Express the trig ratios as fractions in simplest terms.
cOS K =
sin L=
cos K and sin L

The Converse of the Pythagorean Theorem is proved below.

What is Pythagoras Theorem ?

Pythagoras' theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The converse of the Pythagorean theorem states that if a triangle has sides of length a, b, and c, where c is the longest side, and a² + b² = c², then the triangle is a right triangle.

To prove the converse of the Pythagorean theorem, we can use the fact that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. That is, if a triangle has sides of length a, b, and c, where c is the longest side and c² = a² + b², then the triangle is a right triangle.

Proof:

Suppose we have a triangle ABC, with sides of length a, b, and c, where c is the longest side. Assume that a² + b² = c². We want to show that triangle ABC is a right triangle.

We will use the Law of Cosines to show that angle C is a right angle. The Law of Cosines states that for any triangle ABC with sides of length a, b, and c opposite angles A, B, and C, respectively:

c² = a² + b² - 2ab cos(C)

Substituting a² + b² = c² into this equation, we get:

c² = c² - 2ab cos(C)

2ab cos(C) = 0

cos(C) = 0

Since 0 is the cosine of 90 degrees, this implies that angle C is a right angle. Therefore, triangle ABC is a right triangle.

Thus, this completes the proof of the converse of the Pythagorean theorem.

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Using the pοlygοn tοοl a rectangle with length 5 unit and width 3 units has been drawn

A rectangle is a type οf quadrilateral that has its parallel sides equal tο each οther and all the fοur vertices are equal tο 90 degrees. Hence, it is alsο called an equiangular quadrilateral. Since, the οppοsite sides are equal and parallel, in rectangle, therefοre, it can alsο be termed as a parallelοgram.

Using the pοlygοn tοοl a rectangle with length 5 unit and width 3 units has been drawn

check the attachment given belοw tο understand.

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Answer:

183.3

Step-by-step explanation:

Since there were 1 angle and 2 sides, I used law of cosines

Answer: x=0

Step-by-step explanation:

Answer:

30.7

Step-by-step explanation:

sin(71) = 29/x

x = 29/sin71

≈ 30.7

Answer: We can simplify the expression step by step:

3/4 − x(1/2 − 5/8) + (−3/8)x

3/4 − x(-1/8) + (−3/8)x

3/4 + x/8 − 3/8x

Multiplying everything by 8 to get rid of the fractions:

6 − x + (−3x)

6 − 4x

Therefore, the equivalent expression is D) 34 - 14x / (4/3).

Step-by-step explanation:

Answer:

Step-by-step explanation:

[tex]x+y=34\\x-y=8\\2x=42\\x=21\\y=34-21\\y=13\\13(21)=273[/tex]

Approximately 60.68% of callers are on hold between 2.6 and 3.9 minutes.

What Is a Normal Distribution?

A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

First, we standardize the value of 2.6 as follows:

z = (x - μ) / σ

where x is the value of 2.6, μ is the mean of 3.7 minutes, and σ is the standard deviation of 0.6 minutes:

z = (2.6 - 3.7) / 0.6 ≈ -1.83

z = (x - μ) / σ

where x is the value of 3.9, μ is the mean of 3.7 minutes, and σ is the standard deviation of 0.6 minutes:

z = (3.9 - 3.7) / 0.6 ≈ 0.33

We can use a standard normal distribution table or a calculator to find this area. Using a calculator, we can find the area as follows:

P(-1.83 < Z < 0.33) ≈ 0.6068

Therefore, approximately 60.68% of callers are on hold between 2.6 and 3.9 minutes.

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Approximately 60.68% of callers are on hold between 2.6 and 3.9 minutes.

