Answers
Answer: see proof below
Step-by-step explanation:
Given: A + B + C = π → A = π - (B + C)
→ B + C = π - A
Use the Pythagorean Identity: cos² A + sin² A = 1 → sin² A = 1 - cos² A
Use Double Angle Identities: cos 2A = 2 cos² A - 1 → cos² A = (cos 2A + 1)/2
→ cos A = 1 - 2 sin² (A/2)
Use Sum to Product Identity: cos A + cos B = 2 cos [(A + B)/2] · cos [(A - B)/2]
Use Cofunction Identities: cos (π/2 - A) = sin (A)
sin (π/2 - A) = cos A
cos (-A) = cos (A)
Proof LHS → RHS:
[tex]\text{LHS:}\qquad \qquad \sin^2\bigg(\dfrac{B}{2}\bigg)+\sin^2 \bigg(\dfrac{C}{2}\bigg)-\sin^2\bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Pythagorean:}\qquad 1-\cos^2 \bigg(\dfrac{B}{2}\bigg)+1-\cos^2 \bigg(\dfrac{C}{2}\bigg)-\bigg[1-\cos^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =1-\cos^2 \bigg(\dfrac{B}{2}\bigg)-\cos^2 \bigg(\dfrac{C}{2}\bigg)+\cos^2 \bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Double Angle:}\quad 1-\bigg(\dfrac{\cos(2\cdot \frac{B}{2})+1}{2}\bigg)-\bigg(\dfrac{\cos (2\cdot \frac{C}{2})+1}{2}\bigg)+\bigg(\dfrac{\cos (2\cdot \frac{A}{2})+1}{2}\bigg)\\\\\\.\qquad \qquad \qquad =1-\dfrac{\cos B}{2}-\dfrac{1}{2}-\dfrac{\cos C}{2}-\dfrac{1}{2}+\dfrac{\cos A}{2}+\dfrac{1}{2}\\\\\\.\qquad \qquad \qquad =\dfrac{1}{2}[1-(\cos B+\cos C)+\cos A][/tex]
[tex]\text{Sum to Product:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\cos \bigg(\dfrac{B+C}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Given:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\cos \bigg(\dfrac{\pi -A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Cofunction:}\qquad \dfrac{1}{2}\bigg(1-\bigg[2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)\bigg]+\cos A\bigg)[/tex]
[tex]\text{Double Angle:}\qquad \dfrac{1}{2}\bigg[1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)+1-2\sin^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =\dfrac{1}{2}\bigg[2-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)-2\sin^2 \bigg(\dfrac{A}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =1-\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin^2 \bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Factor:}\qquad \qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin \bigg(\dfrac{A}{2}\bigg)\bigg][/tex]
[tex]\text{Given:}\qquad \qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)-\sin \bigg(\dfrac{\pi -(B+C)}{2}\bigg)\bigg][/tex]
[tex]\text{Cofunction:}\qquad 1-\sin \bigg(\dfrac{A}{2}\bigg)\bigg[ \cos \bigg(\dfrac{B-C}{2}\bigg)+\cos \bigg(\dfrac{B+C}{2}\bigg)\bigg][/tex]
[tex]\text{Sum to Product:}\ 1-\sin \bigg(\dfrac{A}{2}\bigg)\cdot 2 \cos \bigg(\dfrac{(B-C)+(B-C)}{2\cdot 2}\bigg)\cdot \cos \bigg(\dfrac{(B-C)-(B+C)}{2\cdot 2}\bigg)\\\\\\.\qquad \qquad \qquad =1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(-\dfrac{C}{2}\bigg)[/tex][tex]\text{Cofunction:}\qquad =1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{LHS = RHS:}\quad \checkmark\\\\\quad 1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)=1-2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \cos \bigg(\dfrac{C}{2}\bigg)\quad[/tex]
Related Questions
Which are the roots of the quadratic function f(b) = 62-75? Select two options,
b=53
b= -573
Ob=35
b = -315
b = 256
Answers
Answer:b=573
Step-by-step explanation:
10 What is the slope of the line that passes through the points
(-6, 13) and
(8, -29)?
-3 3 -1/3 1/3
Answers
Answer:
Slope: -3
Step-by-step explanation:
-29-13=-42
8-(-6)= 14
-42/14= -3
A line is an undefined term because it
Answers
Hope this helps!
