Mathematics High School

Tan α = - 4/3 lies in quad 2, and cos β = 2/3 lies in quad 1 find
a. cos(α + β)
b. sin( α+β)
c. t...

Answers

Answer 1
Recall some identities:


[tex]\cos^2x+\sin^2x=1[/tex]

[tex]\tan^2x+1=\sec^2x=\dfrac1{\cos^2x}[/tex]


[tex]\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta[/tex]


[tex]\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta[/tex]

Not sure what part (c) is asking for, but I assume it's [tex]\tan(\alpha+\beta)[/tex], in which case


[tex]\tan(\alpha+\beta)=\dfrac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)}[/tex]


If [tex]\tan\alpha=-\dfrac43[/tex], then

[tex]\dfrac1{\cos^2\alpha}=1+\left(-\dfrac43\right)^2=\dfrac{25}9[/tex]
[tex]\implies\cos\alpha=\pm\dfrac35[/tex]

We know that [tex]\alpha[/tex] lies in quadrant 2, i.e. [tex]\dfrac\pi2<\alpha<\pi[/tex], so we expect [tex]\cos\alpha<0[/tex]. So we take the negative root. We also find that

[tex]\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}\iff-\dfrac43=\dfrac{\sin\alpha}{-\frac35}\implies\sin\alpha=\dfrac45[/tex]

If [tex]\cos\beta=\dfrac23[/tex], then


[tex]\sin^2\beta=1-\left(\dfrac23\right)^2=\dfrac59\implies\sin\beta=\pm\dfrac{\sqrt5}3[/tex]


Since [tex]\beta[/tex] lies in quadrant 1, i.e. [tex]0<\beta<\dfrac\pi2[/tex], we know that [tex]\sin\beta>0[/tex], so we take the positive root.


Now,

[tex]\cos(\alpha+\beta)=-\dfrac35\cdot\dfrac23-\dfrac45\cdot\dfrac{\sqrt5}3=-\dfrac52-\dfrac4{3\sqrt5}[/tex]


[tex]\sin(\alpha+\beta)=\dfrac45\cdot\dfrac23+\left(-\dfrac35\right)\cdot\dfrac{\sqrt5}3=\dfrac8{15}-\dfrac1{\sqrt5}[/tex]

Then it follows that


[tex]\tan(\alpha+\beta)=\dfrac{\frac8{15}-\frac1{\sqrt5}}{-\frac52-\frac4{3\sqrt5}}=\dfrac{54-25\sqrt5}{22}[/tex]

Related Questions

helpppp! pls! 10 points

Answers

The product of the lengths of the parts of the chord is equal to the product of the lengths of the parts of the chord that divides it.

(7 cm)*(8 cm) = n*(4 cm)
(7*8 cm²)/(4 cm) = n
14 cm = n

[tex]n=\boxed{14}cm[/tex]

Ernesto tried to determine the solution for the system of equations using substitution. His work is shown below. x – y = 7 3x – 2y = 8 Step 1: x = y + 7 Step 2: 3(y + 7) – 2y = 8 Step 3: 3y + 7 – 2y = 8 Step 4: y + 7 = 8 Step 5: y = 1

Answers

The error is in Step 3.

_____
Simplifying 3(y +7) -2y = 8 should result in
  3y +21 -2y = 8

Step 4: y +21 = 8
Step 5: y = -13

A metal washer is made by cutting out a circle 13.2 mm in diameter from a larger circle. To the nearest tenth, what is the area of the washer?

