le \pi.} \end{array}[/tex]


so the expression becomes

[tex]\mathsf{E=sin(\alpha+\beta)}\\\\ \mathsf{E=sin\,\alpha\,cos\,\beta+sin\,\beta\,cos\,\alpha\qquad\quad(ii)}[/tex]


•   Finding [tex]\mathsf{sin\,\alpha:}[/tex]

[tex]\mathsf{sin\,\alpha=sin\!\left[sin^{-1}(x)\right]}\\\\ \mathsf{sin\,\alpha=x\qquad\quad\checkmark}[/tex]


•   Finding [tex]\mathsf{cos\,\alpha:}[/tex]

[tex]\mathsf{sin^2\,\alpha=x^2}\\\\ \mathsf{1-cos^2\,\alpha=x^2}\\\\ \mathsf{cos^2\,\alpha=1-x^2}\\\\ \mathsf{cos\,\alpha=\sqrt{1-x^2}\qquad\quad\checkmark}[/tex]


because [tex]\mathsf{cos\,\alpha}[/tex] is positive for [tex]\mathsf{\alpha\in \left[-\frac{\pi}{2},\,\frac{\pi}{2}\right].}[/tex]


•   Finding [tex]\mathsf{cos\,\beta:}[/tex]

[tex]\mathsf{cos\,\beta=cos\!\left[cos^{-1}(y)\right]}\\\\ \mathsf{cos\,\beta=y\qquad\quad\checkmark}[/tex]


•   Finding [tex]\mathsf{sin\,\beta:}[/tex]

[tex]\mathsf{cos^2\,\alpha=y^2}\\\\ \mathsf{1-sin^2\,\beta=y^2}\\\\ \mathsf{sin^2\,\beta=1-y^2}\\\\ \mathsf{sin\,\beta=\sqrt{1-y^2}\qquad\quad\checkmark}[/tex]


because [tex]\mathsf{sin\,\beta}[/tex] is positive for [tex]\mathsf{\beta\in [0,\,\pi].}[/tex]


Finally, you get

[tex]\mathsf{E=x\cdot y +\sqrt{1-y^2}\cdot \sqrt{1-x^2}}\\\\\\ \therefore~~\mathsf{sin\!\left[sin^{-1}(x)+cos^{-1}(y)\right]=x\cdot y +\sqrt{1-y^2}\cdot \sqrt{1-x^2}\qquad\quad\checkmark}[/tex]


I hope this helps. =)


Tags:   inverse trigonometric trig function sine cosine sin cos arcsin arccos sum angles trigonometry

Related Questions

For a particular event, 845 tickets were sold for a total of $4162. If students paid $4 per ticket and non students paid $6 per ticket, how many student tickets were sold?

Answers

Final answer:

Using a system of equations, we found that 454 student tickets were sold at the event.

Explanation:

To determine how many student tickets were sold for the event, we need to set up a system of equations based on the information given. Let's define x as the number of student tickets sold at $4 each, and y as the number of non-student tickets sold at $6 each.

The total number of tickets sold (student plus non-student) is given as 845 tickets:

x + y = 845

The total amount of money made from selling these tickets is $4162:

4x + 6y = 4162

Now we have a system of two equations:

x + y = 845

4x + 6y = 4162

We can solve this system by multiplying the first equation by 4:

(4)(x + y) = (4)(845)

4x + 4y = 3380

Now, we subtract this new equation from the second equation given by the total amount made to eliminate y:

4x + 6y - (4x + 4y) = 4162 - 3380

2y = 782

Dividing both sides of this equation by 2 gives us:

y = 391

Now, substitute y back into the first equation to find x:

x + 391 = 845

x = 845 - 391

x = 454

So, 454 student tickets were sold.

a construction company is planning to bid on a building contract. the bid costs the company $1900. the probability that the bid is accepted is 1/10. if the bid is accepted, the company will make $94000 minus the cost of the bid.
a. what is the expected value in this situation?

Answers

Final answer:

The expected value for the construction company if they bid on the contract is $7,500, calculated by weighing the potential profit and loss with their respective probabilities.

Explanation:

The question is asking us to calculate the expected value of the construction company's bid on a building contract. To find the expected value, we need to multiply each outcome by its probability and sum these amounts.

There are two possible outcomes:

The bid is accepted, which happens with a probability of 1/10, and the company makes $94,000 minus the $1,900 cost, resulting in a profit of $92,100.The bid is not accepted, which has a probability of 9/10, and the company loses the $1,900 cost.

The expected value (EV) is calculated as follows:
EV = (Probability of acceptance × Net profit when accepted) + (Probability of not being accepted × Loss when not accepted)
EV = (1/10 × $92,100) + (9/10 × -$1,900)
EV = $9,210 - $1,710
EV = $7,500

The expected value for the construction company if they bid on the contract is $7,500.