1. Five more than a number is eight.
Answers
Answer
3, the number is 3
Step-by-step explanation:
work backwards, do 8-5 to get three. you can check to see if you're right by adding 5 to 3, which gives you 8.
Which expression does NOT have a Greatest Common Factor?
Answers
Answer:
4x^2-9
Step-by-step explanation:
i just took the quiz
This is about greatest Common factor (G.C.F) thus Option C and Option D does not have a greatest common factor.
What is Greatest common factor?The Greatest common factor between 2 or more numbers is simply the greatest factor that is common to all the numbers.
Option A; 2x² -18
Factors of 2 are 1, 2
Factors or 18 are; 1, 2, 3, 6, 9, 18
In this option we can see that 2 is the only common factor between all the numbers.
Thus, the GCF = 2 So this has a GCF
Option B: 3x² + 12x + 6
Factors of 3 are 1, 3
Factors of 6 are 1, 3. 6
Factors or 12 are; 1, 2, 3, 4, 6, 12
The highest common factor is 3 Thus, this has GCF.
Option C; 4x² - 9
The factors of 9 are; 1, 3, 9
The factors of 4 are; 1, 2, 4
Since there is no highest common factor here as 1 is not considered to be a highest common factor even though it is a factor.
Option D; 6x² + 2x² + 5x
The factors of 9 are; 1, 3, 9
The factors of 4 are; 1, 2, 4
Since there is no highest common factor here as 1 is not considered to be a highest common factor even though it is a factor.
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Please help please help
Answers
Answer:
I think it will be 16.7 percent, but if I am wrong I am very sorry.
what is 1/5 x 6 ? help ajiwjwkwjwoksjw
Answers
Answer:
6/5 or 1.2
Step-by-step explanation:
put them both as fractions
1/5 x 6/1
6/5=1.2
6/5
Decimal Form:
1.2
Mixed Number Form:
1 1/5
Apply the dilation (x, y) = (3x, 3y). What will be the coordinates of T'?
Answers
Answer:
6,0
Step-by-step explanation:
the dilation moves points from A to B
in this case we're moving point T
point T has coordinates 2,0 to start with (A)
Now let's move it to B
To do so, we need to apply the calculation
(x,y) => (3x,3y)
Substitute in point T:
(2,0) => (3x,3y)
(2,0) => (3(2),3(0))
(2,0) => (6,0)
Can someone help me please
Answers
Answer:
c) 4 or more times
Step-by-step explanation:
Marie was trying to show her friend why -3 -8 equals a positive product. This is what she wrote:
-3 x 0 = 0
-3 × (?) = 0
-3(8) +-3(-8) = 0
-24 + 24 = 0
What should Marie write in place of the question mark?
OA) 8 + 8
OB) -8 + (-8)
OC) 8 + (-8)
OD)-8 + 0
Answers
Answer:
8+8=16
-8 + (-8)= - 16
8+ (-8)=0
-8 + 0 = - 8
F(1)=2
F(n)=-3f(n-1)+[f(n-1)]^2 ?
Answers
Answer:
Step-by-step explanation:
Umm
3³ + 5 ² = 2x
work out the value of x
Answers
Answer:
X = 26
Step-by-step explanation:
3^3 + 5^2 =2x
27 + 25 = 2x
52 = 2x
Divide by 2
X = 26
Please hurry! :) Its a get more math question.
Answers
The sales price is 44. 37 dollars.
Answer: $44.37
Step-by-step explanation:
63.38 x 30/100 (or x 0.3)
You save: $19.01
$63.38 - $19.01 = Final price of $44.37
I really need help on this problem.
Answers
Answer:
a.) 20
b.) 4/3
c.) 3
Step-by-step explanation:
a.) Plug in 5 for x
5(5) - 5
25 - 5
20
b.) Plug in 1/3 for x
5(1/3) - 1/3
5/3 - 1/3
4/3
c.) Set equation equal to 12 to find x input
12 = 5x - x
12 = 4x
3 = x
x = 3
A student has 2/3 quart of yellow paint in a container.The student uses 1/2 quart of the yellow paint to make green paint.How many quarts of yellow paint remains in the container after the student makes green paint?
Answers
Answer:
1/6
Step-by-step explanation:
2/3 - 1/2 = 4/6 - 3/6 = 1/6
1/6 quart remains.
Answer:
There are 1/6 quarts left in the container.