Answers

The area of the washer will be found by calculating the area of the shaded part.
Area of larger circle:
A=πr²
r=4.4+(13.2/2)=11 mm
thus
A=π×11²=380.13

Area of the small circle:
A=π×(13.2/2)²=136.85 mm

The area of the washer will be:
380.13-136.85
=243.28
~243.3 mm²

Find the product. 4(−12)(−3) A) −144 B) −96 C) 144 D) 72

Answers

For this case we have the following expression:
 4 (-12) (- 3)
 Rewriting we have:
 We multiply 4 by -3:
 (-12) (- 12)
 We rewrite:
 (-1) (- 1) (12) (12)
 (1) (12) (12)
 (12) (12)
 144
 Answer:
 the product 4 (-12) (- 3) is:
 C) 144

The area of a parallelogram is 3.36 square feet. the base is 2.8 feet. if the measures of the base and height are each doubled, find the area of the resulting parallelogram

Answers

A =bh

Plug in values and solve for height (h).
A=3.36
b=2.8

3.36 = 2.8h
/2.8      /2.8
1.2 = h

Base and Height doubled:
b = 2.8
h = 1.2

2(2.8)=5.6
2(1.2)=2.4

Plug in new b and h to area equation given to find area:
5.6(2.4) = 13.44

Resulting parallelogram area = 13.44 ft^2

An isoceles triangle has congruent side of 20 cm. The base is 10 cm. Find the height of the triangle

Answers

25+x^2=400
x^2=375
x=sqrt(375)

The probability of picking two black marbles from a box at random without replacement is 1091 . The probability of drawing a black marble first is 514 . What is the probability of picking a second black marble, given that the first marble picked was black?

Answers

The correct answer to the question above is 4/13. Below it shows how it's achieved...

EQUATION: 
P(A|B) = P(A ∩ B)
                 P(B)

Step 1: Identify 
P(B) = 5/14 
P(A ∩ B) = 10/91

Step 2: Insert into the equation and Solve
10/91 ÷ 5/14
Use reciprocal to divide
10/91 × 14/5 
Simplify the answer 
140/455
Simplified (Final Answer)
4/13 






Final answer:

The probability of picking a second black marble, given that the first marble picked was black, is 1.

Explanation:

To find the probability of picking a second black marble, given that the first marble picked was black, we can use the formula for conditional probability: P(A|B) = P(A and B)/P(B).

In this case, A represents the event of picking a black marble on the second draw, and B represents the event of picking a black marble on the first draw.

We are given that P(B) = 514, and the probability of picking two black marbles without replacement is 1091. Therefore, P(A and B) = P(A|B) * P(B) = (1091/514) * (514/1091) = 1.

So, the probability of picking a second black marble, given that the first marble picked was black, is 1.

Difference between ellipse circle parabola and hyperbola equations

Answers

The equation of a circle is:

(x - h)² + (y - k)² = r²

The equation of a parabola is:

(x - h)² = 4p(y - k)

The equation of an ellipse is: 

[tex] \frac{(x-h) ^{2} }{ a^{2} } + \frac{(y-k) ^{2} }{ b^{2} } = 1[/tex]

where variables a and b and the two different measurements of the vertices

The equation of a hyperbola is: 

[tex] \frac{((x-h)^{2} }{ a^{2} } - \frac{(y-k)^{2} }{ b^{2} } = 1[/tex] if it is with a horizontal transverse axis

[tex] \frac{ (y-k)^{2} }{ b^{2} } - \frac{ (x-h)^{2} }{ a^{2} } = 1[/tex] if it is with a vertical transverse axis

Notice these have a subtraction operation, the exact opposite ellipse. 


The solution is: these have a subtraction operation, the exact opposite ellipse.

What is parabola?

The parabola is a plane curve which is mirror symmetrical and is approximately U-shaped. It fits several superficial different mathematical descriptions.

here, we have,

The equation of a circle is:

(x - h)² + (y - k)² = r²

The equation of a parabola is:

(x - h)² = 4p(y - k)

The equation of an ellipse is:

(x - h)²/ a² + (y - k)²/ b² = 1

where variables a and b and the two different measurements of the vertices

The equation of a hyperbola is:

(x - h)²/ a² - (y - k)²/ b² = 1 if it is with a horizontal transverse axis

(y - k)²/ b² - (x - h)²/ a²  = 1 if it is with a vertical transverse axis

Notice these have a subtraction operation, the exact opposite ellipse.

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What is the volume of a cylinder with a diameter of 12 in. and a height of 7 in.?

Answers

Answer

Find out the  volume of a cylinder .

