1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
slava [35]
2 years ago
15

Prove that: (b²-c²/a)CosA+(c²-a²/b)CosB+(a²-b²/c)CosC = 0​

Mathematics
1 answer:
IRISSAK [1]2 years ago
7 0

<u>Prove that:</u>

\:\:\sf\:\:\left(\dfrac{b^2-c^2}{a}\right)\cos A+\left(\dfrac{c^2-a^2}{b}\right)\cos B +\left(\dfrac{a^2-b^2}{c}\right)\cos C=0

<u>Proof: </u>

We know that, by Law of Cosines,

  • \sf \cos A=\dfrac{b^2+c^2-a^2}{2bc}
  • \sf \cos B=\dfrac{c^2+a^2-b^2}{2ca}
  • \sf \cos C=\dfrac{a^2+b^2-c^2}{2ab}

<u>Taking</u><u> </u><u>LHS</u>

\left(\dfrac{b^2-c^2}{a}\right)\cos A+\left(\dfrac{c^2-a^2}{b}\right)\cos B +\left(\dfrac{a^2-b^2}{c}\right)\cos C

<em>Substituting</em> the value of <em>cos A, cos B and cos C,</em>

\longmapsto\left(\dfrac{b^2-c^2}{a}\right)\left(\dfrac{b^2+c^2-a^2}{2bc}\right)+\left(\dfrac{c^2-a^2}{b}\right)\left(\dfrac{c^2+a^2-b^2}{2ca}\right)+\left(\dfrac{a^2-b^2}{c}\right)\left(\dfrac{a^2+b^2-c^2}{2ab}\right)

\longmapsto\left(\dfrac{(b^2-c^2)(b^2+c^2-a^2)}{2abc}\right)+\left(\dfrac{(c^2-a^2)(c^2+a^2-b^2)}{2abc}\right)+\left(\dfrac{(a^2-b^2)(a^2+b^2-c^2)}{2abc}\right)

\longmapsto\left(\dfrac{(b^2-c^2)(b^2+c^2)-(b^2-c^2)(a^2)}{2abc}\right)+\left(\dfrac{(c^2-a^2)(c^2+a^2)-(c^2-a^2)(b^2)}{2abc}\right)+\left(\dfrac{(a^2-b^2)(a^2+b^2)-(a^2-b^2)(c^2)}{2abc}\right)

\longmapsto\left(\dfrac{(b^4-c^4)-(a^2b^2-a^2c^2)}{2abc}\right)+\left(\dfrac{(c^4-a^4)-(b^2c^2-a^2b^2)}{2abc}\right)+\left(\dfrac{(a^4-b^4)-(a^2c^2-b^2c^2)}{2abc}\right)

\longmapsto\dfrac{b^4-c^4-a^2b^2+a^2c^2}{2abc}+\dfrac{c^4-a^4-b^2c^2+a^2b^2}{2abc}+\dfrac{a^4-b^4-a^2c^2+b^2c^2}{2abc}

<em>On combining the fractions,</em>

\longmapsto\dfrac{(b^4-c^4-a^2b^2+a^2c^2)+(c^4-a^4-b^2c^2+a^2b^2)+(a^4-b^4-a^2c^2+b^2c^2)}{2abc}

\longmapsto\dfrac{b^4-c^4-a^2b^2+a^2c^2+c^4-a^4-b^2c^2+a^2b^2+a^4-b^4-a^2c^2+b^2c^2}{2abc}

<em>Regrouping the terms,</em>

\longmapsto\dfrac{(a^4-a^4)+(b^4-b^4)+(c^4-c^4)+(a^2b^2-a^2b^2)+(b^2c^2-b^2c^2)+(a^2c^2-a^2c^2)}{2abc}

\longmapsto\dfrac{(0)+(0)+(0)+(0)+(0)+(0)}{2abc}

\longmapsto\dfrac{0}{2abc}

\longmapsto\bf 0=RHS

LHS = RHS proved.

You might be interested in
A right triangle has a leg measuring 52 m and a hypotenuse measuring 63 m. find the length of the other leg.
tangare [24]
By Pythagoras Theorem,
the length of the other leg= (63^2-52^2)^1/2
hope this helps :)
4 0
3 years ago
I need help with this pls
Vadim26 [7]

Answer:

I tho this was a game at first but gurl i can see the shape and everything but what im i supposed to do?

just tell me the instructions and ill help

8 0
3 years ago
(x²+4x+4)/(x+2)<br>algebraic expressions class 8<br>no spamms​
maks197457 [2]

Answer:

x+2

Step-by-step explanation:

\frac{x ^{2} + 4x + 4 }{x + 2}  \\ \frac{(x) ^{2} + 2 \times x \times 2 + (2) ^{2}  }{x + 2}  \\ \   \frac{(x + 2) ^{2} }{x + 2}  \\  \frac{(x + 2)(x + 2)}{x + 2}  \\ x + 2

5 0
2 years ago
How do you solve 21/4=1/2x+3x and whats the answer?
MissTica
Hi there!

To solve is pretty easy. First, simplify.

21/4=(1/2x+3x) -> Combine Like Terms
21/4=7/2x
FLIP
7/2x=21/4

Multiply both sides by 2/7.
(2/7)*(7/2x)=(2/7)*(21/4)
x = 2/3

Hope this helps!
6 0
3 years ago
What is the triangle proportionality therom
zloy xaker [14]

If a line <span>parallel </span>to one side of a <span>triangle </span>intersects the other two sides of the triangle, then the line divides these two sides proportionally.


5 0
3 years ago
Other questions:
  • Someone help me answer these 3 questions because they are due by 5:00 today plz questions 12-14
    5·1 answer
  • Michael puts 3 beads next to each other on a pice of string to make a bracelet. Each bead has a width of 5/6 inch. Which express
    8·1 answer
  • 2. The sum of the third and the seventh terms
    5·1 answer
  • What is the area of the polygon?
    11·1 answer
  • What is the distance between (-6, 4)(−6,4)left parenthesis, minus, 6, comma, 4, right parenthesis and (-8, 6)(−8,6)left parenthe
    9·1 answer
  • If h(x)=(f o g) (x) and h(x)= ∛x+3, find g( x) if f(x)= ∛x+2 (picture attached)
    11·1 answer
  • Answer these year 7 math questions please ty !
    13·2 answers
  • Help me with this pls !
    11·2 answers
  • A car that originally cost $4,218 in 1955 is valued at $68,000 if in excellent condition. This is 414 times as much as a car in
    9·1 answer
  • What is the solution to the equation 5x=6?
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!