Step-by-step explanation:
You need to subtract 1/2 (the paint used) by 2/3 (the paint in the container)
First, find the LCD (lowest common denominator) if you want to subtract.
3 x 2 = 6; you do the same to the numerator 2 x 2 = 4
4/6 (paint in the container)
2 x 3 = 6; you do the same to the numerator 1 x 3 = 3
3/6 (paint used)
Now, you subtract.
4/6 - 3/6 = 1/6 quarts
Emily is thinking of a number, which she calls n. She finds 1/4 of the number and then subtracts 3.
Answers
Answer:
n/4 - 3 OR 1/4n - 3
Step-by-step explanation:
First, let's assign a variable to Emily's number. Let is be x.
Now all we have to do is go through the problem step by step.
"She finds 1/4 of the number..."
There are two ways you can do this. You can either multiply n by 1/4 (1/4n), or you can divide by 4 (dividing by four is the same as finding 1/4 of a number). For this example let's divide by four
n/4
"...and then subtracts 3."
All we have to do is subtract 3 from n/4
n/4 - 3
REMEMBER: like I said, there are two possible options. You can either have n/4 or 1/4n. They mean the same thing
Answer:
1/4n - 3
Step-by-step explanation:
Find the ratio of the partitioned segments such that C partitions the segment AB where A(-8, -4), B(6, 10), and C(18, 22)
Answers
Answer:
how is your day
Step-by-step explanation:
helpppp please... need to get this doneeee
Answers
Answer:
if this is iready
Step-by-step explanation:
the answer is
the third one
-28 + 2x = -18 what is x?
Answers
Answer: x=5
Step-by-step explanation:
To solve for x, we want to use algebraic properties to isolate x.
-28+2x=-18 [add both sides by 28]
2x=10 [divide both sides by 2]
x=5
Now, we know that x=5.
Answer:
[tex] \boxed{ \bold{ \huge{ \boxed{ \text{x = 5}}}}}[/tex]
Step-by-step explanation:
[tex] \star{ \sf \: - 28 + 2x = - 18 }[/tex]
[tex] \tt{Step \: 1}[/tex] : Move 28 to right hand side and change it's sign
[tex] \hookrightarrow{ \sf{2x = - 18 + 28}}[/tex]
[tex] \tt{Step \:2}[/tex] : Subtract 18 from 28
The negative and positive integers are always subtracted but posses the sign of the bigger integer
[tex] \hookrightarrow{ \sf{2x = 10}}[/tex]
[tex] \tt{Step \: 3}[/tex] Divide both sides by 2
[tex] \hookrightarrow{ \sf{ \frac{2x}{2} = \frac{10}{2}}} [/tex]
[tex] \tt{Step \: 4}[/tex] Calculate
[tex] \hookrightarrow{ \sf{x = 5}}[/tex]
[tex] \tt{∴ \: The \: value \: of \: x \: is \: \bold{5}}[/tex]
[tex] \sf{Hope \: I \: helped !!}[/tex]
[tex] \sf{Best \: regards !!}[/tex]
~[tex] \text{TheAnimeGirl}[/tex]
HELP HELP Point F is the midpoint of segment DE. DF = 2x+6 and DE = 2x+36. Find DE.
Answers
Answer:
since F is midpoint
DF= FE = 2x+6
DF +FE=DE
2x+6 + 2x+6 = 2x+36
4x + 12 = 2x+36
2x = 24
x=12
DE =2x+36 =2*12 + 36 = 60
Step-by-step explanation:
which inequality is true?
1/2 is less than 1/3
3/4 is greater than 2/3
-1/4 is less than -2/3
-1 is greater than 3/4
Answers
The correct option which valid for inequality is the first one that is 3/4 is greater than 2/3.
What is inequality?Inequality is defined as the relation which makes a non-equal comparison between two given functions.
In the given option
3/4 is greater than 2/3 is the right one
1/2 is less than 1/3 is not true because 1/2 is more than 1/3.
-1/4 is less than -2/3 is not true because -1/4 is not less than -2/3,
-1 is greater than 3/4 is not true because -1 is not greater than 3/4s.
Learn more about inequality here;
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Please Help!!! 30 Points!
