Answers
For triangle : A + B + C = π
⇒ A + B = π - C
⇒ cot(A + B) = cot(π - C)
⇒ [tex]\sf\dfrac{cotA\:cotB-1}{cotB+cotA}=-cotA[/tex]
⇒ [tex]\bf{cotA\:cotB+cotB\:cotC+cotC\:cotA=1}[/tex]
Now 1st part of the given expression!
⇒ [tex]\sf\dfrac{cosA}{sinB\:sinC}[/tex]
⇒ [tex]\sf\dfrac{cos(\pi-(B+C))}{sinB\:sinC}[/tex]
⇒ [tex]\sf\dfrac{-cosB\:cosC+sinB\:sinC}{sinB\:sinC}[/tex]
⇒ 1 - cotB cotC
Similarly 2nd part!
⇒ 1 - cotA cotB
Similarly 3rd part!
⇒ 1 - cotC cotA
LHS :
[tex]\circ\:\sf{3-(cotA\:cotB+cotB\:cotC+cotC\:cotA)}[/tex]
[tex]\circ\:\sf{3-1}[/tex]
[tex]\circ\:\bf{2}[/tex] = RHS
Hence Proved!!Answer: see proof below
Step-by-step explanation:
Given: A + B + C = π → A + B = π - C
→ B + C = π - A
→ A + C = π - B
Use the following Double Angle Identity: sin 2A = 2 sin A · cos A
Use the following Cofunction Identity: sin A = cos (π/2 - A)
Use the following Sum to Product Identity:
sin A + sin B = sin [(A + B)/2] · cos [(A - B)/2]
Proof LHS → RHS
[tex]\text{LHS:}\qquad \dfrac{\cos A}{\sin B\cdot \sin C}+\dfrac{\cos B}{\sin C\cdot \sin A}+\dfrac{\cos C}{\sin A\cdot \sin B}\\\\\\.\qquad \quad = \bigg(\dfrac{2\sin A}{2\sin A}\bigg)\dfrac{\cos A}{\sin B\cdot \sin C}+\bigg(\dfrac{2\sin B}{2\sin B}\bigg)\dfrac{\cos B}{\sin C\cdot \sin A}+\bigg(\dfrac{2\sin C}{2\sin C}\bigg)\dfrac{\cos C}{\sin A\cdot \sin B}\\\\\\.\qquad \quad =\dfrac{2\sin A\cdot \cos A+2\sin B\cdot \cos B+2\sin C\cdot \cos C}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Double Angle:}\qquad \qquad \dfrac{\sin 2A+\sin 2B+\sin 2C}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{(\sin 2A+\sin 2B)+\sin 2C}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Sum to Product:}\qquad \dfrac{2\sin (A+B)\cdot \cos (A-B)+\sin 2C}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Given:}\qquad \qquad \qquad \dfrac{2\sin (\pi -C)\cdot \cos (A-B)+\sin 2C}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{2\sin C\cdot \cos (A-B)+\sin 2C}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Double Angle:}\qquad \qquad \dfrac{2\sin C\cdot \cos (A-B)+2\sin C\cdot \cos C}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{2\sin C(\cos (A-B)+\cos C)}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Sum to Product:}\qquad \dfrac{2\sin C(2\cos (\frac{A-B+C}{2})\cdot \cos (\frac{A-B-C}{2})}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{4\sin C\cdot \cos (\frac{A+C}{2}-\frac{B}{2})\codt \cos (\frac{A}{2}-\frac{B+C}{2})}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Given:}\qquad \qquad \dfrac{4\sin C(\frac{\pi-B}{2}-\frac{B}{2})\cdot \cos (\frac{A}{2}-\frac{\pi -A}{2})}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad =\dfrac{4\sin C(\frac{\pi}{2}-B)\cdot \cos (\frac{\pi}{2}-A)}{2\sin A\cdot \sin B\cdot \sin C}[/tex]
[tex]\text{Cofunction:}\qquad \qquad \dfrac{4\sin C\cdot \sin B\sin A}{2\sin A\cdot \sin B\cdot \sin C}\\\\\\.\qquad \qquad \qquad \qquad =2[/tex]
LHS = RHS: 2 = 2 [tex]\checkmark[/tex]
Related Questions
A pack of six cans of coffee cost $12. How much would 19 cans of coffee cost?