A retiree invests $5,000 in savings plan that pays 4% per year. What will the account balance be at the end of the first year ?

Answers

$5000 + 4% = $5000.04
At the end of the first year the account balance would be 200! 5000 x .04 = 200$

If net income is $115,000 and interest expense is $30,000 for 2012 what is the rate earned on total assets for 2012 (round percent to one decimal point)?

Answers

Its (115,000 - 30,000) / 115,000 = 73.9%

If the diameter of the duct is 5.2 inches what's the area of the circular hole

Answers

This should be the answer: A ≈ 21.24 in²

Determine an equation for a function that has a slope of -5 and crosses the x-axis at (3,0)

Answers

Final answer:

To find the equation of the line with a slope of -5 that crosses the x-axis at (3,0), use point-slope form to get y = -5x + 15, which is the required equation.

Explanation:

To determine an equation for a function that has a slope of -5 and crosses the x-axis at (3,0), you can use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Here, the slope m is -5 and the point on the line is (3,0). Plugging these values into the point-slope form, we get:

y - 0 = -5(x - 3)

This simplifies to:

y = -5x + 15

This equation represents a line with a slope of -5 that crosses the x-axis at the point (3,0), meeting all the conditions given in the question.

a car is purchased for 26,000 after each year the resale decrease by 35% what will the resale value be

Answers

26,500 * .25 = 6625
26,500 - 6625 = $19875 ---------> Price after 1 year

Price after 2 years:
19875 * .25 = 4968.75
19875 - 4968.75 = $14,906.3 ---------> Price after 2 years

Price after 3 years:
14,906.3 * .25 = 3726.58
14,906.3 - 3725.58 = $11,179.7 ----------> Price after 3 years

Price after 4 years:
11,179.7 * .25 = 2794.93
11,179.7 - 2794.93 = $8384.77 ------> Price after 4 years, rounded to the nearest dollar = $8385
26,000 * 35% = 910. So $910 would be subtracted from 26,000 for each year

Pamela's age is two times Jiri's age. The sum of their ages is 63 . What is Jiri's age?

Answers

2x + x = 63
3x = 63
x = 21
2x = 43
Jiri is 21
Let x be Pamela's age, and y Jiri's age.
"Pamela's age is 2 times Jiri's age" can be represented by x = 2y
"The sum of their ages is 63" So x + y = 63

We can now create a system of 2 equations:
x = 2y (1)
x + y = 63 (2)

Let's take the algebraic value of x from equation (1) and replace it in equation (2):
x + y = 2y + y = 3y
So 3y = 63
Divide both sides by 3 to isolate y on a side and its value on the other:
(3y)/3 = 63/3
y = 21

So Jiri's age is 21 years old.

Hope this Helps! :)

what is 694 divided by 41 and what is the remainder please help thank you

Answers

We can use long division in finding the remainder.

Place the divisor which is 41, outside the long division sign, and 694 under the long division sign as shown in the diagram.

41 goes into 694 16 times.

Write the 16 on top of the long division sign.

Multiply the 16 by 41, to obtain 656


Write the 656 below the 694 as in the diagram.


Subtract 656 from 694 to obtain 38.

Since 41 is more than 38 , we cannot proceed.

Hence,

[tex]\frac{694}{41}=16\:\:Remainder\:\:38[/tex]

Therefore the remainder is 38


The quotient is 16, and the remainder is 38.

We have,

To find the quotient and remainder when dividing 694 by 41, we can perform the division and observe the results.

Now,

41 x 16 = 656

And,

694 - 656 = 38

This means,

694 divided by 41 is equal to 16 with a remainder of 38.

Therefore,

The quotient is 16, and the remainder is 38.

Learn more about expressions here:

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4 options but only 3 have sea views what fraction has sea view

Answers

3/4, because 3/4 is like saying "3 out of 4", which this is

Ron Co. sold sweatshirts ($20) and baseball caps ($10). If total sales were $2,860 and people bought 6 times as many sweatshirts as caps, how many of each were sold?

Answers

Final answer:

To find the number of baseball caps and sweatshirts sold, we can set up an equation using the given information and solve for the variables. The solution shows that 22 baseball caps and 132 sweatshirts were sold.

Explanation:

Let's solve this problem step by step:

Let x represent the number of baseball caps sold.Since people bought 6 times as many sweatshirts as caps, the number of sweatshirts sold would be 6x.The total sales revenue can be calculated by multiplying the price of each item by the number of items sold. So, the equation becomes: 10x + 20(6x) = 2860.Simplifying the equation gives us 10x + 120x = 2860.Combining like terms, we get 130x = 2860.Dividing both sides of the equation by 130, we find that x = 22. So, 22 baseball caps were sold.Since people bought 6 times as many sweatshirts as caps, the number of sweatshirts sold would be 6 * 22 = 132.