To prove

Formula

[tex]Volume\ of\ cylinder = \pi r^{2}h[/tex]

Where r is the radius and h is the height.

As given

Diameter of 12 in. and a height of 7 in.

[tex]Radius = \frac{Diameter}{2}[/tex]

[tex]Radius = \frac{12}{2}[/tex]

Radius = 6 in

[tex]\pi = 3.14[/tex]

put all the values in the formula

[tex]Volume\ of\ cylinder = 3.14\times 6\times 6\times 7[/tex]

                                          = 791.28 in³

Therefore the volume of the cylinder is 791 .28 in³ .

solve the equation 2 cos x + 1 = 0 , 0 <= x <= 2pi PLEASE HELP! WILL AWARD!

Answers

Im not saying its right but I got 0,2π3,4π3,and2π  f(x)=2cos2x−cosx−1=0
Solve this quadratic equation for cos x.
Since a + b + c = 0, use shortcut.
One real root is cos x = 1 and the other is cosx=ca=−12.
Trig table and unit circle -->
a. cos x = 1 --> arc x = 0, and arc x=2π
b. cosx=−12 --> arc x=±2π3
The arc −2π3 is co-terminal to the arc: 4π3
Answers for (0,2π):
0,2π3,4π3,and2π

The solution of the given trigonometric expression is x = 2π/3.

Use the concept of trigonometric ratio defined as:

Trigonometric ratios depend on the value of the ratio of sides of a triangle which is right-angled and contains the values of all trigonometric functions.

The ratios of a right-angled triangle's sides with regard to a certain acute angle are known as it's trigonometric ratios.

The given expression is:

2 cos x + 1 = 0

Where,

0 ≤ x ≤ 2π

To solve it,

Subtract  both sides,

2 cos x  = - 1

Divide by 2 on both sides,

cos x = -1/2

Take cos inverse on both sides,

[tex]x = cos^{-1} (\frac{1}{2})[/tex]

Since  cos(2π/3) = -1

x = 2π/3

Now since 2π/3 lies between  0 ≤ x ≤ 2π

Hence,

The solution of the given expression is x = 2π/3

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Johnny flips a coin three times. what is the probability he lands on heads every time

Answers

there is a 1/2 probability of landing on heads each flip

he flips it 3 times

so 1/2 x 1/2 x 1/2 = 1/8 probability of getting heads every time.

[tex] |\Omega|=2^3=8\\
|A|=1\\\\
P(A)=\dfrac{1}{8}=12.5\% [/tex]

An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. they report the angles of elevation as 14° and how high is the airplane? round to the nearest hundredth of a mile.

Answers

The complete question: 
An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. they report the angles of elevation as 14° and 30° how high is the airplane? round to the nearest hundredth of a mile.

Check the diagram of the situation in the picture attached.

First, we are going to use the law of sines to find the measure of side [tex]a[/tex] in triangle ABC:
[tex] \frac{a}{sin(14)} = \frac{5}{sin(16)} [/tex]
[tex]a= \frac{5sin(14)}{sin(16)} [/tex]
[tex]a=4.39[/tex]

Next, we are going to use the trig function Sine in triangle BCD to find the height, [tex]h[/tex], of the plane:
[tex]sin(30)= \frac{h}{4.39} [/tex]
[tex]h=4.39sin30[/tex]
[tex]h=2.19[/tex]

We can conclude that the plane is 2.19 miles high. 

The dollar amount (d) that erin earns varies directly as the number of hours (t) that she works. if she earns $206.25 in 25 hours, how many hours did she work if she earned $264?

Answers

d=t*m
d=dollar amount
t=hours
m=how much she earns per hour

Plug in the first set of numbers
$206.25=(25hr)*m
the amount she earns per hour is
$206.25/25hr = m = $8.25 per hr

So

$264=t*$8.25
264/8.25 = t
t = 32 hr




use the distance formula and the pythagorean theorem to find the distance, to the nearest tenth, from T (4,-2) to (-2,3).

