Very quickly: since tan(30°) = 1/√3,
(1 + tan(30°)) / (1 - tan(30°)) = (1 + 1/√3) / (1 - 1/√3)
… = (√3 + 1) / (√3 - 1)
… = (√3 + 1)² / ((√3)² - 1²)
… = ((√3)² + 2√3 + 1²) / (3 - 1)
… = (3 + 2√3 + 1) / 2
… = (4 + 2√3) / 2
… = 2 + √3
On the other side, we have sin(30°) = 1/2 and sin(60°) = √3/2, so
(1 + sin(60°)) / (1 - sin(30°)) = (1 + √3/2) / (1 - 1/2)
… = (1 + √3/2) / (1/2)
… = 2 (1 + √3/2)
… = 2 + √3
and we're done, both sides are equal.
We can also prove this algebraically:
Force the denominator on the left side into a difference of squares:
(1 + tan(30°)) / (1 - tan(30°)) × (1 + tan(30°)) / (1 + tan(30°))
… = (1 + tan(30°))² / (1 - tan²(30°))
Expand the numerator:
… = (1 + 2 tan(30°) + tan²(30°)) / (1 - tan²(30°))
Recall that 1 + tan²(x) = sec²(x) :
… = (sec²(30°) + 2 tan(30°)) / (1 - tan²(30°))
Since sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x), multiply through every term by cos²(30°) :
… = (sec²(30°) + 2 tan(30°)) / (1 - tan²(30°)) × cos²(30°) / cos²(30°)
… = (1 + 2 sin(30°) cos(30°)) / (cos²(30°) - sin²(30°))
Recall that sin(2x) = 2 sin(x) cos(x) and cos(2x) = cos²(x) - sin²(x) :
… = (1 + sin(2×30°)) / cos(2×30°)
… = (1 + sin(60°)) / cos(60°)
Finally, recall that cos(90° - x) = sin(x), so cos(60°) = sin(30°) = 1/2, and we can write 1/2 = 1 - 1/2, so
… = (1 + sin(60°)) / (1 - sin(30°))
as required.
Melanie invested $63,000 in an account paying an interest rate of 7 1/8% compounded
quarterly. Lillian invested $63,000 in an account paying an interest rate of 7 3/8%
compounded continuously. After 15 years, how much more money would Lillian have
in her account than Melanie, to the nearest dollar?
Answer: $8727- thsi should be the correct answer
HELP I NEED HELP ASAP
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Answer:
formula
Step-by-step explanation:
GIVING BRAINLIEST PLEASE HELP!!!
Answer:
52° is correct answer of this question
hope it helps
thanks for free points
Step-by-step explanation:
option a.
38°+x=90°
x=90°-38°
x=52°
will give brainliest if you help with this question.
Answer: (-5, 2). Please mark me brainliest
Step-by-step explanation: So we are finding the coordinates of point A first we have to mark the x-coordinate (-5) so we have to look at the horizontal line (x-coordinate) and the A is an -5 so the first part should be -5 next we have to do the vertical line (y-coordinate) and it is on the 2 so 2 so the answer is (-5,2).
Write the rule to describe this transformation.
Need help please!
Answer:
Figure P'Q'R'S' has been dilated by a scale factor = 1.5 to form image quadrilateral P"Q"R"S".
Step-by-step explanation:
From the graph attached,
Coordinates of point T' → (2, -2)
Coordinates of point T'' → (3, -3)
As it's clear form the picture, quadrilateral P'Q'R'S' has been dilated to form the image quadrilateral P"Q"R"S" by a scale factor = k
By the rule,
T'(x, y) → T"(kx, ky)
Therefore, T'(2, -2) → T"(2k, -2k)
T"(2k, -2k) → T"(3, -3)
By comparing x-coordinates,
2k = 3
k = [tex]\frac{3}{2}[/tex]
k = 1.5
Therefore, figure P'Q'R'S' has been dilated by a scale factor = 1.5 to form quadrilateral P"Q"R"S".
