1. Prove the following identity:
—> sin^2 theta (1+ 1/tan^2 theta) =1

Answers

Answer 1

9514 1404 393

Explanation:

  [tex]\sin^2(\theta)\times\left(1+\dfrac{1}{\tan^2(\theta)}\right)=\\\\\sin^2(\theta)\times\left(1+\dfrac{\cos^2(\theta)}{\sin^2(\theta)}\right)=\\\\\dfrac{\sin^2(\theta)\cdot(\cos^2(\theta)+\sin^2(\theta))}{\sin^2(\theta)}=\\\\\cos^2(\theta)+\sin^2(\theta)=1\qquad\text{Q.E.D.}[/tex]


Related Questions

After how many years, to the nearest whole year, will an investment of $100,000 compounded quarterly at 4% be worth
$213,022?
Provide your answer below:

Answers

9514 1404 393

Answer:

  19 years

Step-by-step explanation:

The compound interest formula tells you the future value of principal P invested at annual rate r compounded n times per year for t years is ...

  A = P(1 +r/n)^(nt)

Solving for t, we get ...

  t = log(A/P)/(n·log(1 +r/n))

Using the given values, we find t to be ...

  t = log(2.13022)/(4·log(1 +0.04/4)) ≈ 19.000

The investment will be worth $213,022 after 19 years.

A car insurance company has determined that6% of all drivers were involved in a car accident last year. If14drivers are randomly selected, what is the probability of getting at most 3 who were involved in a car accidentlast year

Answers

Answer:

[tex]P(x \le 3) = 0.9920[/tex]

Step-by-step explanation:

Given

[tex]p = 6\%[/tex] --- proportion of drivers that had accident

[tex]n = 14[/tex] -- selected drivers

Required

[tex]P(x \le 3)[/tex]

The question is an illustration of binomial probability, and it is calculated using:

[tex]P(x ) = ^nC_x * p^x * (1 - p)^{n-x}[/tex]

So, we have:

[tex]P(x \le 3) = P(x = 0) +P(x = 1) +P(x = 2) +P(x = 3)[/tex]

[tex]P(x=0 ) = ^{14}C_0 * (6\%)^0 * (1 - 6\%)^{14-0} = 0.42052319017[/tex]

[tex]P(x=1 ) = ^{14}C_1 * (6\%)^1 * (1 - 6\%)^{14-1} = 0.37578668057[/tex]

[tex]P(x=2 ) = ^{14}C_2 * (6\%)^2 * (1 - 6\%)^{14-2} = 0.15591149513[/tex]

[tex]P(x=3 ) = ^{14}C_3 * (6\%)^3 * (1 - 6\%)^{14-3} = 0.03980719024[/tex]

So, we have:

[tex]P(x \le 3) = 0.42052319017+0.37578668057+0.15591149513+0.03980719024[/tex]

[tex]P(x \le 3) = 0.99202855611[/tex]

[tex]P(x \le 3) = 0.9920[/tex] -- approximated

A popular beach erodes 4 inches per year on average.

An eroding beach.
A. How many years will it take for the coastline to erode one foot?

Answers

Answer:

3 years

Step-by-step explanation:

4 inches per year on average

1 foot = 12 inches

12 divided by 4 equals 3

therefore it is 3 years

Which two shapes make up the digital camera below?

Answers

Rectangular prism and cylinder

Rectangular prism and cylinder make up a camera.

What is rectangular prism and cylinder?

A cube is a rectangular prism with all of its sides being the same length, a triangular prism has a triangle as its base, and a rectangular prism has a rectangle as its foundation. Another form of right prism that has a circle as its basis is a cylinder.

A rectangular prism includes a total of 6 faces, 12 sides, and eight vertices. Like a cuboid, it contains three dimensions- the base width, the height, and the length. The top and base of the rectangular prism exist rectangular. The pairs of opposite faces of a rectangular prism exist as identical or congruent.

A cylinder contains traditionally been a three-dimensional solid, one of the most essential curvilinear geometric shapes. Geometrically, it includes been regarded as a prism with a circle as its base.