Answers

Answer:
distance = [tex] \sqrt{61} [/tex] which is 7.8 to the nearest tenth

Explanation:
The distance between two points can be calculated using the following formula:
distance = [tex] \sqrt{(x2 - x1)^2 + (y2 - y1)^2} [/tex]

In the given, we have:
point (4,-2) representing (x1,y1)
point (-2,3) representing (x2,y2)

Substitute with the givens in the above formula to get the distance as follows:
distance = [tex] \sqrt{(-2-4)^2 + (3--2)^2} = \sqrt{61} [/tex] which is 7.8 to the nearest tenth

Hope this helps :)
T=(4, -2)=(xt, yt)→xt=4, yt=-2
P=(-2, 3)=(xp, yp)→xp=-2, yp=3

d=sqrt[ (xp-xt)^2+(yp-yt)^2 ]
d=sqrt[ (-2-4)^2+(3-(-2))^2 ]
d=sqrt[ (-6)^2+(3+2)^2 ]
d=sqrt[ 36+(5)^2 ]
d=sqrt( 36+25 )
d=sqrt(61)
d=7.810249676
To the nearest tenth
d=7.8

Answer: The distance, to the nearest tenth, from T (4,-2) to (-2,3) is 7.8


Amy owns a graphic design store. She purchases a new printer to use in her store.
The printer depreciates by a fixed rate of 14% per year. The function
V = 2,400(1 – 0.14)t can be used to model the value of the printer in dollars after
t years.

Question:

Part A: Explain what the parameter 2,400 represents in the equation of the function.
Part B: What is the factor by which the printer depreciates each year?
Part C: Amy also considered purchasing a printer that costs $4,000 and depreciates by 25% each year. Which printer will have more value in 5 years? Justify your answer by showing/explaining your work.

i really need help :( im working hard to get all my school work done and this one thing is making my head hurt :(

Answers

The parameter 2,400 in the depreciation function represents the initial value of Amy's printer. The printer depreciates by a factor of 0.86 each year. After calculations, the first printer has more value ($1,004.40) after 5 years compared to the second printer ($949.24).

Part A: The parameter 2,400 represents the initial purchase price or the original value of the printer that Amy owns in her graphic design store.

Part B: The factor by which the printer depreciates each year is 0.86 (which is calculated as 1 - 0.14, meaning the value retains 86% of its worth each subsequent year).

Part C: To compare which printer will have more value in 5 years, we need to calculate the future value of both printers using their respective depreciation rates:

First printer: V = 2,400(1 - 0.14)^5Second printer: V = 4,000(1 - 0.25)^5

After calculating:

Value of the first printer after 5 years: V \\approx 2,400(0.86)^5 \\approx 2,400(0.4185) \\approx $1,004.40Value of the second printer after 5 years: V \\approx 4,000(0.75)^5 \\approx 4,000(0.2373) \\approx $949.24

Therefore, the first printer will have more value after 5 years.

PROBABILITY PROBLEM! NEED HELP AND EXPLAIN/SHOW WORK

Answers

10 + 5 + 1 = 16 - all balls
The probability that the first ball is red:
[tex]P(R)=\dfrac{10}{16}=\dfrac{5}{8}[/tex]
The probability that the second ball is black:
[tex]P(B)=\dfrac{1}{15}[/tex]
Together:
[tex]P(R)\cdot P(B)=\dfrac{5}{8}\cdot\dfrac{1}{15}=\dfrac{1}{24}[/tex]

Rewrite in simplest radical form x56x16

Answers

I must assume that by "x56" you actually meant "x^56," or "x to the 56th power.

If that's the case, then  x^56 * x^16 = x^(56+16) = x^72, or "x to the 72nd power".

If at all possible, see a teacher regarding your "x56x16."  It's very ambiguous.

the rational roots of a polynomial function F(x) can be written in the form p/q,
where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient

TRUE OR FALSE??

Answers

ANSWER

True

EXPLANATION

According to the Rational Roots Theorem,the rational roots of a polynomial function F(x) can be written in the form

[tex] \frac{p}{q} [/tex]

,

where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient.