Answers
Answer:
228
Step-by-step explanation:
12 times 19
Sanjay and Felipe are putting together a 500-piece puzzle. Sanjay could do the entire puzzle by himself in 6 hours, while it would take Felipe 8 hours to do the puzzle alone. A table showing Rate in part per hour, Time in hours, and Part of puzzle complete. The first row shows Sanjay, and has blank, t, t. The second row shows Felipe, and has blank, t, t. Use the table to write and solve an equation to determine the approximate amount of time it would take Sanjay and Felipe to do the puzzle if they were working together. 0.8 hours 1.7 hours 3.4 hours 7 hours
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Answer:
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Step-by-step explanation:
Answer:
C 3.4 hours
Step-by-step explanation:
graph the function and its parent function on the calculator . Then describe the transformation.
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-3 ÷ -1 − -1 show your work step by step
Answers
Answer:
4
Step-by-step explanation:
-3/(-1)-(-1)
simplify this first
-3/(-1) + 1
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-3/-1 = 3 since negative ÷ negative = positive
3+1
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Hope this helps :)
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Answer:
The answer for this question is. D
Answer:
D just read each of the selections very carefully and look for keywords like equals or more than, per..., and etc pls mark me brainliest
Step-by-step explanation:
Given f(x) = 2x2 – 1 and g(x) = x + 3, find f(g(-5))
Answers
Answer:
7
Step-by-step explanation:
f(x) = 2x2 – 1 and g(x) = x + 3,
f(g(-5))
First find g(-5)
g(-5) = -5 + 3
g(-5) = -2
Then substitute into f(x)
f(-2) = 2 ( -2) ^2 -1
= 2 ( 4) -1
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what are the coordinates of the midpoint between the points (-4,-9) and (-8,-5)?
Answers
Answer:
(-6,-7)
Step-by-step explanation:
Apply the midpoint formula!
Simplify by combining like terms.
1. 5a-a
Answers
HELP ME FOR BRAINETEST PLS
Answers
Answer: the first one is when you multiply it, the second is where you divide it
Step-by-step explanation:
Because they are different equations that are similar but one just uses multiplication and the other uses division.
(-3-5/)(-3+5/)=
34
34-30i
-16
Answers
Answer:
(m+ni)2= -16 - 30i
2m + 2ni = -16-30i
.)2m=-16 .)2n=-30
m= -8 n= -15
por lo tanto.....
m2 + n2
(-8)2 + (-15)2
-16 + (- 30 )
-16 - 30
-46
respuesta..... - 46
Step-by-step explanation:
Answer:
34
Step-by-step explanation:
Given
(- 3 - 5i)(- 3 + 5i) ← expand using FOIL
= 9 - 15i + 15i - 25i² [ i² = - 1 ]
= 9 + 25
= 34
our solutions to elimin:
ate
2. 3+5x = 3x - 9
Answers
Answer:
x=−6
Step-by-step explanation:
Let's solve your equation step-by-step.
3+5x=3x−9
Rotate the shape 180 degrees counterclockwise around B
Answers
Answer:
see attached
Step-by-step explanation:
Solve for x: -2(x + 3) = -2(x + 1) - 4. (1 point)
a
2.
