SATHEE: Trigonometrical Equations 1 Question 15

Trigonometrical Equations 1 Question 15

15. In a $△ A B C$ , angle $A$ is greater than angle $B$ . If the measures of angles $A$ and $B$ satisfy the equation $3 \sin x - 4 \sin^{3} x - k = 0, 0 < k < 1$ , then the measure of $∠ C$ is

(1990, 2M)

(a) $\frac{π}{3}$

(b) $\frac{π}{2}$

(d) $\frac{5 π}{6}$

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Answer:

Correct Answer: 15. $(c)$

Solution:

Given, $3 \sin x - 4 \sin^{3} x = k, 0 < k < 1$ which can also be written as $\sin 3 x = k$ .

It is given that $A$ and $B$ are solutions of this equation. Therefore,

$\begin{array}{lrl} \sin 3 A = k and \sin 3 B = k, where 0 < k < 1 \\ \Rightarrow & 0 < 3 A < π and 0 < 3 B < π \\ Now, & \sin 3 A = k and \sin 3 B = k \\ \Rightarrow & \sin 3 A - \sin 3 B = 0 \\ \Rightarrow & 2 \cos \frac{3}{2} (A + B) \sin \frac{3}{2} (A - B) = 0 \end{array}$

$\Rightarrow \cos 3 \frac{A + B}{2} = 0, \sin 3 \frac{A - B}{2} = 0$

But it is given that, $A > B$ and $0 < 3 A < π, 0 < 3 B < π$ .

Therefore, $\sin 3 \frac{A - B}{2} \neq 0$

Hence, $\cos 3 \frac{A + B}{2} = 0$

$\begin{aligned} \Rightarrow 3 \frac{A + B}{2} = \frac{π}{2} \\ \Rightarrow A + B = \frac{π}{3} \\ \Rightarrow C = π - (A + B) = \frac{2 π}{3} \end{aligned}$

Trigonometrical Equations 1 Question 15

15. In a $△ A B C$ , angle $A$ is greater than angle $B$ . If the measures of angles $A$ and $B$ satisfy the equation $3 \sin x - 4 \sin^{3} x - k = 0, 0 < k < 1$ , then the measure of $∠ C$ is

Answer:

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Trigonometrical Equations 1 Question 15

15. In a △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 ∠C is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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15. In a $△ A B C$ , angle $A$ is greater than angle $B$ . If the measures of angles $A$ and $B$ satisfy the equation $3 \sin x - 4 \sin^{3} x - k = 0, 0 < k < 1$ , then the measure of $∠ C$ is