Solve for all solutions

Question

Solve for all solutions

Solve 3sin(2w)+4sin(w)=0 for all solutions 0≤w<2π

Solve 3cos(2t)=3sin2(t)+2 for all solutions 0≤t<2π

Answer 1

Hi Yazmine W.,

For 3sin(2w) + 4sin(w) = 0

3[2sin(w)cos(w)] + 4sin(w) = 0, [sin(2u) = 2sin(u)cos(u)]

6sin(w)cos(w) + 4sin(w) = 0

2sin(w)[3cos(w) + 2] = 0,

2sin(w) = 0 and 3cos(w) + 2 = 0

sin(w) = 0 and cos(w) = -2/3

sin(w) = 0 at 0, π for 0 ≤ w < 2π

cos(w) = -2/3 at 2.301 radians (cos^-1(-2/3) = 2.031 - Quadrant II) and at 3.983 radians (2π - 2.301 - Quadrant III) for 0 ≤ w < 2π.

For 3cos(2t) = 3sin²(t) + 2, [ I'll assume sin2(t) is sin²(t) ]

cos(2t) = sin²(t) + 2/3

1 - 2sin²(t) = sin²(t) + 2/3, [ cos(2u) = 1 - 2sin²(u) ]

3sin²(t) = 1/3

sin²(t) = 1/9

sin(t) = ± 1/3

Therefore t = 0.340 radians (sin^-1(1/3) = 0.3398 - Quad I), 2.802 radians (π - 0.3398 - Quad II), 3.481 radians (π + 0.3398 - Quad. III), 5.943 radians (2π - 0.3398 - Quad IV), for 0 ≤ t < 2π.

I hope this helps, Joe.

Answer 2

3 sin 2w - 4 sin w = 0

3 (2 sin w cos w) - 4 sin w = 0

6 sin w cos w - 4 sin w = 0

sin w (6 cos w - 4) = 0

sin w = 0

or

6 cos w - 4 = 0

Can you solve for w and answer?

Answer 3

3sin(2w)+4sin(w)=0

3sin(2w)=-4sin(w)

sin(2w)=-4/3sin(w)

w=.01°