ABC is a triangle such that sin(2A + B) = sin(C - A) = -sin(B + 2C) = 1/2. If A, B and C are in arit

Q: Wait, why do we reject the negative root for $A$?

In geometry, specifically within the context of a triangle, all interior angles must be strictly positive ($> 0^\circ$). If we solve $2A + 60^\circ = 30^\circ$, we get $2A = -30^\circ$, which implies $A = -15^\circ$. Since a triangle cannot have a negative angle, this solution is physically impossible and must be discarded.

Q: Why can't we just use energy conservation or some other law?

This is a pure geometry and trigonometry problem, not a mechanics problem! There is no energy to conserve here. We rely on the fundamental properties of triangles (the angle sum property) and the algebraic constraints provided by the Arithmetic Progression condition.

Q: How do I know which quadrant to pick for the sine equation?

Great question! The equation $\sin(2A + B) = \frac{1}{2}$ has solutions at $30^\circ$ and $150^\circ$ (and others, but these are the relevant ones). Since $A$ and $B$ are angles of a triangle, $2A + B$ must be less than $360^\circ$. By testing both $30^\circ$ and $150^\circ$, we ensure we don't miss any valid geometric configurations.

Question

$ABC$ is a triangle such that $\sin(2A + B) = \sin(C - A) = -\sin(B + 2C) = 1/2$. If $A, B$ and $C$ are in arithmetic progression, determine the values of $A, B$ and $C$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We begin with the most fundamental truth of Euclidean geometry: in any triangle $ABC$, the sum of the interior angles is always $180^\circ$. We write this as $A + B + C = 180^\circ$.

The problem introduces a beautiful constraint: the angles $A, B,$ and $C$ are in an Arithmetic Progression (AP). In the language of algebra, this means the middle term $B$ is the arithmetic mean of $A$ and $C$.

We express this as $2B = A + C$. By substituting $A + C = 2B$ into our angle sum equation, we get $(A + C) + B = 180^\circ$, which simplifies to $2B + B = 180^\circ$.

Suddenly, the complexity vanishes. We find that $3B = 180^\circ$, or **$B = 60^\circ$**. We have unlocked the first piece of our puzzle.

### The Trigonometric Gatekeeper

Now that we know $B = 60^\circ$, we turn our attention to the trigonometric equation provided: $\sin(2A + B) = \frac{1}{2}$. This looks intimidating, but it is merely a gatekeeper.

By substituting our known value of $B$, the equation becomes:

$$\sin(2A + 60^\circ) = \frac{1}{2}$$

We must ask ourselves: for which angles is the sine value equal to $\frac{1}{2}$? We know from our trigonometric tables that $\sin(30^\circ) = \frac{1}{2}$ and $\sin(150^\circ) = \frac{1}{2}$.

Therefore, the argument $(2A + 60^\circ)$ must be either $30^\circ$ or $150^\circ$. This leads us to two distinct cases.

### The Path to the Solution

Let us test the first case: $2A + 60^\circ = 30^\circ$. Subtracting $60^\circ$ from both sides gives $2A = -30^\circ$, which means $A = -15^\circ$.

As we discussed, a triangle cannot have a negative angle. We must reject this path.

Now, let us test the second case: $2A + 60^\circ = 150^\circ$. Subtracting $60^\circ$ from both sides yields $2A = 90^\circ$, which gives us **$A = 45^\circ$**.

This is a valid, positive angle! With $A = 45^\circ$ and $B = 60^\circ$ in hand, finding $C$ is trivial. We return to our initial sum: $A + B + C = 180^\circ$.

Substituting our values, we get $45^\circ + 60^\circ + C = 180^\circ$, which simplifies to $105^\circ + C = 180^\circ$. Thus, **$C = 75^\circ$**.

### The Final Verification

In the world of JEE, verification is not just a suggestion; it is a habit of excellence. Let us check our work against the other given equations.

We are told $\sin(C - A) = \frac{1}{2}$. With $C = 75^\circ$ and $A = 45^\circ$, we have:

$$\sin(75^\circ - 45^\circ) = \sin(30^\circ) = \frac{1}{2}$$

It matches perfectly! We also check $-\sin(B + 2C) = \frac{1}{2}$. Substituting our values, we get:

$$-\sin(60^\circ + 150^\circ) = -\sin(210^\circ)$$

Since $\sin(210^\circ) = -\frac{1}{2}$, the expression becomes $-(-\frac{1}{2}) = \frac{1}{2}$. Everything is consistent.

We have successfully navigated the constraints to find the angles **$A = 45^\circ, B = 60^\circ,$ and $C = 75^\circ$**.

Animated Solution for Mathematics - Trigonometry: $A B C$ is a triangle such that $sin (2 A + B) = sin (C - A) = - sin (B + 2 C) = 1/2$ . If $A, B$ and $C$ are in arithmetic progression, determine the values of $A, B$ and $C$ .

Visualized Solution

Angle Sum Property

Angles in Arithmetic Progression

Substituting AP Condition

Calculating Angle $B$

Using the Sine Equation

Evaluating the Sine Argument

Checking the First Case

Solving for Angle $A$

Setting up for Angle $C$

Calculating Angle $C$

Verification and Final Answer

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Trigonometric Gatekeeper

The Path to the Solution

The Final Verification

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Visualized Solution

Angle Sum Property

Angles in Arithmetic Progression

Substituting AP Condition

Calculating Angle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Using the Sine Equation

Evaluating the Sine Argument

Checking the First Case

Solving for Angle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A

Setting up for Angle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Calculating Angle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Verification and Final Answer

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Trigonometric Gatekeeper

The Path to the Solution

The Final Verification

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