In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B satisfy the eq

Q: Wait, why can't we just solve for $x$ directly using the quadratic formula or something?

Because $k$ is a parameter, not a specific number! We don't know the value of $k$, so we cannot isolate $x$ numerically. We must treat the equation as a functional relationship and use the properties of the roots.

Q: Why did we reject the case where $\sin(\frac{3(A-B)}{2}) = 0$?

Great catch! If that factor were zero, then $3(A-B)/2$ would have to be a multiple of $\pi$. However, $A$ and $B$ are angles of a triangle, and we are given $A > B$. The difference is constrained by the geometry of the triangle, making that specific case impossible for the given conditions.

Q: Is it possible for $\cos(\frac{3(A+B)}{2})$ to be something other than $0$?

In a general trigonometric equation, yes. But here, $A$ and $B$ are constrained by the geometry of a triangle. We are looking for the specific solution that fits the physical reality of the triangle, which leads us directly to the value that satisfies the angle sum property.

Question

In a triangle $ABC$, 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 angle $C$ is

Options:

(A) $\pi/3$

(B) $\pi/2$

(C) $2\pi/3$

(D) $5\pi/6$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, open field, holding a triangle $ABC$. You know the basics: the sum of the angles is $\pi$, and the sides are connected by the Law of Sines.

We are given a trigonometric equation: $3 \sin x - 4 \sin^3 x - k = 0$. At first glance, this looks like a standard, perhaps even intimidating, cubic equation in terms of $\sin x$.

However, the expression $3 \sin x - 4 \sin^3 x$ is the triple angle identity for sine, $\sin 3x$. By recognizing this, we transform the complex cubic into the elegant statement:

$$\sin 3x = k$$

### Connecting Algebra to Geometry

The problem states that $A$ and $B$ satisfy this equation. This means $A$ and $B$ are the roots of $\sin 3x = k$, giving us the relations $\sin 3A = k$ and $\sin 3B = k$.

Since both are equal to $k$, we set them equal to each other: $\sin 3A = \sin 3B$. Bringing them to one side, we obtain:

$$\sin 3A - \sin 3B = 0$$

We now apply the sum-to-product identity, $\sin C - \sin D = 2 \cos(\frac{C+D}{2}) \sin(\frac{C-D}{2})$. Applying this to our equation, we get:

$$2 \cos\left(\frac{3A+3B}{2}\right) \sin\left(\frac{3A-3B}{2}\right) = 0$$

### The Trapdoor and the Solution

We have a product of two terms equal to zero. We are given that $A > B$, which implies $A - B > 0$.

Because $A$ and $B$ are angles of a triangle, their difference is constrained such that the term $\sin(\frac{3(A-B)}{2})$ cannot be zero. We can safely discard this factor.

This leaves us with the cosine term:

$$\cos\left(\frac{3(A+B)}{2}\right) = 0$$

For the cosine of an angle to be zero, the angle itself must be an odd multiple of $\frac{\pi}{2}$. In the context of our triangle, we set:

$$\frac{3(A+B)}{2} = \frac{\pi}{2}$$

### Final Calculation

With a quick algebraic simplification, we cancel the denominators to find that $3(A+B) = \pi$, which implies:

$$A+B = \frac{\pi}{3}$$

We know the fundamental property of any triangle: $A + B + C = \pi$. Substituting our value for $A+B$, we get:

$$\frac{\pi}{3} + C = \pi$$

Solving for $C$, we find the final result:

**$C = \frac{2\pi}{3}$**

Animated Solution for Mathematics - Trigonometry: In a triangle $A B C$ , 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 angle $C$ is

Select Answer:

Visualized Solution

Triangle Setup

The Trigonometric Equation

Multiple Angle Identity

Substituting the Roots

Equating the Expressions

Sum-to-Product Transformation

Checking the Constraints

Setting Cosine to Zero

Finding the Sum of Angles A and B

The Triangle Angle Sum

Calculating Angle C

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

Connecting Algebra to Geometry

The Trapdoor and the Solution

Final Calculation

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