The polynomial must be written in standard form.

This means the polynomial must be written in descending powers of x.

The leading term is the first term of a polynomial written in standard form.

The constant term is the term independent of x.

Lines b and c are parallel. What is the measure of <6?
m<6 = 45
m<6 = 54
m<6 = 117
m<6 = 126

Answers

Answer:

m<6 = 54

St,ep-by-step explanation:

Lines b and c are parallel.

13x +9 and 5x+9 are consecutive exterior angles.

Consecutive exterior angles are supplementary.

[tex]13x+9+5x+9= 180[/tex]

Now we solve for x. combine like terms

[tex]18x+18= 180[/tex]

Subtract 18 from both sides

[tex]18x=162[/tex]

Divide both sides by 18

x=9

So the value of x=9

angle 6 and 5x+9 are vertical opposite angles . they are equal

m<6= 5x+9

Plug in 9 for x

m<6= 5(9)+9=54

m<6 = 54

Answer:

m<6 = 54

Step-by-step explanation:

Another geometry question! Please help!

Answers

An angle of depression is measured down from the horizontal. The angle of depression from B to C is ...
  ∠3

.48 divided by 9.23 rounded by nearest tenth

Answers


Answer
48/9.23=0.052 
0.05

0.48/9.23 =0.05

THIS IS ROUNDED

Which of the following sets shows all the numbers from the set {2, 3, 4, 5} that are part of the solution to the inequality 4n + 6 > 22?

Answers

5 is the only number to satisfy the equation. If you were to plug in 2, 3, or 4, you'd get these results.

[tex]4(2) + 6 > 22 \\ 8 + 6 > 22 \\ 14 > 22 \\ \: which \: is \: not \: true[/tex]
[tex]4(3) + 6 > 22 \\ 12 + 6 > 22 \\ 18 > 22 \\ \: also \: not \: true[/tex]

Answer:

the answer is  

{2, 3, 4}

Step-by-step explanation:

if 5 is the only one that works and it has to be inequality  so  

{2, 3, 4} can be the only answer

Graphic lesions to solve the system Y equals 1/3 X +6 and why equals 1/3 X +4

Answers

For this case we have the following functions:
 y = (1/3) x + 6
 y = (1/3) x + 4
 We note that both lines have the same slope in this case given by:
 m = 1/3
 Therefore, the lines are parallel.
 As they are parallel, they are not cut at any point.
 Thus, the system of equations has no solution.
 Answer:
 The system has no solution (parallel lines)
Graph the equations to solve the system Y equals 1/3 X +6 and y equals 1/3 X +4
Following the formula: y=mx+b,where:m= slopeb=y-interceptAs observed in the given equations, they have the same slope which is equal to 1/3 but each have different y-intercept which is 6 and 4.
Note that two lines are parallel if their slopes are equal and they have different y-intercepts. 
Upon graphing, it had seen that the two functions are parallel.See attached picture for the graph.

four-barrel of colored marbles you randomly select three blue to Yellow 7 red eight green and two purple marbles. find the experimental probability of randomly selecting either green or purple marble

Standard number cube is rolled 180 times. Predict how many times 3 or 5 will be the result

The movie open the other day. So far 500,000 people have seen it. the producers of the movie needed to know if the people liked it. a random sample of 8000 people were asked as they were leaving the theater if they like the movie. of those interviewed 4,200 enjoyed the movie. predict the total number of people who have enjoyed the movie.

Jamestown Builders has a development of new homes. there are 6 different floor plans 7 exterior color than an option of either one car or two car garage . How many choices are there for one home?

if no digit may be used more than once how many 5 digits can be formed using only three digits. 8 1 2 5 and7?

Answers

only  two 56 32 hope it helps

The answer provides the experimental probability of selecting green or purple marbles, predicts the occurrences of rolling 3 or 5 on a number cube, and estimates the total number of people who enjoyed the movie.