Therefore, 22 baseball caps and 132 sweatshirts were sold.

Final answer:

The problem is solved by setting up equations based on the given information. Ron Co. sold 132 sweatshirts at $20 each and 22 baseball caps at $10 each to make a total of $2,860 in sales.

Explanation:

To solve the problem, let's define the variables representing the number of sweatshirts and caps sold. Let's say 's' equals the number of sweatshirts and 'c' equals the number of baseball caps.

We are given two pieces of information:

The total sales were $2,860.People bought 6 times as many sweatshirts as caps, so s = 6c.

The price of a sweatshirt is $20, and the price of a cap is $10. The total sales can be expressed as the sum of the sales from sweatshirts and caps, which gives us the equation 20s + 10c = 2860.

Using the information that s = 6c, we can substitute 6c for s in the total sales equation to get 20(6c) + 10c = 2860. Simplifying this equation, we get 120c + 10c = 2860, which leads to 130c = 2860. Dividing both sides by 130 gives us c = 22.

Now, we can find the number of sweatshirts sold by substituting c into the equation s = 6c. So, s = 6(22) = 132.

Therefore, Ron Co. sold 132 sweatshirts and 22 baseball caps.

Terry added 3 and 7 he got the sum of 9 his answer is is not correct described how Terry can find the correct sum

Answers

He is not correct because 3+7=10 and not 9

A hacker is trying to guess someone's password. The hacker knows (somehow) that the password is 15 characters long, and that each character is either lowercase, an uppercase letter or a numerical digit. Assume that the hacker makes a random guess.

What is the probability that the hacker guesses the password on his first try? Enter your answer as a decimal or a fraction.

Answers

Final answer:

The probability that the hacker guesses the password correctly on the first try is 1 / 62^15.

Explanation:

To calculate the probability that the hacker guesses the password correctly on the first try, we need to determine the total number of possible passwords and divide it by the total number of possible guesses.

Since the password is 15 characters long and each character can be a lowercase letter, an uppercase letter, or a numerical digit, there are 26 + 26 + 10 = 62 possible options for each character.

Therefore, the total number of possible passwords is 62^15 (62 raised to the power of 15).

The probability of the hacker guessing the password correctly on the first try is 1 divided by the total number of possible passwords.

So, the probability is 1 / 62^15. This can be left as a fraction or converted to a decimal.

solution for 5y+2y = -24

Answers

7y=-24
y= -24/7
 You have to add the two vaiables and then divide to get the y by itself.
Simplifying
5y+2y= -24

Combine Like Terms
Y
7y=-24

Solving
7y=-24

Solving for variable 'y'

Move all terms containing y to the left, all other terms to the right.

Divide each side by '7'
Y=-3.428571429

Answer

Y= -3.428571429

if Q(x)=4x^2 -1, find Q (-7)

Answers

4(-7)²−1
4(49)-1
196-1
=195

Tell whether the two ratios form a proportion. Explain. 25/80 and 5/16

Answers

Can you give me brainliest answer? 2 more for a level up!

Okay, 25/80 and 5/16.

Divide 25/80 by 5.

25/5 = 5
80/5 = 16

5/16

The other fraction is 5/16, so the ratios are proportional!

The Martin family's truck gets an average of 25 miles per gallon. Predict how many miles they can drive using 7 gallons of gas.

Answers

I believe they can drive 3.57 miles.
175 miles because 1gallon = 25 miles
                                  7 gallons = x
25*7 = 1x
175 = x

175 miles.



If u bring 10 empty bottles to the bottle shop, the bottle shop will change 10 empty bottles to 1 drink. If u bring 100 bottles to the shop. How many drinks will you get?

Answers

In order to calculate this, we can set up a simple ratio. We say that 10 empty bottles : 1 drink 100 empty bottles : x drinks We cross multiply to form the equation: 10x = 100 Now, we can simply solve for x to determine the number of drinks we would get for 100 bottles. x = 10 So we will get 10 drinks if we bring 100 bottles to the bottle shop.Hope this helps. Let me know if you need additional help!

By bringing 100 bottles to the shop, you will receive 10 drinks.

The students question relates to a mathematical conversion problem, where empty bottles are exchanged for drinks at a bottle shop. If the conversion rate is 10 empty bottles for 1 drink, then by bringing 100 bottles, we can set up a simple division:

Number of bottles: 100

Conversion rate: 10 bottles per drink

To find out how many drinks you can get for 100 bottles, we divide the total number of bottles by the conversion rate:

Divide 100 (bottles) by 10 (bottles per drink).

