What is the length of PR?

The trigonometric function gives the ratio of different sides of a right-angle triangle. The given problems can be solved as given below.

The trigonometric function gives the ratio of different sides of a right-angle triangle.

[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}\\\\\\Cosec \theta=\dfrac{Hypotenuse}{Perpendicular}\\\\\\Sec \theta=\dfrac{Hypotenuse}{Base}\\\\\\Cot \theta=\dfrac{Base}{Perpendicular}\\\\\\[/tex]

where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.

1st.) x = 5 /Sin(30°)

x = 10

!) sin(45°) = 4/x

x = 4/sin(45°)

x = 4√2

I) Cos(45°) = √3 / x

x = √3 / Cos(45°)

x = √6

E) Tan(60°) = 3√3 / x

x = 3√3 / 3

W) For isosceles right-triangle, the angle made by the legs and the hypotenuse is always 45°.

x = 45°

N) x² + x² = (7√2)²

x = 7

V) Tan(60°) = 7 / x

x = 7√3/3

K) x² + x² = (9)²

x = 9/√2

Y) Sin(60°) = 7√3/x

x = 14

M) Sin(30°) = x/11

x = 11/2

T) Sin(45°) = x/√10

x = √5

A) x + 2x + 90° = 180°

x = 30°

O) Sin(45°) = √2 / x

x = 2

R) Tan(30°) = x / 4

x = 4/√3 = 4√3 / 3

S) Sin(60°) = x / (10/3)

x = 5√3 / 3

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The equation of the line passing through the points is y = x + 2

The points are (-1, 1) and (4, 6).

The slope is calculated using:

m= (y2 - y1)/(x2 - x1)

So, we have:

m = (6 - 1)/(4 + 1)

Evaluate

m = 1

The equation of the line is then calculated as:

y = m(x -x1) + y1

This gives

y = 1 *(x + 1)  + 1

Evaluate

y = x + 2

Hence, the equation of the line passing through the points is y = x + 2

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[tex]\therefore \sf{ LHS = \dfrac{tan \theta}{1 – cot \theta } + \dfrac{cot \theta}{1 – tan \theta}}[/tex]

[tex]\\[/tex]

[tex]\therefore\sf{ LHS = \dfrac{tan \theta}{1 – \dfrac{1}{tan \theta}} + \dfrac{ \dfrac{1}{tan \theta}}{1 – tan \theta} }[/tex]

[tex]\\[/tex]

[tex]\therefore\sf{ LHS = \dfrac{ tan^{2} \theta }{tan \theta – 1} + \dfrac{1}{tan \theta (1–tan \theta) } }[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{tan^{2} \theta}{tan \theta–1} – \dfrac{1}{tan \theta(tan \theta – 1 )}}[/tex]

[tex]\\[/tex]

[tex]\therefore \sf{LHS = \dfrac{tan^{3}\theta–1}{tan \theta(tan \theta –1)}}[/tex]

[tex]\\[/tex]

[tex]\therefore\sf{LHS = \dfrac{tan^{3}\theta–(1)^{3}}{tan\theta(tan\theta–1)}}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{ (tan \theta -1)(tan^{2}\theta+tan\theta .1 + (1)^{2})} {tan \theta (tan\theta(tan \theta –1 )}}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{(tan^{2} \theta×tan\theta .1+1)}{tan \theta}}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{(sec^{2}\theta+tan\theta)}{tan\theta}}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{sec^{2}\theta}{tan\theta} + \dfrac{tan \theta}{tan\theta}}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = sec^{2}\theta \times cot \theta + 1 }[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \dfrac{1}{cos^{2}\theta} \times \dfrac{cos \theta }{sin\theta}+1}[/tex]

[tex]\\[/tex]

[tex] \therefore\sf{ LHS = \left( \dfrac{ 1}{cos \theta} \times \dfrac{1}{sin \theta} \right) + 1}[/tex]

We have proven that the trigonometric identity [(tan θ)/(1 - cot θ)] + [(cot θ)/(1 - tan θ)] equals 1 + (secθ * cosec θ)

We want to prove the trigonometric identity;