Find the midpoint between -5 and 37
Answer:
16
Step-by-step explanation:
5+37= 42. 42 is the distance between the 2.
divide that by 2- 21
then, subtract 5 to get to the midpoint. 21-5= 16
Sakal and Roy each have a sack of 50 apples. All apples are either red or green. If Sakal has three times as many red apples as Roy does and half as many green apples as Roy does, how many red apples does Sakal have?
Answer:
Step-by-step explanation:
Sakal and Roy each have a sack of 50 apples.
All apples are either red or green.
Sakal = x
Roy = y
If Sakal has three times as many red apples as Roy does
x = 3y
and half as many green apples as Roy does,
x = 1/2y
how many red apples does Sakal have?
due soon pls help will give brianliest
Find the length of the third side. If necessary, round to the nearest tenth. 5 7.
Answer:
mmmmmmmmm
Step-by-step explanation:
why everyone is asking about maths........
I am a ...........
PLEASE PLEASE PLEASE FIND THE UNKNOWNS WILL GIVE BRAIN TYSM!!
Answer:
Sum of the opposite pair of a cyclic quadrilateral is 180°.
c= 180-102= 78°
c= 180-102= 78°d= 180-92= 88°
intercepted arc
=>2(102°)= 140+a°
a=64°
Simillarly
=> 2(d°)= a°+b°
b°= 112°
so ....
a=64°
a=64°b=112°
a=64°b=112°c=78°
a=64°b=112°c=78°d=88
hope it helps you ....
Which table represents linear function?
Answer:
does not represent a linear function.
Step-by-step explanation:
For the function to be linear, the slope must be the same for every poknt along the function. If is not, then is not a linear function.
Let us now calculate the slope;
[tex]m = \frac{y2 - y1}{x2 - x1} [/tex]
We shall take the first two points
[tex]m = \frac{4 - 2}{2 - 1} \\ m = \frac{2}{1} \\ m = 2[/tex]
Next we shall take the 2nd and 3rd points
[tex]m = \frac{8 - 4}{3 - 2} \\ m = \frac{4}{1} \\ m = 4[/tex]
The slope is not the same for these two different point, hence this table does not represent a linear function.
A storage shed is a rectangular prism and has dimensions of 6 meters by 5 meters by 12 meters. If Jean were to double these dimensions, she believes she would only double the volume. Is she correct? Explain why or why not. Include a drawing in your explanation.
Answer:
Hence doubling the dimension gives a volume greater than doubling the original volume.
Step-by-step explanation:
Given The dimension :
Original dimension : 6 meters by 5 meters by 12 meters
Volume of original dimension :
6 * 5 * 12 = 360 m³
If dimension is doubled :
2 * 6 = 12
2 * 5 = 10
2 * 12 = 24
Volume = (12 * 10 * 24) = 2880 m³
2 * original volume = 2 * 360 = 720 m³
Hence doubling the dimension gives a volume greater than doubling the original volume.
If I was asked to think of a number and multiply it by 2, I could
write this algebraically as 2x.
Write the following algebraically, using x as your unknown.
"I think of a number, multiply it by 2, subtract 3 and multiply the result by 4."
Answer:
4(2x - 3) is how I would write it algebraically.
Step-by-step explanation:
The number is x, as it is unknown. 2x means you multiply the unknown number by 2.
2x
Now, you subtract 3.
2x - 3
And then you multiply the result of that by 4, hence the parentheses.
4(2x - 3)
To solve, just distribute 4. 4*2x = 8x. 4*-3 = -12.
8x - 12
Answer:
8x-12
Step-by-step explanation:
(2x-3)* 4 = 8x-12
Q. A box of cookies has 5 chocolate chip, 5 peanut butter, and 5 sugar in it.
When choosing a cookie at random, which answer does NOT show the
probability of getting a peanut butter cookie?
