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Volgvan
2 years ago
10

If x = a sin α, cos β, y = b sin α.sin β and z = c cos α then (x²/a²) + (y²/b²) + (z²/c²) = ?​

Mathematics
1 answer:
Oduvanchick [21]2 years ago
5 0

\large\underline{\sf{Solution-}}

<u>Given:</u>

\rm \longmapsto x = a \sin \alpha  \cos \beta

\rm \longmapsto y = b \sin \alpha  \sin \beta

\rm \longmapsto z = c\cos \alpha

Therefore:

\rm \longmapsto \dfrac{x}{a}  = \sin \alpha  \cos \beta

\rm \longmapsto \dfrac{y}{b}  = \sin \alpha  \sin \beta

\rm \longmapsto \dfrac{z}{c} = \cos \alpha

Now:

\rm =  \dfrac{ {x}^{2} }{ {a}^{2}} +  \dfrac{ {y}^{2} }{ {b}^{2} } +  \dfrac{ {z}^{2} }{ {c}^{2} }

\rm =  { \sin}^{2} \alpha  \cos^{2}  \beta   +  { \sin}^{2} \alpha  \sin^{2} \beta  +  { \cos}^{2} \alpha

\rm =  { \sin}^{2} \alpha  (\cos^{2}  \beta   +  \sin^{2} \beta  )+  { \cos}^{2} \alpha

\rm =  { \sin}^{2} \alpha \cdot1+  { \cos}^{2} \alpha

\rm =  { \sin}^{2} \alpha + { \cos}^{2} \alpha

\rm = 1

<u>Therefore:</u>

\rm \longmapsto\dfrac{ {x}^{2} }{ {a}^{2}} +  \dfrac{ {y}^{2} }{ {b}^{2} } +  \dfrac{ {z}^{2} }{ {c}^{2} }  = 1

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Answer:

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We are given the following function in the question:

f(x) = \ln (3x^2)

We have to derivate the given function.

Formula:

\dfrac{d(\ln x)}{dx} = \dfrac{1}{x}\\\\\dfrac{d(x^n)}{dx} = nx^{n-1}

The derivation takes place in the following manner

f(x) = \ln (3x^2)\\\\\dfrac{d(f(x))}{dx} = \displaystyle\frac{d(\ln(3x^2))}{dx}\\\\=\frac{1}{3x^2}\times \frac{d(3x^2)}{dx}\\\\= \frac{1}{3x^2}\times (6x)\\\\=\frac{6x}{3x^2}\\\\=\frac{2}{x}

\dfrac{d(f(x))}{dx} = \dfrac{2}{x}

5 0
2 years ago
If the value of y varies directly with the value of x. If the value of y is 20 when x is 1/2, what is the value of y when did 4
inysia [295]
If i understand the question correctly then the answer you are looking for is 9
6 0
3 years ago
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Of the 32 students in a certain class, 15 are in a music club and 20 are in a dance club. if 10 of the students are not in eithe
mel-nik [20]
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3 years ago
URGENT!!!!!!!!
LuckyWell [14K]

Answer:

x+y=24 so we can say

x=24-y making 3x+5y=100 become

3(24-y)+5y=100

72-3y+5y=100

72+2y=100

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y=14, since x=24-y

x=10

So there are 10 3-point questions and 14 5-point questions.

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2 years ago
The length of the base of a triangle is twice its height. if the area of the triangle is 225 square kilometers, find the height
Anon25 [30]
Let
x----------> length of the base
y----------> height

we know that
[the area of triangle]=[length of the base]*[height]/2
x=2y
A=225 km²
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the answer is
the height is 15 km

3 0
3 years ago
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