Hence, Rectangular prism and cylinder make up a camera.

To learn more about rectangular prisms and cylinders refer to:

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prove that 2^n+1>(n+2).sin(n)​

Answers

Step-by-step explanation:

F(n)=|sin(n)|+|sin(n+1)|

then

F(n+π)=|sin(n+π)|+|sin(n+π+1)|=|sin(n)|+|sin(n+1)|=F(n)

and

F(π−n)=|sin(π−n)|+|sin(π−n+1)|=|sinn|+|sin(n−1)|≠F(n)

so we must prove when n∈(0,π), have

F(n)>2sin12

when n∈(0,π−1),then

F(n)=sinn+sin(n+1)=sinn(1+cos1)+sin1cosn

and n∈(π−1,π),then

F(n)=sinn−sin(n+1)

How prove it this two case have F(n)>2sin12? Thank you

and I know this well know inequality

|sinx|+|sin(x+1)|+|sin(x−1)|≥2sin1,x∈R

Give example to verify if the given statement is true or false
1) if two numbers are co-primes , at least one of them should be prime number.

Answers

Answer:

no

Step-by-step explanation:

if two numbers are co-prime that is not necessary that one of them must be a prime number

five brothers of 4, 9, 11, 13 and 16 years respectively, receive an inheritance of 1,500,000, the will stipulated that that amount must be shared by the heirs so that, placed the shares in a bank, they would result in equal capitalized amounts, when each one reached 21, could raise his share. Knowing that the bank charges an interest rate of 9% per year, what is the amount of each share?

Answers

9514 1404 393

Answer:

Youngest to oldest:

160,406.86246,805.83293,230.01348,386.58451,170.72

Step-by-step explanation:

At 9% interest per year, the present value of 1 at age 20 is ...

  p(a) = 1.09^(a-20)

Adding the present values for the different ages, we get a total of about 2.35528984846. Dividing the inheritance by that amount gives the multiplier for each of the present value numbers. The result is the list of shares shown above. At age 20, each brother will inherit about 636,864.29.

__

Additional comment

This is the sort of question that suggests the use of a graphing calculator or spreadsheet for doing the tedious number crunching.

(We assume the bank pays 9% per year, rather than charges 9% per year.)

The polygons in each pair are similar. Find the missing side length.

Answers

Let missing side be x

If both polygons are similar

[tex]\\ \sf\longmapsto \dfrac{3}{4}=\dfrac{18}{x}[/tex]

[tex]\\ \sf\longmapsto 3x=4(18)[/tex]

[tex]\\ \sf\longmapsto 3x=72[/tex]

[tex]\\ \sf\longmapsto x=\dfrac{72}{3}[/tex]

[tex]\\ \sf\longmapsto x=24[/tex]

12/1,000 into decimal​

Answers

0.012 is the answer!

I hope this helps you out! :D

[tex]\\ \sf\longmapsto \dfrac{12}{1000}[/tex]

1000 has 3zeros hence decimal will go 3 points left

[tex]\\ \sf\longmapsto 0.012[/tex]

More:-

[tex]\\ \sf\longmapsto \dfrac{1}{10}=0.1[/tex]

[tex]\\ \sf\longmapsto \dfrac{1}{100}=0.01[/tex]

Please help explanation if possible

Answers

Answer:

18.84 feet.  let me know if you have ay other questions.

Step-by-step explanation:

The way to find the formula for circumference is kinda complicated so it is best to ust memorize the formula, which is 2πr.  or 2 times pi times the radius.  

Your problem gives you the formula, but instead of 2 and r in it you have d, which is the diameter.

The diameter of the circle is 2 times the radius, so that's why it is replaced.

the radius is the distance from the center fo the circle to one edge, and the diameter is the distance through the circle passing through its center.  so it's the center to one end plus the center to another end.  or r+r which is also 2r.

So d = 2r, so in this problem d =6 feet.