Experimental Probability Calculation: In the given scenario, there are 8 green marbles and 2 purple marbles, making a total of 10 marbles. The probability of selecting a green or purple marble is the sum of the individual probabilities, which is (8/10) + (2/10) = 0.8.

Expected Outcome of Rolling a Number Cube: Since a standard number cube has numbers 1 to 6, each number has an equal probability of 1/6. Predicting how many times 3 or 5 will be rolled out of 180 rolls can be calculated by (1/6 + 1/6) x 180 = 60 times each for 3 and 5.

Total Number of People Who Enjoyed the Movie: Based on the sample data, 4,200 out of 8,000 people enjoyed the movie. Extrapolating this to the total viewers, the predicted number of people who enjoyed the movie would be: (4,200/8,000) x 500,000 = 262,500 people.

I think its B
A researcher finds a negative correlation between the total number of people driving for pleasure and the sales of electric cars.

Which variable is most likely the lurking variable that explains the correlation?



population

gas prices

amount of rainfall

water prices

Answers

The correct answer is:


gas prices


Explanation:


As the number of people driving for pleasure goes down, the sales of electric cars go up. This is likely due to gas prices. As gas prices go up, fewer people drive for pleasure and more people purchase electric cars. Both are ways to reduce the amount of money spent on gasoline.

The option 'gas prices' variable is most likely the lurking variable that explains the correlation.

Given that a researcher finds a negative correlation between the total number of people driving for pleasure and the sales of electric cars.

It is required to find which variable is most likely the lurking variable that explains the correlation.

What is correlation?

It is defined as the relation between two variables which is a quantitative type and gives an idea about the direction of these two variables.

We have a negative correlation between:

The total number of people driving for pleasure and the sales of electric cars.

We know that the negative correlation means one variable decreases and as the other increases and vice versa.

If the sales of electric cars increase the total number of people driving for pleasure goes down.

This can happen if the gas prices go up fewer people drive for pleasure and more people buy an electric car.

Thus, the option 'gas prices' is correct.

Learn more about the correlation here:

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Write the sum in sigma notation. 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50

Answers

The sum in sigma notation for the sequence will be as follows:
From
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50
first term=5
common difference=5
number of terms=10
n=nth term
thus the sum will be:
(i=2 to 10)∑(5(n-1)+5)

Please mark brainliest

The sum in sigma notation for the sequence as follows:

From

5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50

first term=5

common difference=5

number of terms=10

n=nth term

therefore the total will be:

(i=2 to 10)∑(5(n-1)+5

Write an inequality for the given statement: The sum of –18 and a number is more than 22. A. 18 – n < 22 B. 18 – n > 22 C. –18 + n < 22 D. –18 + n > 22

Answers

- 18 + n > 22

Obviously the answer is D

What is the value of z in the equation 3z − 7 = 14? (4 points)

Answers

good day ^-^ ///////////////

Answer:

7

Step-by-step explanation:

A tractor was purchased for $30,000 six years ago. what is the value of the tractor today if the depreciation rate was 2.5%?

Answers

Initial cost = $30,000
Depreciation rate = 2.5%

Depreciation expense per year = 30,000*2.5/100 = $750
In six years,
Depreciation = 6*750 = $4,500

Value of the tractor = Initial cost - Depreciation = $30,000 - $4,500 = $25,500

Rewrite the radicand as two factors, one of which is a perfect cube.

3/250

Answers

[tex]\bf ~~~~~~~~~~~~\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} \qquad \qquad \cfrac{1}{a^n}\implies a^{-n} \qquad \qquad a^n\implies \cfrac{1}{a^{-n}} \\\\ -------------------------------\\\\ \cfrac{3}{250}\qquad \begin{cases} 250=2\cdot 5\cdot 5\cdot 5\\ \qquad 2\cdot 5^3 \end{cases}\implies \cfrac{3}{5^3\cdot 2}\implies \cfrac{1}{5^3}\cdot \cfrac{3}{2} \\\\\\ 5^{-3}\cdot \cfrac{3}{2}\implies (5^{-1})^3\cdot \cfrac{3}{2}[/tex]
Other Questions
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