Trigonometric EquationHard
Question
In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B satisfy the equation 3sin x - 4sin3 x - k = 0,0 < k < 1, then the measure of angle C is
Options
A.
B.
C.
D.
Solution
Given, 3sin x - 4sin3 x = k,0 < k < 1 which can also be written as sin 3x = k.
It is given that A and B are solutions of this equation.
Therefore, sin3A = k and sin3B = k, where 0 < k < 1
⇒ 0 < 3A < π and 0 < 3B < π
Now, sin3A = k and sin3B = k
⇒ sin3A - sin3B = 0
⇒ 2 cos
(A + B) sin (A - B) = 0
⇒
But it is given that, A > B and 0 < 3A < π, 0 < B < π
Therefore, sin3
Hence,
⇒
⇒ A + B =
⇒ C = π -(A + B) =
It is given that A and B are solutions of this equation.
Therefore, sin3A = k and sin3B = k, where 0 < k < 1
⇒ 0 < 3A < π and 0 < 3B < π
Now, sin3A = k and sin3B = k
⇒ sin3A - sin3B = 0
⇒ 2 cos
(A + B) sin (A - B) = 0⇒
But it is given that, A > B and 0 < 3A < π, 0 < B < π
Therefore, sin3
Hence,
⇒
⇒ A + B =
⇒ C = π -(A + B) =
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