Answer:
nithen dior becausee full sorry baby
PLEASE HELP I WILL MARK BRAINEST JUST HELP PLEASE!!!!!
Answer:
1/4
Step-by-step explanation:
Answer:
scale factor = [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
The scale factor is the ratio of corresponding sides, image to original, that is
scale factor = [tex]\frac{1.5}{6}[/tex] = [tex]\frac{1}{4}[/tex]
Find the circumference use 3.14 for the value of
Answer:
the answer is 69.3
Step-by-step explanation:
Every time Vince orders Chinese food, he gets egg drop soup. So, he decides to learn how to make it himself. His recipe calls for 1.5 liters of broth, but he can only find 500-milliliter cartons at the store. How many cartons should he buy?
Answer: 3
Step-by-step explanation:
Vince will have to buy three cartons of 500 ml each.
What is expression in mathematics?An expression in mathematics is a combination of terms both constants and variables.
Given is that every time Vince orders Chinese food, he gets egg drop soup. So, he decides to learn how to make it himself. His recipe calls for 1.5 liters of broth, but he can only find 500 milliliter cartons at the store.
We know that -
1 Liter = 1000 milliliters
1.5 liter = 1500 liters
Now -
1 carton 500 ml.
Then, 1 ml will be in (1/500) carton.
So, 1500 ml will be in (1/500 x 1500) = 3 cartons
Therefore, Vince will have to buy three cartons of 500 ml each.
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Solve system of linear equations by graphing.
-2x - 5y = -10
3x + 6y= 18
Answer:
this is the answer bb
Step-by-step explanation:
HELP PLS QUICKLY !!!
Factor as the product of two binomials.
x^2 + 3x + 2 ?
Answer:
x^2+3x+2 = (x+1)(x+2)
Step-by-step explanation:
Is q=9 true or false
Answer:
If there is an expression along with it I'd like to see it to determine if its true or false.
But yes a variable always has a value so im pretty sure its true. Post another question with the expression please or any other information given.
For this item, enter the answers in the spaces provided.
Consider the equation representing a customer purchasing 3 boxes of cereal
at $x each and 4 apples at $y each at a grocery store for a total of $12.
3x + 4y = 12
The equation when solved for y is y y=__x+__
Answer: I think 3-1x not 100% sure on that one though been a while since I've done this stuff
Step-by-step explanation:
PLEASE ANSWER FAST!
A water tank contains exactly 1,000 gallons. A pump fills the remainder of the tank using
a logistic function. If the volume of the tank is 15,000 gallons and the pump fills the tank at a rate of 10.4% per minute, explain how to find the value of a for the logistic function that describes this scenario.
Answer: V ( t ) = 1 + 1 4 e − 0 . 1 0 4 t 1 5 , 0 0 0
The required value for the logistic function describing the scenario of the water tank being filled at a rate of 10.4% per minute.
What is the logistic function?The logistic function has the form:
[tex]f(x) = L / (1 + a \times e^{(-kx)})[/tex]
Where:
f(x) is the volume of the tank at time x
L is the maximum volume the tank can hold
a is a constant that determines the growth rate of the function
k is a constant that determines the rate at which the growth slows down as the tank approaches the maximum volume
In this case, the maximum volume of the tank is 15,000 gallons, so L = 15,000.
The pump fills the tank at a rate of 10.4% per minute, so k = 0.104.
To find the value of a, we need to solve the logistic equation for a given the values of f(x), L, and k.
We can rearrange the logistic equation to solve for a:
[tex]a = (1 / f(x)) - (1 / L) \times e^{(kx)}[/tex]
Substituting the known values into this equation, we get:
[tex]a = (1 / 1,000) - (1 / 15,000) \times e^{(0.104x)}[/tex]
This equation can be used to solve for the value of a for any given value of x (time). For example, if we want to know the value of an after 10 minutes, we can substitute x = 10 into the equation to get:
[tex]a = (1 / 1,000) - (1 / 15,000) \times e^{(1.04)}[/tex]
This gives us the value of a for the logistic function that describes the scenario of the water tank being filled at a rate of 10.4% per minute.
