Let f(θ) = sin θ (sin θ + sin 3θ). Then f(θ) is (A) ≥ 0 only when θ ≥ 0 (B) ≤ 0 for all real θ (C) ≥

Q: Wait, why can't I just expand $\sin 3\theta$ using the triple angle formula? Wouldn't that work?

You absolutely could! Expanding $\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$ is a valid algebraic path. However, in JEE Advanced, time and elegance are your best friends. Using the sum-to-product identity is much faster and less prone to algebraic errors. Always look for the 'path of least resistance' in trigonometry!

Q: Why is the answer $\ge 0$ and not just $> 0$? Can the function be zero?

That is a fantastic observation! The function $f(\theta) = \sin^2 2\theta$ is a square. Since the square of any real number is non-negative, it is always $\ge 0$. It equals zero whenever $\sin 2\theta = 0$, which happens at $\theta = \frac{n\pi}{2}$. So, it touches the x-axis but never dips below it.

Q: Is there a specific domain for $\theta$ I should worry about?

The question asks for the behavior of $f(\theta)$ for 'all real $\theta$'. Since $\sin^2 2\theta$ is defined for all real numbers and its range is $[0, 1]$, there are no domain restrictions to worry about. You are safe to proceed for all $\theta \in \mathbb{R}$.

Question

Let $f(	heta) = \sin 	heta (\sin 	heta + \sin 3	heta)$. Then $f(	heta)$ is

Options:

(A) $\ge 0$ only when $	heta \ge 0$

(B) $\le 0$ for all real $	heta$

(C) $\ge 0$ for all real $	heta$

(D) $\le 0$ only when $	heta \le 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers! Today, we are going to peel back the layers of a trigonometric expression that, at first glance, might seem like a standard textbook problem.

We are analyzing the function:
$$f(	heta) = \sin 	heta (\sin 	heta + \sin 3	heta)$$

The immediate instinct for many students is to reach for the triple angle formula for $\sin 3	heta$. While that is a valid algebraic maneuver, it is often messy and increases the probability of a sign error. Instead, let us embrace the elegance of trigonometric identities.

### Phase 1: The Sum-to-Product Transformation

Look closely at the term inside the bracket: $(\sin 3	heta + \sin 	heta)$. Whenever you see a sum of two sines, your mathematical intuition should trigger the sum-to-product identity:
$$\sin C + \sin D = 2 \sin\left(\frac{C+D}{2}ight) \cos\left(\frac{C-D}{2}ight)$$

By setting $C = 3	heta$ and $D = 	heta$, we simplify the geometric complexity of the function. Substituting these values, we get:
$$\sin 3	heta + \sin 	heta = 2 \sin\left(\frac{3	heta + 	heta}{2}ight) \cos\left(\frac{3	heta - 	heta}{2}ight)$$

Simplifying the angles, we find $\frac{4	heta}{2} = 2	heta$ and $\frac{2	heta}{2} = 	heta$. Thus, the bracketed expression collapses beautifully into:
$$2 \sin 2	heta \cos 	heta$$

### Phase 2: The Double Angle Realization

Now, let us reconstruct our original function with this new, simplified form:
$$f(	heta) = \sin 	heta \cdot (2 \sin 2	heta \cos 	heta)$$

If we rearrange these terms, we get:
$$f(	heta) = (2 \sin 	heta \cos 	heta) \cdot \sin 2	heta$$

The term $(2 \sin 	heta \cos 	heta)$ is the classic double angle identity for sine, which is $\sin 2	heta$. By substituting this back, our function becomes:
$$f(	heta) = \sin 2	heta \cdot \sin 2	heta = \sin^2 2	heta$$

### Phase 3: The Final Conclusion

We have arrived at the destination. The function **$f(	heta) = \sin^2 2	heta$** is a perfect square.

In the realm of real numbers, the square of any quantity is always non-negative. Therefore, **$\sin^2 2	heta \ge 0$** for all real values of $	heta$.

The graph of this function will oscillate between $0$ and $1$, touching the x-axis but never crossing into the negative region. This is the power of trigonometric identities—they transform a complex-looking expression into a simple, intuitive truth.

Animated Solution for Mathematics - Trigonometry: Let $f (θ) = sin θ (sin θ + sin 3 θ)$ . Then $f (θ)$ is

Select Answer:

Visualized Solution

Analyzing the Function $f (θ)$

The Sum-to-Product Identity

Applying the Identity

Simplifying the Angles

Reconstructing $f (θ)$

The Double Angle Formula

Final Simplified Form

Analyzing the Range

Final Conclusion

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

Phase 1

Phase 2

Phase 3

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(θ)=sinθ(sinθ+sin3θ). Then f(θ) is