[(tan θ)/(1 - cot θ)] + [(cot θ)/(1 - tan θ)] = 1 + sec θ

The left hand side can be expressed as;

[(tan θ)/(1 - (1/tan θ)] + [(1/tan θ)/(1 - tan θ)]

⇒ [tan²θ/(tanθ - 1)] - [1/(tan θ(tanθ - 1)]

Taking the LCM and multiplying gives;

(tan³θ - 1)/(tanθ(tanθ - 1))

This can also be expressed as;

(tan³θ - 1³)/(tanθ(tanθ - 1))

By expansion of algebra this gives;

[(tanθ - 1)(tan²θ + tanθ.1 + 1²)]/[tanθ(tanθ(tanθ - 1))]

Solving Further gives;

(sec²θ + tanθ)/tanθ

⇒ sec²θ * cotθ + 1

⇒ (1/cos²θ * cos θ/sin θ) + 1

⇒ (1/cos θ * 1/sin θ) + 1

⇒ 1 + (secθ * cosec θ)

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The answer choice which represents a solution to the system of equations is; (–2, 10).

The system of equations as given in the task content is;

-6x - (2/5)y = 8

(1/2)x +3y = 29

Hence, by making x subject in equation 1 and substitution; we have;

x = (-0.4y -8)/6

Hence; (0.4y -8)/12 + 3y = 29.

(0.4y -8) + 36y = 348

35.6y = 356

y = 10

and x = (-0.4(10) -8)/6 = -2.

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The ordered pair that is on g(x) is (2, -3/400)

determine the ordered pair:

The function is given as:

f(x)=3/4(10)∧-x

The rule of reflection across the x-axis is:

g(x) = -f(x)

So, we have:

g(x) = -3/4(10)∧-x

Set x = 2.

So, we have:

g(2) = -3/4(10)∧-2

Evaluate the exponent

g(2) = -3/4 * 1/100

Evaluate the product

g(2) = -3/400

This means that:

(x, y) = (2, -3/400)

Hence, the ordered pair that is on g(x) is (2, -3/400)

Ordered pairs are pairs of two numbers (or variables) that are enclosed in parentheses and separated by commas. For example, (1, 2) is an ordered pair. Coordinate geometry represents points, and set theory represents elements of relations / Cartesian products.

Ordered pairs are pairs of numbers in a particular order. For example, (1, 2) and (-4, 12) are ordered pairs. The order of the two numbers is important. (1, 2) is not equivalent to (2, 1)-(1, 2) ≠ (2, 1).

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Answer:

I think it's 87.5 or 87½

Step-by-step explanation:

I hope it helps you.

By the converse of the basic proportionality theorem, we can say DE║AB.

We need to check whether DE is parallel to AB.

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

In triangle ABC, we can see on side AC, D is midpoint (∵AD=DC) and on side BC, E is midpoint (∵BE=CE).

Therefore, by the converse of the basic proportionality theorem, we can say DE║AB.

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The proof is shown below:

The line segment joining the mid points of any two sides of a triangle is parallel to the third side.

△ABC in which D and E are the mid points of AB and AC,.

If D and E are the mid points of AB and AC,

then AD=DB and AE=EC.

Thus,

AD/DB = AE/ EC

​

by the converse of Thales theorem, DE∥BC.

As, by theorem we can say that DE∥BC.

So, there is no other side parallel to DE.

​

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The number of people coming in during the lunch hour is 160

The complete question is in the attached image

The random numbers are given as:

18, 38, 87, 16, 56, and 5

Regroup the numbers

Number           Group       Probability

5, 16, 18           1 - 20             3/6

38, 56             21 - 60           2/6

56                    61 - 90          1/6

Based on the individual numbers and the probability table, we have:

X      20      30     40

P     3/6     2/6    1/6

The expected value is:

[tex]E(x) = \sum xp[/tex]

This gives

E(x) = 20 * 3/6 + 30 * 2/6 * 40 * 1/6

Evaluate

E(x) = 160/6

There are six ten-minute intervals.

So, we have:

People = 6 * 160/6

Evaluate

People = 160

Hence, the number of people coming in during the lunch hour is 160

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The possible values is  θ = π/4 and 5π/4

The branch of mathematics concerned with specific functions of angles and their application to calculations.