100 / 10 = 10

Therefore, by bringing 100 bottles to the shop, you will receive 10 drinks.

explain how you can determine if four points lie on a single parabola

Answers

four points in the plane determine two conjugate parabolas, each of which ... system we can fit a second degree polynomial to the three points P1,P3,P4. ... of the quadratic give the two orientations in which the four points lie on a single ... shows that if a parabola with vertical axis intersects a horizontal line at points P,Q
1.) Plug them into the equation

2.) They should all be symmetric to some line. If this is a slant asymptote, it's much harder to determine. However, if you are being asked this question, I would guess you're taking either Algebra I or II. In which case there should be a vertical or horizontal line they are symmetric to... and there should be 2 sets of points with equivalent X OR Y values.... determine the line. If you're able to, they're on a parabola....if not, they probably aren't.

Solve for B
6=10B+12C

Answers

I hope this helps you


10B=6-12C


B=2 (3-6C)/10


B =3-6C/5
6 = 10b + 12c
6 - 12c = 10b
(6 - 12c) / 10 = b or 3/5 - 6/5c = b

How many times do you need to multiply by ten to get from 0.3637 to 363.7

Answers

3 times, or you could multiply 0.3637 by 1000 which is quicker. Hope this helps!

the sales tax rate for the state of washington was 5.1% . What is the sales tax on a $5900 car in Washington?

Answers

$300.9. I got that by multiplying $5900 by 0.051.
5900$/5.1%
1156.86
5900+1156.86
7056.86$


Brandy wants to buy a digital camera that costs $300. Suppose she saves $15 each week. In how many weeks will she have enough money for the camera?

Answers

It would take 19 weeks exactly.

It will take Brandy 20 weeks to save enough money to buy the digital camera.

To find out how many weeks it will take for Brandy to save enough money for the camera, we can set up an equation:

Let x be the number of weeks it will take for Brandy to save enough money.

$15 * x = $300

Solving for x:

x = $300 / $15

x = 20

Therefore, it will take Brandy 20 weeks to save enough money to buy the digital camera.

In the following distribution of the variable semesters completed, which is the mode: 4, 3, 1, 0, 3, 3, 4, 0, 3, 2?

Answers

In this set of data your mode would be 3 because it occurs most often in the set.

The weight of water is 62 1/2 lb per cubic foot. What is the weight of 1 1/9 cubic feet of water?

Answers

62 1/2 = improper fraction 125/2
1 1/9 = improper 10/9

125/2 times 10/9 = 652/9 = 69 whole 4/9 lb 

Final answer:

To determine the weight of 1 1/9 cubic feet of water, convert the mixed number to an improper fraction and multiply by the weight per cubic foot, resulting in approximately 77.16 pounds.

Explanation:

The student is asking to calculate the weight of 1 1/9 cubic feet of water given that the weight of water is 62 1/2 lb per cubic foot. To find the weight of the water, we multiply the volume by the weight per unit volume:

Convert 1 1/9 to an improper fraction: 1 1/9 = 10/9.

Multiply 10/9 cubic feet by 62.5 lb/ft³ (weight of water per cubic foot).

Calculate: (10/9) × 62.5 = 694.44/9 ≈ 77.16 lb.

Hence, the weight of 1 1/9 cubic feet of water is approximately 77.16 pounds.

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that   [tex] \frac{ \sqrt{x^2-36} }{x} [/tex]     0 ≤ θ < π/2.   
x=6secФ

Answers

After substituting:
[tex]\frac{\sqrt{36sec^2 \theta - 36}}{6 sec \theta}[/tex]
Factor out the 36, allowing the 6's to cancel
[tex]\frac{ \sqrt{sec^2 \theta - 1}}{sec \theta} [/tex]
Use trig identity:
sec^2 - 1 = tan^2
[tex]\frac{\sqrt{tan^2 \theta}}{sec \theta} = \frac{tan \theta}{sec \theta} = sin \theta[/tex]

A particular restaurant can legally have only 150 people in it at one time. The tables in the restaurant can seat 4 people at a time. The number of tables, t, in the restaurant can be represented by the inequality 4t < 150. What is the maximum number of tables the restaurant can have?

Answers

37.5 tables can be in the restaurant at onces.
the restaurant can have 37.5 tables

Area model division 246÷3

Answers

246 divided by 3 should equal 82

Answer:

it is right

Step-by-step explanation:

identify the decimal labeled with the letter A on the scale below

Answers

It equals 0.6 just count the lines :)

The decimal labeled with the letter A on the scale is 0.75.

What is number line?

A number line can be defined as a straight line with numbers placed at equal intervals or segments along its length.

Here, the number starts with zero.

There are 8 division between 0 and 1.

So, first one represent 1/8=  0.125

second one =0.25

third one= 0.375

fourth one= 0.50

fifth one= 0.625

sixth one= 0.75

Hence, the decimal labeled with the letter A on the scale is 0.75.

learn more about number line here:

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what is 84 divide by 4

Answers

The answer to 84 divided by 4=21.
It's 21. If you multiply 4 by 21 you get 84. Not good at giving math explanations sorry.
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