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Please help it’s due todayI will give brainliest!!
Answer:
i dont know sorry
Step-by-step explanation:
9 because if you divide 414 by 46 you get 9
Why is the numbers of ways of arranging the letters of CALL is less than the number of ways of arranging the letters of BARK? Explain your answer
Answer:
See Explanation
Step-by-step explanation:
Given
(1) CALL and (2) BARK
Required
Why is the number of arrangement of (1) less than (2)
The reason is that in (1), letter L occurred twice. When there is a repetition of letters, the number of ways of arrangement reduces
For (1)
There are 4 letters and L is repeated twice.
[tex]Arrangement = \frac{4!}{2!}[/tex]
[tex]Arrangement = \frac{24}{2}[/tex]
[tex]Arrangement = 12[/tex]
For (2)
There are 4 letters and no repetition
[tex]Arrangement = 4![/tex]
[tex]Arrangement = 24[/tex]
The reason why the number of ways of CALL is less than the number of ways of arranging the letters of BARK should be explained below.
Calculation of the number of ways:The reason should be that in (1), letter L occurred twice. At the time When there is a repetition of letters, the number of ways of arrangement decreases.
So here arrangement be like
When there is 4 letters and L is repeated twice.
=4!/2!
= 24/2
= 12
And,
When there is 4 letters and no repetition
= 4!
= 24
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Lilian uses 3/4 calories per minute just by sitting. She uses 1 more calorie per minute by walking. Lilian a uses a total of 12 1/4 calories walking to the park.
Lilian a uses the equation, d (3/4 + 1) = 12 1/4 to represent the situation.
what does variable d represent in the equation?
Answer:
Variable d represents the time per minute Lilianna spends in the park.
Step-by-step explanation:
Answer:
Number of minutes Lilianna walked
Step-by-step explanation:
Hope this helps!
What is the volume of a cylinder, in cubic centimeters, with a height of 18 centimeters and a base diameter of 8 centimeters? Round to the nearest tenths place
Answer:
The formula for the volume of a cylinder = height x π x (diameter / 2)^2
volume = 18 x π x (8/2)^2
volume = 502.654825 cm^3
volume = 502.7 cm^3
Answer:
904.8
Step-by-step explanation:
delta math
a girl 2.2m tall observes that the angle of elevation of the top of a building 16m away from her is 45° find the height of the building
Answer:
18.2 m = height of the building
Step-by-step explanation:
Let x = the distance from the top of the girl's head to the top of the building
tan 45 = x/16
x = 16 tan 45
x = 16 m
16 + 2.2 = 18.2 m = height of the building
AYUDA PORFAVOR
En el par ordenado (m,n) ¿cual letra representa la abcisa y cuan la ordenada?
Answer:
El eje horizontal es el Eje de Abscisas y se representa con la letra “x”, el eje vertical es el Eje de Ordenadas y se representa con la letra “y”.
Step-by-step explanation:
The Frenchtown Roller Rink charges a $5 entrance fee and an hourly rate for roller skating. The total cost for roller skating depends on the number of hours a
person skates. The table below represents the total cost of skating for different numbers of hours.
ROLLER SKATING COST
Number
of Hours
Total Cost
in Dollars
0
5
1
8
2.
11
3
14
4
17
Which equation represents the relationship between the cost, and the number of hours, h?
A
C= 8h
B
C= 5h+3
C c = 2h +7
D = 3h5
Answer:
Step-by-step explanation:
the total cost for roller skating depends on the number of hours a person skates. which equation represents the relationship btw the cost c and the ...
1 answer
·
20 votes:
To answer this item, we let n be the number of hours that a person spend on th
easy slope question need answer FAST
Answer:
Assuming that the scale is 1, x int: (-4,0), y int: (0,-3)
Step-by-step explanation:
Hope this helps :)