Given:

sin θ = cos θ

The above condition satisfied for θ = π/4 and 5π/4

sin π/4 = cos π/4 = 1/√2

sin  5π/4 = cos 5π/4 = -1/√2

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The value of m is 14.5 and the three angles of the triangle are 32.5°, 57°, and 90.5°.

There are three angles in a triangle. Consider as ∠A, ∠B, and ∠C.

The sum of all the three angles gives 180° in a triangle. I.e.,

∠A + ∠B + ∠C = 180°.

The given triangle has three angles,

(m + 18)° , (2m + 28)°, and (3m + 47)°

On adding all these and equating to 180°,

(m + 18)° + (2m + 28)° + (3m + 47)° = 180°

⇒ 6m + 93 = 180

⇒ 6m = 180 - 93

⇒ 6m = 87

∴ m = 14.5

Then, the values of the three angles are:

(m + 18)° = (14.5 + 18)° = 32.5°

(2m + 28)° = (2 × 14.5 + 28)° = 57°

(3m + 47)° = (3 × 14.5 + 47)° = 90.5°

Therefore, the angles of the given triangle are 32.5°, 57°, and 90.5°.

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Similar triangles are those triangles whose corresponding sides are in the same ratio. The triangles that are similar to ΔABC are ΔDEF and ΔJKL.

Similar triangles are those triangles whose corresponding sides are in the same ratio. And the corresponding angles measure the same.

For two triangles to be similar there sides should be in the same ratio.

For ΔABC ~ ΔMNO,

AB/MN should be equal to BC/NO,

AB/MN = 15/35 = 0.4286

BC/NO = 8/12 = 0.66

Thus, there are not similar triangles.

For ΔABC ~ ΔDEF,

AB/MN should be equal to BC/NO,

AB/DE = 15/22.5 = 0.66

BC/EF = 8/12 = 0.66

Thus, there are similar triangles.

For ΔABC ~ ΔGHI,

AB/GH should be equal to BC/HI,

AB/MN = 15/77 = 0.1948

BC/NO = 8/36 = 0.222

Thus, there are not similar triangles.

For ΔABC ~ ΔJKL,

AB/MN should be equal to BC/NO,

AB/JK = 15/60 = 0.250.25

BC/KL = 8/32 = 0.66

Thus, there are similar triangles.

Hence, the triangles that are similar to ΔABC are ΔDEF and ΔJKL.

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Answer:

9x - 2

Step-by-step explanation:

f(x) = 4x - 8

g(x) = 5x + 6

because it's (f + g)(x), we can simply add the functions together

{note: don't get thrown off by the (x) being separate parentheses--distributing the x first can help it look familiar:

(x)(f + g)

fx + gx,

it's the same thing}

(4x - 8) + (5x + 6)

here, we want to combine like terms to simplify:

4x + 5x - 8 + 6

4x + 5x  -   2

9x - 2

So, (f + g)(x) = 9x - 2

hope this helps!! have a lovely day :)

Answer:

26.6 cubic yards

Step-by-step explanation:

Remark

What you have is a rectangle prism. It has an easy enough formula.

Formula

V = L *w * h

Givens

L = 10 yards

w =  8 yards

h = 1/3 yards

Solution

V = L*w*h

V = 10 * 8 * 1/3

V = 26.66 yds^3

The computation shows that the total number of chocolate bars when there are 280 orange flavored bars will be

The total number of chocolate bars when there are 280 orange flavored bars will be:

= Coconut + Honeycomb + Coffee + Orange + Strawberry

= (4/6 × 280) + (5/6 × 280) + (1/6 × 280) + 280 + (4/6 × 280)

= 1033

The total number when there are 960 coconut bars will be:

= Coconut + Honeycomb + Coffee + Orange + Strawberry

= 960 + (5/4 × 960) + (1/6 × 960) + (6/4 × 960) + 960

= 960 + 1200 + 160 + 1440 + 960

= 4720

The mean of 7, 2, and 9 will be:

= (7 + 2 + 9)/3

= 6

The mean of 5, 6, 6, and 5 will be:

= (5 + 6 + 6 + 5)/4

= 22/4

= 5.5

The mean of 2, 1, 2, 3, 3, 1 will be:

= (2 + 1 + 2 + 3 + 3 + 1) / 6

= 15/6

= 2.5

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The Area of triangle is A= 58.02 (ft)^2

According to statement

Let

B is the measure of the vertex angle of the isosceles triangle ABC

A and C is the two congruent sides of the isosceles triangle ABC

we know that

The area of a triangle applying the law of sines is equal to the

A= 1/2 *AC *SinB

we have

A=C= 17.91 ft

And

B= 21.21

Substitute these values in the formula

A= 1/2* 17.91* 17.91 * Sin(21.21)

A= 58.02 (ft)^2

So, the Area of triangle is A= 58.02 (ft)^2

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Step-by-step explanation:

[tex]sin2x=sinx\\sin2x-sinx=0\\2*sinx*cosx-sinx=0\\sinx*(2*cosx-1)-0\\sinx=0 \\x=\pi n\ \ \ \ \ x\in[0;2\pi ]\ \ \ \ \Rightarrow\\x_1=0\ \ \ \ x_2=\pi .\\2*cosx-1=0\\2*cosx=1\ |:2\\cosx=\frac{1}{2} \\x=б\frac{\pi }{3} +2\pi n\ \ \ \ \ x\in[0;2\pi ]\ \ \ \ \Rightarrow\\x_3=\frac{\pi }{3} \ \ \ \ \ x_4=\frac{5\pi }{3}.\\[/tex]

[tex]Answer: x=0,\ \frac{\pi }{3} ,\ 1\pi ,\ \frac{5\pi }{3} .[/tex]

Answer:

4.09 seconds

Step-by-step explanation:

simplify miles per hour to feet per second how many feet is a mile and how many seconds are in an hour you end up with 88 fps from 60 mph

now simply a football field is 360 feet so 360/88 is 4.09090909 etc

Your eyes off the road for 4.09 seconds.

The rate of change of position of an object in any direction is called the speed.

Given that, Taking your eyes off the road for how many seconds at 60 miles per hour means you have travelled blindly for half the length of a football field,

Converting 60 mph into fps

60 mph = 88 fps

The length of a football field = 360 ft

Time = 350/88

= 4.09 seconds.

Hence, you travelled for 4.09 seconds.

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yd=

yd=

16x2+12x+18=3x=2x2+6x+9=3x=

sorry i try

Answer:

D

Step-by-step explanation:

Answer:

v = 2u

Step-by-step explanation:

note that

2u = 2 (5, - 2 ) ← multiply each component by 2

    = (10, - 4) = v , then

v = 2u

Answer:

415

Step-by-step explanation:

20x20=400+15= 415

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\mathbf{20^2 + 15^1}[/tex]

[tex]\huge\textbf{Solving:}[/tex]

[tex]\mathbf{20^2 + 15^1}[/tex]

[tex]\mathbf{= 20\times 20 + 15\times 1}[/tex]

[tex]\mathbf{= 400 + 15}[/tex]

[tex]\mathbf{= 415}[/tex]

[tex]\huge\textbf{Answer:}[/tex]

[tex]\huge\boxed{\mathsf{415}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

~[tex]\frak{Amphitrite1040:)}[/tex]

The probability of both events occurring is  4/15

The given parameters are:

P(A) = 4/5

P(B) = 1/3

Because the events are independent, we have:

p(A and B) = P(A) * P(B)

This gives

P(A and B) = 4/5 * 1/3

Evaluate

P(A and B) = 4/15

Hence, the probability of both events occurring is  4/15

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Answer:

1066

Step-by-step explanation:

5 mins = 410 words

410/5 = 82 so she is typing 82 words a minute

Then do 82 times 13

The volume of the rectangular prism is 1/10 cubic units

The formula for calculating the volume of a rectangular prism is expressed as;

V = Bh

B is the base area

h is the height

Given the following parameters

B = 3/20 units²

h = 2/3 units

Substitute

V = 3/20 * 2/3

V = 2/20

V = 1/10 cubic units

Hence the volume of the rectangular prism is V = 1/10 cubic units

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Answer:8 1/2^3

Step-by-step explanation:

Answer: 28.40 cans of soda

Step-by-step explanation:

Volume of cylinder: pi * r^2 * h

Volume of the can of soda = pi * 3.25^2 * 12 = 126.75 pi

Volume of the cylindrical pitcher = pi * 10^2 * 36 = 3600 pi

3600/126.75: 28.40 cans of soda

Answer:

28 whole soda cans or 28.4 soda cans (rounding to nearest tenth)

Step-by-step explanation:

The soda cans and the pitcher can be modeled as cylinders.

