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Eduardwww [97]
3 years ago
14

Prove Sin3theta = 3sintheta - 4sin^3theta

Mathematics
1 answer:
trapecia [35]3 years ago
3 0

Express the left hand side as

sin3theta=sin(theta+2theta)

now expand the right side of this equation using color(blue)"Addition formula"

color(red)(|bar(ul(color(white)(a/a)color(black)(sin(A±B)=sinAcosB±cosAsinB)color(white)(a/a)|)))

rArrsin(theta+2theta)=sinthetacos2theta+costhetasin2theta.......(A)

color(red)(|bar(ul(color(white)(a/a)color(black)(cos2theta=cos^2theta-sin^2theta=2cos^2theta-1=1-2sin^2theta)color(white)(a/a)|)))

The right hand side is expressed only in terms of sintheta's

so we use cos2theta=1-2sin^2theta........(1)

color(red)(|bar(ul(color(white)(a/a)color(black)(sin2theta=2sinthetacostheta)color(white)(a/a)|)))........(2)

Replace cos2theta" and " sin2theta by the expansions (1) and (2)
into (A)

sin(theta+2theta)=sinthetacolor(red)((1-2sin^2theta))+costhetacolor(red)((2sinthetacostheta)

and expanding brackets gives.

sin(theta+2theta)=sintheta-2sin^3theta+2sinthetacos^2theta....(B)

color(red)(|bar(ul(color(white)(a/a)color(black)(cos^2theta+sin^2theta=1rArrcos^2theta=1-sin^2theta)color(white)(a/a)|)))

Replace cos^2theta=1-sin^2theta" into (B)"

rArrsin(theta+2theta)=sintheta-2sin^3theta+2sintheta(1-sin^2theta)

and expanding 2nd bracket gives.

sin(theta+2sintheta)=sintheta-2sin^3theta+2sintheta-2sin^3theta

Finally, collecting like terms.

sin3theta=3sintheta-4sin^3theta="R.H.S hence proven"
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The ratio of the length and breadth of a rectangle is 5:4 and its area is 180 cm
Fed [463]

Answer:

Length = 15 cm, width = 12 cm, Perimeter = 54 cm

Step-by-step explanation:

The ratio of the length and breadth of a rectangle is 5:4 and its area is 180 cm².

Length = 5x

Breadth = 4x

Area of rectagle,

A = bl

180 = 5x × 4x

180 = 20x²

x² = 9

x = 3

Length = 5x = 5(3) = 15 cm

Width = 4x = 4(3) = 12 cm

Perimeter of rectangel,

P = 2(l+b)

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Hence, this is the required solution.

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3 years ago
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Klio2033 [76]
Remember that x/4 and 1/4*x is the same thing.

1/4*52 is therefore the same as 52/4. Fractions represent division, so 52 divided by 4 is also correct.
0.25*x is the same as 1/4*x because 1/4=0.25
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Final answer: B
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Mortgage option (3) would be best suited for them.

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Now, in the option (3) the interest is to be paid for 8 years and the annual rate of interest is also relatively low (only 4%) and if they pay about $18,000 annually with a down-payment of $ 34,000 and  repay the rest of the amount at the end of 8 years,(which would be less than $ 35,000)  they can easily clear their mortgage. Hence, for option (3) they would need to pay lowest total amount and for lowest time to clear the mortgage among the three options. Hence, this would be best suited option for them.

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