in a triangleABC,the angleA>angleB.if the values of anglesAandBsatisfy the eq: 3sinx-4sin^3x-k=0,find angleC

Question

1)pi/3
2)pi/2
3)2pi/3
plese xplain ur ans

Answer 1

It's hard to use clear notation without a special program for typing Maths, but I take it that sin^3x means "sine cubed x",
i.e. (sin x)^3. In that case, the first two terms of this equation are equal to sin 3x (standard formula), and so the equation is

sin 3x - k = 0.

Since A and B satisfy the equation, then sin 3A and sin 3B must each be equal to k.
i.e. sin 3A = sin 3B

However, 3A > 3B, and so the possibilities are

(1) 3A = pi - 3B, and so
A + B = pi/3
whence
C = pi - (A + B)
= 2pi/3

(2) 3A = 2pi + 3B
A = 2pi/3 + B
whence
C= pi - (A + B)
= pi - (2pi/3 + 2B)
= pi/3 - 2B.
Of course A, B, C must all be positive, and so B must be less than pi/6 (so that pi/3 - 2B will be positive) and C must be less than pi/3 (since B is positive). So any value of C:
0 < C < pi/3, is a correct solution.

(3) 3A = 3pi - 3B i.e. A = pi - B is not really a possibility as it leaves C=0.

So, unfortunately, I disagree with the problem as set, agreeing only with the third answer provided, and finding an infinite set of others.

If C were equal to pi/3, then A + B would be 2pi/3, and so
3A + 3B would be 2pi. But sin (2pi - 3A) = -sin 3A, which can equal sin 3A only if they are both zero. Since B > 0, the only possibility is 3A = pi = 3B, which contradicts A>B.

If C = pi/2, then 3A + 3B = 3pi/2, and so

sin 3B = sin(3pi/2 - 3A)
= - cos 3A, which can equal sin 3A only if 3A =3pi/4, and then
3B = 3pi/2 - 3pi/4
= 3pi/4,
again contradicting A>B.

Answer 2

The equation -4u^3 + 3u - k = 0 has discriminant 4 + k^2, which is positive; therefore there is only one real solution. Therefore the two solutions A and B have sines equal to the same number u, and

A = pi/2 - B

Since A + B + C = pi, we must have C = pi/2

Answer 3

Since angle A and angle B don't seem to show up anywhere in the equation you gave (and k seems to have just 'popped up' out of nowhere), I'd say you're just SOL.


Doug

Answer 4

my cat's breath smells like cat food...