A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

First, we standardize the value of 2.6 as follows:

z = (x - μ) / σ

where x is the value of 2.6, μ is the mean of 3.7 minutes, and σ is the standard deviation of 0.6 minutes:

z = (2.6 - 3.7) / 0.6 ≈ -1.83

z = (x - μ) / σ

where x is the value of 3.9, μ is the mean of 3.7 minutes, and σ is the standard deviation of 0.6 minutes:

z = (3.9 - 3.7) / 0.6 ≈ 0.33

We can use a standard normal distribution table or a calculator to find this area. Using a calculator, we can find the area as follows:

P(-1.83 < Z < 0.33) ≈ 0.6068

Therefore, approximately 60.68% of callers are on hold between 2.6 and 3.9 minutes.

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AFE triangle,

FE= 6cos60

    = 3

dc= 3

ae= 6sin60

   = 3*squareroot3

Area= 1/2(10+4)*3*squareroot3

       = 7*3*squareroot3

        = 21*squareroot3

        = 21*1.73

         = 36.33square units

The true statement about the parallelogram is (c) Yes, because adjacent sides are congruent.

Given that

Figure ABCD is a parallelogram

Such that

Side lengths = 5 cm

This means that

The figure is a square

As a general rule

All squares can be classified as rhombus

This is because a rhombus is a quadrilateral with all four sides of equal length and square is a type of rhombus in which all four sides are equal in length

Hence, the true statement is (c)

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Answer:

28x + 40

Step-by-step explanation:

Since all four sides of a square are equal, we can make an equation:

[tex]4(7x + 10) = 28x + 40[/tex]

The value of tangent (90°) is 1, so the rise in distance for the submarine to pass over the sunken ship is 1500 m.

Value is the worth of something as measured by its usefulness, importance, or desirability. It is something that is sought after and appreciated by individuals and society. Value is subjective and can vary among people and cultures, but can be broadly defined as the quality of something that makes it worth having or doing.

To calculate the distance the submarine must rise to pass over the sunken ship, we must use the tangent function. The tangent of an angle is equal to the opposite side (in this case, the rise in distance for the submarine) divided by the adjacent side (in this case, the 1500 m distance to the sunken ship).

Therefore, the formula to calculate the rise in distance for the submarine is:
Tangent (90°) = Rise / 1500 m

Solving for the rise, we get:
Rise = 1500 m x Tangent (90°)

Using a calculator, the value of tangent (90°) is 1, so the rise in distance for the submarine to pass over the sunken ship is 1500 m.

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Because the tangent (90°) value is 1, the distance travelled by the submarine to cross over the sunken ship is increased by 1500 m.

Trigonometric angles are the angles in a right-angled triangle that can be used to illustrate various trigonometric functions. Standard angles in geometry include 0°, 30°, 45°, 60°, and 90°. The trigonometric functions are real functions that link the angle of a right-angled triangle to side length ratios. The sine, cosine, and tangent angles are the main classification of trigonometric functions. The primary functions can be used to determine the three functions cotangent, secant, and cosecant.

According to the question draw the graph we should solve the length of BC

tan A = BC/AC

tan90° = BC/1500

BC = 1500 tan90°

  = 516.5 (m)

The value of BC is 516.5 m

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Answer:

Step-by-step explanation:

104.4 divided by 4 = 26.1

a. The probability of rolling a number greater than 10 is 1/6.

b. The probability of rolling a number less than 5 is 1/3.

c. The expected number of times a 4, 6, or 9 will be rolled in 200 rolls is 50

a. The probability of rolling a number greater than 10 is equal to the number of faces with numbers greater than 10 (i.e., 11 and 12) divided by the total number of faces. Thus, P(number greater than 10) = 2/12 = 1/6.

b. The probability of rolling a number less than 5 is equal to the number of faces with numbers less than 5 (i.e., 1, 2, 3, and 4) divided by the total number of faces. Thus, P(number less than 5) = 4/12 = 1/3.

c. The probability of rolling a 4, 6, or 9 is equal to the number of faces with those numbers (i.e., 1 each) divided by the total number of faces. Thus, the probability of rolling a 4, 6, or 9 is 3/12 = 1/4.