So now the formula πd = 3.14*6 feet = 18.84 feet

look at the image below

Answers

Answer:

201.1 km²

Step-by-step explanation:

Surface area of a sphere= 4πr², where r = radius

so,

4πr²

= 4×π×4²

= 64π

= 201.1 km² (rounded to the nearest tenth)

factorize : ( p- q ) cube​

Answers

Answer:

[tex]( {p - q}^{3} ) \\ = {p}^{2} - 3 {p}^{2} q + 3p {q}^{2} - {q}^{3} [/tex]

Hope my answer helped u :)

This graph represents which of these expressions?

Answers

Answer:

x > 43

Step-by-step explanation:

Open circle at 43, which means it is not equal to 43

Line goes to the right, which means x is greater than

x > 43

Answer:

B

Step-by-step explanation:

Starting with a fresh bar of soap, you weigh the bar each day after you take a shower. Then you find the regression line for predicting weight from number of days elapsed. The slope of this line will be:__________.

Answers

Answer:

The slope will be negative

Step-by-step explanation:

The slope of the regression line tells us about the relationship or behavior of the dependent and independent variables. In the scenario above, where the weight is being compared with the number of days elapsed. What is expected of the weight and quantity of a bar soap each time it is used for a shower is that it will decrease in weight. Therefore, as the number of days increases, and hence, number of showers rise, the weight of soap will decrease. Hence, we'll obtain a negative slope, one in which the increase in a variable leads to decrease in the other.

Jack’s backpack weighs 15 pounds. Fernando’s backpack weighs less than Jack’s. Which graph shows how much Fernando’s backpack can weigh?

Answers

Answer:

A

Step-by-step explanation:

c and d out of the question

b has its circle filled in meaning it could be 15lbs, which it's not

A correct answer by default

Answer:b

Step-by-step explanation: it has a filled in diamond which mean it's that...

Simplify the following expression

Answers

Answer:

[tex]\frac{98p^{6}}{q}[/tex]

Step-by-step explanation:

Distribute the exponents

[tex](\frac{(7^{-2}p^{-6}q^{-8})}{2q^{-9}} )^{-1}[/tex]

[tex](\frac{q}{98p^{6}} )^{-1}[/tex]

Distribute the -1

[tex]\frac{98p^{6}}{q}[/tex]

Triangle ABL is an isosceles triangle in circle A with a radius of 11, PL = 16, and ∠PAL = 93°. Find the area of the circle enclosed by line PL and arc PL. Show all work and round your answer to two decimal places.

Answers

The area bounded by a chord and arc it intercepts is known as a segment of a circle segment of a circle

The area of the circle enclosed by line PL and arc PL is approximately 37.62 square units

The reason the above value is correct is as follows:

The given parameters in the question are;

The radius of the circle, r = 11

The length of the chord PL = 16

The measure of angle ∠PAL = 93°

Required:

The area of part of the circle enclosed by chord PL and arc PL

Solution:

The shaded area of the given circle is the minor segment of the circle enclosed by line PL and arc PL

The area of a segment of a circle is given by the following formula;

Area of segment = Area of the sector - Area of the triangle

Area of segment = Area of minor sector APL - Area of triangle APL

Area of minor sector APL:

Area of a sector = (θ/360)×π·r²

Where;

r = The radius of the circle

θ = The angle of the sector of the circle

Plugging in the the values of r and θ, we get;

Area of the minor sector APL = (93°/360°) × π × 11² 98.2 square units

Area of Triangle APL:

Area of a triangle = (1/2) × Base length × Height

Therefore;

The area of ΔAPL = (1/2) × 16 × 11 × cos(93°/2) ≈ 60.58 square units

Required shaded area enclosed by line PL and arc PL:

Therefore, the area of shaded segment enclosed by line PL and arc PL is found as follows;

Area of the required segment PL (98.2 - 60.58) square units = 37.62 square units

The area of the circle enclosed by line PL and arc PL ≈ 37.62 square units

Learn more about the finding the area of a segment can be found here:

https://brainly.com/question/22599425

The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.

The calculation of the area between line segment PL and circle arc PL is described below:

1) Calculation of the area of the circle arc.

2) Calculation of the area of the triangle.

3) Subtracting the area found in 2) from the area found in 1).

