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hichkok12 [17]
3 years ago
10

Show that cos(3x) = cos3(x) − 3 sin2(x)cos(x). (Hint: Use cos(2x) = cos2(x) − sin2(x) and the cosine sum

Mathematics
1 answer:
otez555 [7]3 years ago
4 0

<u>Step-by-step explanation:</u>

To prove:

\cos 3x=\cos^3x-3\sin^2 x\cos x

Identities used:

\cos(A+B)=\cos A\cos B+\sin A\sin B      ......(1)

\sin 2A=2\sin A\cos A        ........(2)

\cos 2A=\cos^2A-\sin^2 A     .......(3)

Taking the LHS:

\Rightarrow \cos 3x=\cos (x+2x)

Using identity 1:

\Rightarrow \cos (x+2x)=\cos x\cos 2x-\sin x\sin 2x

Using identities 2 and 3:

\Rightarrow \cos x(\cos ^2x-\sin^2 x)-\sin x(2\sin x\cos x)\\\\\Rightarrow \cos^3x-\sin^2x\cos x-2\sin^2 x\cos x\\\\\Rightarrow \cos^3x-3\sin^2x\cos x

As, LHS = RHS

Hence proved

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Answer:

2/3

Step-by-step explanation:

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3 years ago
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Perimeter of a rectangle with these dimentions w--12 l--10
GuDViN [60]
Perimeter = 2w + 2l = 2(12) + 2(10) = 44 
4 0
3 years ago
Complete the square on the following quadratic x2+10x+9=0
kakasveta [241]
Given a quadratic equation x^{2} +ax+c=0

1. the first thing we do when we want to compete the square, is write the coefficient of x as 2 times a number.

In our case the coefficient of x is 10, so we write 10 as 2*5

2. then we write + 5^{2} and - 5^{2} to the expression:

x^{2} +2*5*x+ 5^{2} -5^{2}+9=0

(x^{2} +2*5*x+ 5^{2}) -25+9=0

(x+5)^{2}-16=0



Answer: (x+5)^{2}-16=0
8 0
3 years ago
A particle begins at point (1, 2) and is moving along the line segment joining (1, 2) to (3, 4). Initially, what is the rate of
Bumek [7]

Answer:

Initially, that is, at (1,2), the rate of increase of the function in the direction given = 0

Hence the function is neither increasing nor decreasing initially in the given direction.

Step-by-step explanation:

f(x,y) = 2x² - y²

Rate of Change of the function = ∇f = (fₓ, fᵧ)

And we're told to find the rate of change in a particular direction = ∇f.û

We first obtain the unit vector in that direction, û

Direction = (3,4) - (1,2) = (2,2)

Uni vector = vector/magnitude

Vector = 2î + 2j

magnitude = √(2² + 2²) = √8 = 2 √2

(2î + 2j)/(2√2) = (1/√2)î + (1/√2)j = û

Unit vector in the direction = (1/√2, 1/√2)

f = 2x² - y²

At (1,2)

fₓ = ∂f/∂x = 4x = 4

fᵧ = ∂f/∂y = -2y = -4

∇f = (4,-4)

∇f.û = (4î - 4j).((1/√2)î + (1/√2)j) = (4/√2) - (4/√2) = 0

If it was positive, then the function is increasing, if it was negative, the function is decreasing. Zero means neither of them.

Hence the function is neither increasing nor decreasing initially in the given direction.

7 0
3 years ago
Solve the simultaneous equations<br>8x – 3y = 30<br>3x + y = 7​
hodyreva [135]
Solve for y in 3x+7=y

y=7−3x

Substitute y=7−3x

y=7−3x into 8x−3y=30

8x−3y=30.

17x−21=30

Solve for x

x in 17x−21=30

17x−21=30.

X=3

Substitute x=3 into y=7−3x

y=-2

Therefore,

x=3

y=−2

​

6 0
3 years ago
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