Therefore, the expected number of times a 4, 6, or 9 will be rolled in 200 rolls is:

(expected number of times) = (probability of rolling a 4, 6, or 9) x (total number of rolls)

= (1/4) x (200)

= 50

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The tοtal price οf the game including tax is $29.96.

Physical assets such as prοperty and transactiοns such as the sale οf stοck οr a hοme are subject tο taxatiοn. Taxes include incοme, cοrpοrate, capital gains, prοperty, inheritance, and sales taxes.

Because the tax rate is 7%, the tax amοunt is 7% οf the purchase price. Tο calculate the tax, multiply the purchase price by the tax rate expressed as a decimal:

0.07 x $28 = $1.96 in taxes

As a result, the purchase tax is $1.96.

Tο calculate the tοtal price, add the purchase price and the tax:

Purchase price + Tax = Tοtal Price

Tοtal cοst: $28 + $1.96 = $29.96

As a result, the tοtal cοst οf the game, including tax, is $29.96.

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Answer:

146.82 yards is the distance from tee to hole.

Step-by-step explanation:

Let A is the position of tee and where the golfer is standing.

C is the position of hole.

CB is the vertical distance between greens and tee.

Angle of elevation,

To find: side AC.

Using trigonometric functions, we know that value of sine is:

[tex]sin\theta=\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}[/tex]

Here [tex]\theta=\angle BAC=15^\circ[/tex]

Perpendicular is side CB.

Hypotenuse is side AC.

[tex]\text{sin}12^\circ=\dfrac{\text{CB}}{\text{AC}}[/tex]

[tex]\implies\text{CB}=\dfrac{38}{\text{sin}15^\circ}[/tex]

[tex]\implies\text{CB}=146.82[/tex]

Hence, 146.82 yards is the distance from tee to hole.

Answer:

44 students

Step-by-step explanation:

1200 for the first year

a third of that is 400

400 for the second year

a third of that is 133.4

133.4 for the third year

a third of that is 44

(a) The value of sin 2θ is -3√(91)/50.

(b)  θ/2 is located in the third quadrant.

(c) The value of cos θ/2 is -√65/10.

We know that cos θ = 3/10, so we can find sin θ using the Pythagorean identity:

sin² θ + cos² θ = 1

sin²θ + (3/10)² = 1

sin²θ = 1 - (3/10)² = 91/100

sin θ = ±√(91)/10

Since 3π/2 < θ < 2π,

we know that sin θ < 0, so sin θ = -√(91)/10.

a) To find sin 2θ, we can use the double angle formula:

sin 2θ = 2 sin θ cos θ

sin 2θ = 2 (-√(91)/10) (3/10)

sin 2θ = -3√(91)/50

b) To find the quadrant in which θ/2 is located, we first need to find θ/2:

θ/2 = (3π/2 + 2π)/2 = 5π/4

5π/4 is in the third quadrant, so θ/2 is located in the third quadrant.

c) To find cos θ/2, we can use the half angle formula:

cos θ/2 = ±√((1 + cos θ)/2)

Since 3π/2 < θ < 2π, we know that cos θ < 0, so we take the negative square root:

cos θ/2 = -√((1 + 3/10)/2)

cos θ/2 = -√(13/20)

Simplifying the radical by dividing both numerator and denominator by 4:

cos θ/2 = -√(13)/2√5

Multiplying numerator and denominator by √5 to rationalize the denominator:

cos θ/2 = -√65/10

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Answer:

The percent decrease in population can be calculated by dividing the difference between the two populations by the original population and then multiplying by 100. In this case, the percent decrease is ((137000-117820)/137000)*100 = 14%.