Step 1:

The area of a circle arc is determined by the following formula:

[tex]A_{ca} = \frac{\alpha\cdot \pi\cdot r^{2}}{360}[/tex] (1)

Where:

[tex]A_{ca}[/tex] - Area of the circle arc.

[tex]\alpha[/tex] - Arc angle, in sexagesimal degrees.

[tex]r[/tex] - Radius.

If we know that [tex]\alpha = 93^{\circ}[/tex] and [tex]r = 11[/tex], then the area of the circle arc is:

[tex]A_{ca} = \frac{93\cdot \pi\cdot 11^{2}}{360}[/tex]

[tex]A_{ca} \approx 98.201[/tex]

Step 2:

The area of the triangle is determined by Heron's formula:

[tex]A_{t} = \sqrt{s\cdot (s-l)\cdot (s-r)^{2}}[/tex] (2)

[tex]s = \frac{l + 2\cdot r}{2}[/tex]

Where:

[tex]A_{t}[/tex] - Area of the triangle.

[tex]r[/tex] - Radius.

[tex]l[/tex] - Length of the line segment PL.

If we know that [tex]l = 16[/tex] and [tex]r = 11[/tex], then the area of the triangle is:

[tex]s = \frac{16+2\cdot (11)}{2}[/tex]

[tex]s = 19[/tex]

[tex]A_{t} = \sqrt{19\cdot (19-16)\cdot (19-11)^{2}}[/tex]

[tex]A_{t} \approx 60.399[/tex]

Step 3:

And the area between the line segment PL and the circle arc PL is:

[tex]A_{s} = A_{ca}-A_{t}[/tex]

[tex]A_{s} = 98.201 - 60.399[/tex]

[tex]A_{s} = 37.802[/tex]

The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.

According to this diagram, what is tan 62°?
62°
17
18
280
90°
15
O A.
8
17
OB.끝
O c. 1
8
15
D.
15
8
O
E.
17
15
F.
15
17

Answers

Answer:

15/8

Step-by-step explanation:

tan(62)=P/B

tan(62)=15/8

The segments shown below could form a triangle.
A
C
7
9
12
B
А
a
A. True
B. False

Answers

Answer:

TRUE

Step-by-step explanation:

I SEEN SOME ONE ELSE WIT 5 STARS SAY SO(:

The given segment can form triangle. Therefore, the given statement is true.

What is triangle?

A polygon has three edges as well as three vertices is called a triangle. It's one of the fundamental geometric shapes. In Euclidean geometry, each and every three points that are not collinear produce a distinct triangle and a distinct plane. In other words, every triangle was contained in a plane, and there is only single plane that encompasses that triangle.

All triangles are enclosed in a single plane if all of geometry is the Euclidean plane, however this is no longer true in higher-dimensional Euclidean spaces. Unless when otherwise specified, this article discusses triangles within Euclidean geometry, namely the Euclidean plane. The given segment can form triangle.

Therefore, the given statement is true.

To know more about triangle, here:

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HELPP plz don't send a file whats the answer and explanation plz if you can

Answers

Answer:

5/0 is not a rational number

Step-by-step explanation:

you cant divide by 0

The​ half-life of a radioactive substance is 20 years. If you start with some amount of this​ substance, what fraction will remain in 180 years?

Answers

Answer:

1/512

Step-by-step explanation:

Let staring fraction = x

Half-life = 20 years ; this is the time taken for an element to decrease to half of its original size

Hence,

After 20 years - - - > x/2

After 40 years - - - - > x/2 ÷ 2 = x/2 * 1/2 = x /4

After 60 years - - - - > x/4 ÷ 2 = x/4 * 1/2 = x/8

After 80 years - - - - -> x/8 ÷ 2 = x/8 * 1/2 = x / 16

After 100 years - - - > x/16 * 1/2 = x/32

After 120 years - - - - > x/32 * 1/2 = x/64

After 140 years - - - - -> x / 64 * 1/2 = x / 128

After 160 years - - - - - > x / 128 * 1/2 = x/256

After 180 years - - - - > x/256 * 1/2 = x / 512

Hence, the fraction after 180 years = 1/512

Shawn has 4 times as many candies as Jason, who has a third as many candies as
lan. If Shawn has 64 candies, how many candies does Ian have?