CA
3
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All real numbers
Od
No solution
Answers
Answer:
no solution
Step-by-step explanation:
[tex]-2 ( x + 3) = -2 (x + 1) -4 \\-2x + 3 = -2x -2 -4\\-2x + 3 = -2x - 6[/tex]
add [tex]2x[/tex] to both sides
[tex]-2x + 3 + 2x = -2x - 6 + 2x \\3 = -6[/tex]
That's impossible
so the equation no solution
Answer:
All real numbers ( -∞ ,∞)
Step-by-step explanation:
Simplify [tex]-2(x+3)[/tex]
Apply the distributive property.
[tex]-2x-6=-2(x+1)-4[/tex]
Simplify
[tex]-2(x+1)-4[/tex]
[tex]-2x-6=-2x -6[/tex]
Move all terms containing x to the left side of the equation.
Add 2 x to both sides of the equation.
[tex]-2x-6+2x=0[/tex]
Combine the opposite terms in
[tex]-2x-6+2x[/tex]
add [tex]-2x[/tex] and [tex]2x[/tex]
[tex]0-6= -6[/tex]
[tex]-6= -6[/tex]
Since [tex]-6= -6[/tex] , the equation will always be true for any value of x . All real numbers
The result can be shown in multiple forms.
All real numbers
Interval Notation:
( − ∞ , ∞ )
I need help. Someone pls answer 50 and 40. I'm giving 25 brainly
Answers
Answer:
I'm not sure if you're supposed to answer the question in numbers, i think you move the boxes "rational" and "irrational" in the rectangles provided below each question.
3(32 − 4 + 6 )− (82 + + 3) ?
simpilyfy
plzz
Answers
Answer:
17 is your answer
Step-by-step explanation:
Answer:
17
Step-by-step explanation:
3(32-4+6) --> 102 & (82+3) --> 85
102-85 =17
Stephen had $101 to buy 9 pairs of PEMDAS socks. After buying them, he had 11 dollars
left. How much did each pair of socks cost?
Answers
Answer:
They are 10 dollars a piece
What is the equivalent fraction of 0.95
Answers
.95 as a fraction would be 95/100
In simplest form this would be 19/20
-2x + 3y = 9, solve for x
Answers
Answer:
If your going to graph it , it would be x=3/2y+ -9/2
Step-by-step explanation:
Help please 50 points!
Answers
Answer:
84%
Step-by-step explanation:
The mean has 50% below the mean and 50 % above the mean
The mean is 103.47
The standard deviation is 4.07
Add the mean and the standard deviation
103.47+4.07 = 107.54
We want to know how much data is below the mean plus one standard deviation
Within one standard deviation is 68% , but that is on both sides, we are only needed on the right side so we need 1/2 of the 68 = 34
50+34 = 84
The stander deviation 4.07
1/2*68=34
The percent of time will be 80
34+50=80
Sam sharpens pencils for his after school job. He can sharpen 26 pencils in 6 minutes. If he keeps this rate, how many pencils can he sharpen in 15 minutes?
Answers
Answer:
234
Step-by-step explanation:
multiply 26 x 9
PLEASE HELP QUICKLY! I KNOW THE ANSWER IS 65 BUT WHAT DOES F REPRESENT??
The equation 201.50 = f + 6.50(21) represents the cost of printing the shirts at a second printing company. Find the solution to the equation and state what the solution represents in this situation.
Answers
Answer:
maybe it means how many shirts
Answer:
F= 65
Step-by-step explanation:
Hope it helps
Can someone please assist me with this? If so praise God :)
Answers
Answer:
option 1
Step-by-step explanation:
the square root of 6 is 2.44949, rounded it's 2.4
the square root of 42 is 6.48074, rounded it's 6.5
in option 1 the 6 is placed after the 2 because it is point 4, not quite 2 but a bit bigger than 2 but less than 3.
same thing for 42, it is placed after 6 because it is not quite 6 but not over 7. i hope this helped
0.63 repeating as a fraction
Answers
Answer:
63 100
Step-by-step explanation:
To convert the decimal 0.63 to a fraction, just follow these steps:
Step 1: Write down the number as a fraction of one: 0.63 = 0.63 1.