Step-by-step explanation:

Answer:

3x² -2xy + 4y² at x = -1, y = -2  = 15

Step-by-step explanation:

Given function is

f(x, y) = 3x² -2xy + 4y²

To find At x = - 1 and y - 2 just plug these values for x and y into f(x, y)

f(-1, -2) = 3(-1)² - 2(-1)(-2) + 4(-2)²

(-1)² = 1 ==>   3x²  = 3(-1)² = 3 x 1 = 3

(-1) (-2) = 2   ==>   2xy = 2(-1)(-2) = 2 x 2 = 4

(-2)² = 4  ==> 4y² = 4 (-2)² = 4 x 4 = 16

f(-1 1, - 2) = 3x² -2xy + 4y²
= 3 - 4 + 16

= 15

The minimum amount of money to be deposited in the bank account is $16,921. The solution has been obtained by using compound interest.

What is compound interest?

Compound interest is the type of interest that is calculated using both the principal and the interest that has accumulated over the preceding period. In contrast to simple interest, compound interest does not factor in the principal when calculating the interest for a subsequent period. In mathematics, compound interest is frequently referred to as C.I.

We are given that we need $25,000 eight years from now and the account pays an annual percentage rate (APR) of 5% which is compounded annually.

So, using the compound interest formula, we get the principal as

⇒ P = [tex]\frac{A}{(1 + \frac{r}{n} )^{nt} }[/tex]

⇒ P = [tex]\frac{25000}{(1 + \frac{0.05}{1} )^{8} }[/tex]

⇒ P = $16,920.98

Hence, the minimum amount of money to be deposited in the bank account is $16,921.

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Answer:

1/2

Step-by-step explanation:

20/√5 = 4√5

4√5 * (5√2 / 4√5) = 5√2

8 / 4 = 2

4 / (2^3) = 1/2

|1/2| = 1/2

The first term of this sequence is equal to 1/2

Suppose the initial term of a geometric sequence is a and the term by which we multiply the previous term to get the next term is r

Then the sequence would look like

[tex]a, ar, ar^2, ar^3, \cdots[/tex]

Thus, the nth term of such sequence would be; [tex]T_n = ar^{n-1}[/tex](you can easily predict this formula, as for nth term, the multiple r would've multiplied with initial terms n-1 times).

We are given that The 4th term of a different geometric sequence is

4, The 6th term of this sequence is 5√2

20/√5 = 4√5

4√5 (5√2 / 4√5) = 5√2

8 / 4 = 2

Therefore,

4 / (2^3) = 1/2

|1/2| = 1/2

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Answer:  114,510,000

Delete the commas if needed.

============================================

Explanation:

The given number is 114,509,772

The 0 is in the ten thousandths place value. Just to the right of it is 9, which fits the criteria of "5 or larger". So we'll bump the "509" up to "510". Everything to the right of 9 will be replaced with zeros.

That's how we go from 114,509,772 to 114,510,000

-------------

Put another way:

Temporarily ignore the 114 up front and focus on the remaining portion 509,772

This number can be verbalized as "509 thousand" in an approximate sense. The 772 at the end means it's much closer to 510 thousand than it is to 509 thousand. That's why the 509 bumps up to 510.

Therefore, we have 509,772 round to 510,000

Overall, we have 114,509,772 round to 114,510,000

Answer: yes correct i love math

Step-by-step explanation:

Answer:

hope its right

Step-by-step explanation:

Answer:

To determine the lowest amount that agent C could have earned in 2014, we need to know the number of tanks that agent C sold in 2014 and the commission rate per tank.

Let's assume that the commission rate per tank for agent C is R x. We need to find the lowest value of R that would result in the agent earning a minimum commission of R713.76 per tank.

We know that the commission earned per tank is the product of the commission rate and the selling price of the tank. Let's assume that the selling price of each tank is SP.

So, the commission earned per tank is R x SP.

We also know that the minimum commission per tank is R713.76. Therefore,

R x SP >= R713.76

Solving for R, we get:

R >= R713.76 / SP

This means that the commission rate per tank must be at least R713.76 / SP in order for the agent to earn a minimum commission of R713.76 per tank.