Answers

Ian has 48 candies. hope that helps!
Ian has 4 candies...........

please helpppp i need it by tonight its very important

Answers

Answer:

m<1=145

m<2=35

m<3=35

Step-by-step explanation:

measure one is supplementary(the angles add to 180) to measure four.

so we do 180-35=145

measure 2 is congruent to measure four because they are corresponding angles

so measure 2=35

and measure 3 is also congruent to measure 4 because the are corresponding angles

so m<3=35

write your answer as an integer or as a decimal rounded to the nearest tenth​

Answers

Answer:

FH ≈ 6.0

Step-by-step explanation:

Using the sine ratio in the right triangle

sin49° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{FH}{FG}[/tex] = [tex]\frac{FH}{8}[/tex] ( multiply both sides by 8 )

8 × sin49° = FH , then

FH ≈ 6.0 ( to the nearest tenth )

Answer:

6

Step-by-step explanation:

sin = opposite/hypotenuse

opposite = sin * hypotenuse

sin (49) = 0,75471

opposite = 0,75471 * 8 = 6,037677 = 6

Factors and rewrite the expression 25x-15​

Answers

Answer:

5(5x-3)

Step-by-step explanation:

The common factor in this expression is 5 so divide 5 to all the values

25/5=5

-15/5= -3

Put these values into parenthesis and leave the 5 on the left side and out of the parenthesis

5(5x-3)

Answer:

5(5x - 3)

Step-by-step explanation:

The greatest common factor here is 5. Divide each term by 5 and simplify.

25x/5 = 5x

15/5 = 3

Therefore, the answer is 5(5x - 3).

Can someone do #4 and #5

Answers

Answer:

First, find two points on the graph:

(x₁, y₁) = (0, 2)(x₂, y₂) = (2, 8)

Slope = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1}} = \frac{8-2}{2-0} =\frac{6}{2}=3[/tex]

16 + (-3) = 16 - 3 = 13

5 A machine puts tar on a road at the rate of 4 metres in 5 minutes.
a) How long does it take to cover 1 km of road
b) How many metres of road does it cover in 8 hours?​

Answers

Answer:

5 a) Total = 20.83 hrs = 20 hrs and 50 mins  (1250mins total)

5 b) Total = 96 meters. = 0.096km in 8 hrs.

Step-by-step explanation:

1km = 1000 meters

5 mins = 4 meters

1000/4 = 250 multiplier

250 x 5mins = 1250 minutes

1250/60 = 20hrs + 50 minutes

50 / 60 =  0.83 = 20.83hrs

b)  8 hrs = 8 x 60 = 480 minutes

480/5 = 24 multiplier of 4 meters

24 x 4 = 96 meters

Convert 0.450 to a proper fraction

Answers

Answer:

9/20

Step-by-step explanation:

450/1000

this is not the answer, because it is not simplified

so here we have to find common factor and simplifying

________________________________________________

450/1000 is simplified to 9/20, and it can no longer be simplified.

Please help me with this on the picture

Answers

9514 1404 393

Answer:

  (-5, 4)

Step-by-step explanation:

The inside corner moves from (2, -2) to (-3, 2). That is 5 is subtracted from the x-coordinate, and 4 is added to the y-coordinate. (x, y) ⇒ (x -5, y +4)

The translation vector can be written horizontally as (-5, 4), or vertically as ...

  [tex]\displaystyle\binom{-5}{4}[/tex]

2(4×+2)=10
[tex]6 \times + 4 = 10 \\ \\ 6 \times = 10 - 4 \\ \\ 6 \times = 6 \\ \\ [/tex]
that is the answer

Answers

Answer:

x = 3/4

Step-by-step explanation:

2(4x + 2) = 10           Remove the brackets

8x + 4 = 10               Subtract 4 from both sides

8x = 6                       Divide by 8

x = 6/8

x = 3/4

Check

2(4*3/4 + 2) =?10

2( 3 + 2) = 10

2*5 = 10

10 = 10

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