Step 2: Multiply both top and bottom by 10 for every number after the decimal point:
Answer:
7/11
Step-by-step explanation:
if g(x) = x^2 -1, find g(-3)
Answers
Step-by-step explanation:
>>▪︎g(x)=x²-1▪︎<<>>▪︎g(3)=3²-1▪︎<<>>▪︎=9-1▪︎<<>>▪︎8▪︎<<Given f(x)=x−1−−−−√+2, what is the relationship between f(x) and f−1(x)? Drag and drop an inequality into each box to correctly complete each statement.
Answers
Answer:
The domain of f(x) is x>-2, so the range of f-1(x) is y>-2
The range of f(x) is y>-1, so the domain of the f-1(x) is x>-1
Step-by-step explanation:
To solve this question, we need to know the definition of domain and range.
The domain of a function is defined as all the possible input x-values while the range is defined as all the possible output y-values.
For f(x), domain for x is; x ≥ 1 and range is; y ≥ 2
For f⁻¹(x); domain is; x ≥ 2 and range is; x ≥ 1
We are given the function;
f(x) = [tex](\sqrt{x - 1} ) + 2[/tex]
Now, when x = 1, we will have zero inside the square root sign and this means that any x-value less than 1 will make the value inside the square root symbol to be negative which will result in an imaginary root.
Thus, we can say that, the domain for x is; x ≥ 1
Putting x = 1, we can find the least range value;
f(x) = 0 + 2
f(x) = 2
Thus, the range will be; y ≥ 2
Now, we want to find the range of f⁻¹(x)
This means an inverse of the original equation.
Thus;
f⁻¹(x) = [tex](x - 2)^{2} + 1[/tex]
Let's put x = 2;
f⁻¹(x) = (2 -2)² + 1
f⁻¹(x) = 1
Since x=2 gives the bracket to be zero, it means the domain is; x ≥ 2 and range is; x ≥ 1
Read more at; brainly.com/question/24522291
5 | 9 - 5n|-7 = 38
I need helppp.
Answers
Answer:
I got 0 or 18/5
Step-by-step explanation:
Does anyone know how to solve this?
Answers
Answer:
14.76
Step-by-step explanation:
First find the length of the diagonal across the bottom using the Pythagorean theorem
a^2 + b^2 = c^2 where a and b are the legs and c is the hypotenuse
12^2+5^2 = c^2
144+25 = c^2
169 = c^2
Take the square root of each side
sqrt(169) = sqrt(c^2)
13 = c
Then find the length of the diagonal across the middle using the Pythagorean theorem
7^2 + c^2 = d^2
7^2 + 13^2 = d^2
49+ 169 = d^2
218 = d^2
Take the square root of each side
sqrt(218) = sqrt(d^2)
14.76482306 = d
To the nearest thousandth
14.76
4y-x=6 solve the equation for x
Answers
Answer:
x=6
Step-by-step explanation:
you have to find the y value which should be 3 then subtract the x value. Which should be 6
the area of a square is 36 cm^2? what is the side length
Answers
Answer:
6cm
Step-by-step explanation:
[tex]\sqrt[]{36} =6[/tex] Add the units, and there is your answer.
la cancha de futbol del colegio es de forma rectangular, pedro es el cargado de colocar un lazo alrededor para que nadie entre ala cancha si el ladao largo mide 100 mertros y el lado mas corto mide 64 metros ¿ cuanto debe medir el lazo
Answers
Answer:
328 metros
Step-by-step explanation:
Aquí, estamos interesados en colocar un bucle alrededor de un campo de fútbol de la escuela de manera que sea imposible que alguien pueda acceder al campo.
Matemáticamente, la longitud de este bucle sería igual al perímetro del campo.
El perímetro del campo rectangular se puede calcular usando la fórmula; 2 (L + B)
En este caso tenemos; P = 2 (100 + 64)
P = 2 (164)
P = 328 metros La longitud del bucle es de 328 metros.