Answers
Answer:
ABC is congruent to A' B'C'
Step-by-step explanation:
All three sides are equal on triangle ABC and they are equal to the sides on A'B'C'
All three angles are equal on triangle ABC and they are equal to the angles on A'B'C'
We know that since the sides and the angles are equal
ABC is equal to A'B'C'
ABC is congruent to A' B'C'
-1/2x+3=-x+7 what does x equal
Answers
Answer:
x = 8
Step-by-step explanation:
-1/2x+3=-x+7
Add x to each side
-1/2x+x+3=-x+x+7
1/2x +3 = 7
Subtract 3 from each side
1/2x +3-3 = 7-3
1/2x = 4
Multiply each side by 2
1/2x *2 = 4*2
x = 8
Find the distance between each pair of points (-6,-10) , (-2,-10)
Answers
Answer:
4 units
Step-by-step explanation:
Use the distance formula, d = [tex]\sqrt{(x2 - x1)^2 + (y2 - y1)^2}[/tex], where (x1, y1) and
(x2, y2) are 2 points
Plug in the points given
d = [tex]\sqrt{(x2 - x1)^2 + (y2 - y1)^2}[/tex]
d = [tex]\sqrt{(-6 + 2)^2 + (-10 + 10)^2}[/tex]
d = [tex]\sqrt{(-4)^2 + (0)^2}[/tex]
d = [tex]\sqrt{16 + 0}[/tex]
d = [tex]\sqrt{16}[/tex]
d = 4
So, the distance between the points is 4 units
The distance between each pair of points (-6,-10) , (-2,-10) is 4 units after using the distance formula.
What is a distance formula?It is defined as the formula for finding the distance between two points. It has given the shortest path distance between two points.
The distance formula can be given as:
[tex]\rm d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
It is given that:
The points are (-6,-10) , (-2,-10)
Using distance formula:
[tex]\rm d=\sqrt{(-2-(-6))^2+(-10-(-10))^2}[/tex]
d = √16
d = 4 units
Thus, the distance between each pair of points (-6,-10) , (-2,-10) is 4 units after using the distance formula.
Learn more about the distance formula here:
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Mrs. Ivy is grading math homework papers. For a certain problem, the correct answer equals −0.5. Choose all of the students that give a correct answer. Student 1: 1 −2 Student 2: − 1 2 Student 3: −1 2 Student 4: −1 −2
Answers
Answer:
student.1 1/-2 student.2 - 1/2 student. 3 1-/2
Step-by-step explanation:
brainly please
Which expression is equivalent to (X^5 +2X^2−4)−(−5X^5−3X^2+2X−7)?
Answers
Answer:
6[tex]x^{5}[/tex] + 5x² - 2x + 3
Step-by-step explanation:
Given
([tex]x^{5}[/tex] + 2x² - 4) - (-5[tex]x^{5}[/tex] - 3x² + 2x - 7)
Remove the parenthesis on the first factor and distribute the second by - 1
= [tex]x^{5}[/tex] + 2x² - 4 + 5[tex]x^{5}[/tex] + 3x² - 2x + 7 ← collect like terms
= 6[tex]x^{5}[/tex] + 5x² - 2x + 3 ← in standard form
2xy when x = 5 and y = -3
Answers
Answer:
the answer is -30
Step-by-step explanation:
2xy=2*5*-3
= -30
The value of the expression 2xy will be -30.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given expression is 2xy and x = 5 y = -3. To calculate the value put the value of x and y in the expression.
2xy=2 x 5 x -3
2xy= - 30
Therefore, the value of the expression 2xy will be -30.
To know more about an expression follow
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find the percent decrease from 40 to 24
Answers
-40
Explanation:
A change from 40 to 24 represents a negative change (decrease) of -40
24 - 40
|40|
x 100 = -16
40
x 100 = -40 % (decreased)
Combine like terms to reach the top of the pyramid by clicking and dragging
the expressions into the correct spots. The 1st one is done for you.
Answers
Sorry answered for more points for more questions I'm so sorry.
A line passes through the points (3,7) and (5,19) Part A: find the slope
Part B: what is a possible equation of a line parallel to the line through (3,7) and (5,19)
Y=mx + b
Answers
Answer:
The slope is 6. For the first line you could do y=6x+7, and the line parallel to that could be y= 6x+3
Step-by-step explanation:
For it to be parallel the slopes have to be the same and the y intercepts (b) have to be different. And I found the slope by doing y-y/x-x
plug in you numbers 19-7/5-3= 12/2 = 6