Now, let's assume that agent C sold a total of T tanks in 2014. The total commission earned by agent C in 2014 would be:

Total Commission = R x SP x T

From the above equation, we can see that the total commission earned by agent C depends on the commission rate per tank, the selling price per tank, and the number of tanks sold.

To find the lowest amount that agent C could have earned in 2014, we need to minimize the total commission earned by agent C subject to the constraint that the commission rate per tank is at least R713.76 / SP.

Assuming that the selling price per tank is fixed, the lowest amount that agent C could have earned in 2014 would be when the commission rate per tank is equal to R713.76 / SP. In this case, the total commission earned by agent C would be:

Total Commission = R713.76 x T

Therefore, the lowest amount that agent C could have earned in 2014 is R713.76 x T, where T is the number of tanks sold by agent C in 2014.

We use the arctangent function to find the angle [tex]theta[/tex]

[tex]\theta= 2(3 \sqrt{2},-3 \sqrt{2}) \approx 225.000^{\circ}[/tex]

Inverse tangent (notation: arctan or tan-1) is the inverse of the tangent. That is: Arctangent and tangent are inverse functions, so their composition is an identity. 

To find [tex]z_{-} 1+z_{-} 2[/tex], we add the complex numbers in rectangular form:

[tex]$$z_1+z_2=6\left(\cos 55^{\circ}+i \sin 55^{\circ}\right)+6\left(\cos 135^{\circ}+i \sin 135^{\circ}\right)$$[/tex]

Simplifying, we get:

[tex]$$z_1+z_2=6\left(\cos 55^{\circ}+\cos 135^{\circ}+i \sin 55^{\circ}+\sin 135^{\circ}\right)$$[/tex]

Using the trigonometric identities [tex]\backslash \cos (\backslash p i-\backslash$ theta) =- $\backslash \cos \backslash$ theta\$ and $\$ \backslash \sin (\backslash p i-$ $\backslash$ theta) $=\backslash \sin \backslash$ theta[/tex], we can simplify this to:

[tex]$$z_1+z_2=6\left(-\frac{\sqrt{2}}{2}+i \frac{\sqrt{2}}{2}\right)=-3 \sqrt{2}+3 i \sqrt{2}$$[/tex]

Therefore, [tex]z_{-} 1+z_{-} 2=-3 \backslash s q r t\{2\}+3 i \backslash s q r t\{2\}[/tex]

To find [tex]r[/tex] and [tex]theta[/tex] such that [tex]z_{-} 1+z_{-} 2=$ rlangle $\backslash$ theta[/tex], we first calculate [tex]r[/tex] as the magnitude of [tex]z_{-} 1+z_{-} 2[/tex] :

[tex]$$|r+s i|=\left|z_1+z_2\right|=\sqrt{(-3 \sqrt{2})^2+(3 \sqrt{2})^2}=3 \sqrt{2} .$$[/tex]

Next, we use the arctangent function to find the angle [tex]theta[/tex]

[tex]\theta= 2(3 \sqrt{2},-3 \sqrt{2}) \approx 225.000^{\circ}[/tex]

Hence, we can say the values of both the z_{-} 1+z_{-} 2 and r is [tex]z_{-} 1+z_{-} 2=-3 \backslash s q r t\{2\}+3 i \backslash s q r t\{2\}[/tex] and [tex]\theta= 2(3 \sqrt{2},-3 \sqrt{2}) \approx 225.000^{\circ}[/tex] respectively.

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Answer:

27/70

Step-by-step explanation:

To find an area you have to multiply [tex]length*width[/tex] In this problem, the length of the striped rectangle is 3/7  and the width is 9/10. So you would multiply them together. After multiplying you get the answer of 27/70, most likely Khan Academy will ask for a simple answer so the answer is 27/70 (it is already in its simplest form)