Volume of a cylinder

  [tex]\sf Volume=\pi r^2h[/tex]

where:

Diameter of circle

[tex]\sf d= 2r[/tex]

[tex]\sf \implies r=\dfrac{1}{2}d[/tex]

Volume of Soda can

Given values:

[tex]\begin{aligned}\sf \implies Volume & = \sf \pi (3.25)^2 \cdot 12\\ & = \sf 126.75 \pi \:\:cm^3\end{aligned}[/tex]

Volume of Pitcher

Given values:

[tex]\begin{aligned}\sf \implies Volume & = \sf \pi (10)^2 \cdot 36\\ & = \sf 3600 \pi \:\:cm^3\end{aligned}[/tex]

To calculate how many cans of soda the pitcher will hold, divide the volume of the pitcher by the volume of one soda can:

[tex]\begin{aligned}\implies \textsf{Number of soda cans} & = \dfrac{\textsf{Volume of Pitcher}}{\textsf{Volume of one soda can}}\\\\&= \sf\dfrac{3600 \pi}{126.75 \pi}\\\\& = \sf 28.402366...\end{aligned}[/tex]

Therefore, the pitcher can hold 28 soda cans (nearest whole can), or 28.4 soda cans (nearest tenth).

Slope of AB = 4/3, Slope of BC = -2/7, Slope of CD = 4/3, Slope of AD = -2/7 and Quadrilateral ABCD is a parallelogram because both pairs of opposite sides are parallel.

We are given the coordinates of the quadrilateral ABCD as;

A(−5, 1), B(−2, 5), C(5, 3), and D(2, −1).

Slope of AB = (5 - 1)/(-2 + 5) = 4/3

Slope of BC = (3 - 5)/(5 + 2) = -2/7

Slope of CD = (-1 - 3)/(2 - 5) = 4/3

Slope of AD = (-1 - 1)/(2 + 5) = -2/7

Thus we can see that both pairs of slope are equal which denotes that both pairs are parallel.

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7

Step-by-step explanation:

p = 1, q = 72 =49, q = 7

ookrndodnepwkpqq

[tex]\huge\textbf{Hey there!}[/tex]

[tex]\mathbf{-3pq^2 + q^3}[/tex]

[tex]\mathbf{= -3(2.9)(-1.3)^2 + (-1.3)^3}[/tex]

[tex]\mathbf{= -3(2.9)(-1.3)^2 - 1.3^3}[/tex]

[tex]\mathbf{= -8.7(-1.3)^2 - 1.3^3}[/tex]

[tex]\mathbf{= -8.7(1.69) - 2.197}[/tex]

=[tex]\mathbf{ -14.703 -2.197}[/tex]

[tex]\mathbf{= -16.9}[/tex]

[tex]\huge\textbf{Thus, your answer should be: \boxed{\mathsf{-16.9}}}}\huge\checkmark[/tex]

[tex]\huge\textbf{Good luck on your assignment \& enjoy}\\\huge\textbf{your day!}[/tex]

Answer:

110

Step-by-step explanation:

Sally takes $110 from her bank account if she spent $54.00 for a round-trip ticket.

It is defined as the money spends on the utility, the amount of money is required to buy something, in other words, it is the outflow of money from the sole earner's income or the money incurred by any organization.

We have:

Sally took money from her bank account to go out of town for an audition.

The expeses = (7.90 + 15.10) ÷ (1 - 1/2) = $46

Total money = 54 + 46

Total money = $110

Thus, Sally takes $110 from her bank account if she spent $54.00 on a round-trip ticket.

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Answer:

c. x=8

Step-by-step explanation:

make me a brain list.