ii) Reduce the given statement when we replace n by ( n) ? [1] iii) What does 'r' represent in the above statement ? [1] iv) What does ' ' represent in the above statement ? [1] v) Write any one…

The question refers to the trigonometric identity for sine of sum/difference of angles: $$\sin (n+r)=\sin n\cos r+\cos n\sin r$$

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ii) Reduction when $n$ is replaced by $(-n)$

When $n$ is replaced by $(-n)$, the identity becomes: $$\sin (-n+r)=\sin (-n)\cos r+\cos (-n)\sin r$$ Using the properties of trigonometric functions: $$\sin (-n)=-\sin n,\cos (-n)=\cos n$$ Thus, the expression simplifies to: $$\sin (r-n)=-\sin n\cos r+\cos n\sin r$$ This can be rewritten as: $$\sin (r-n)=\sin r\cos n-\cos r\sin n$$ This matches the sine of difference formula: $$\sin (r-n)=\sin r\cos n-\cos r\sin n$$

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iii) What does '$r$' represent?

In the given trigonometric identity: $$\sin (n+r)=\sin n\cos r+\cos n\sin r$$

• $r$ represents an angle (in radians or degrees) that is added to another angle $n$.
• It is typically a fixed or arbitrary angle used to expand the sine of a sum of two angles.
• In applications, $r$ can represent a phase shift, a known angle in a right triangle, or a variable angle in harmonic motion.

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iv) What does '$\theta$' represent?

(Assuming the question meant $\theta$ instead of $r$ or refers to a general angle in the identity.) If the identity were written as: $$\sin (n+\theta )=\sin n\cos \theta +\cos n\sin \theta$$

• $\theta$ represents an angle (often a known or reference angle) that is added to $n$.
• It can denote:
  • A phase angle in wave functions (e.g., $\theta =\omega t$ in harmonic motion).
  • A fixed angle in trigonometric expansions (e.g., $\theta =30^{\circ },45^{\circ },$ etc.).
  • A variable angle in trigonometric identities or proofs.

(If the question strictly refers to the original identity with $r$, then $\theta$ is not present. Clarification may be needed, but the above covers the general interpretation.)

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v) One application of the identity

The sine of sum/difference identity has wide applications in physics, engineering, and mathematics. One key application is:

1. Harmonic Motion and Wave Superposition

In physics, when two waves of the same frequency but different phases interfere, their resultant displacement can be expressed using this identity. For example:

• Let $y_1=A\sin (\omega t+{\varphi }_1)$ and $y_2=B\sin (\omega t+{\varphi }_2)$ be two waves.
• The sum $y=y_1+y_2$ can be rewritten using the sine addition formula to find the amplitude and phase of the resultant wave.

2. Electrical Engineering (Phasor Addition)

In AC circuits, voltages or currents with phase differences are added using trigonometric identities. For instance:

• If $V_1=V_0\sin (\omega t)$ and $V_2=V_0\sin (\omega t+\theta )$, their sum $V=V_1+V_2$ can be simplified using: $$V=V_0[\sin (\omega t)+\sin (\omega t+\theta )]=2V_0\sin \left(\omega t+\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)$$ This helps in analyzing circuit behavior.

3. Navigation and Astronomy

• In celestial navigation, the sine formula is used to calculate angles between stars, planets, or the horizon.
• For example, if an observer measures the angle $\alpha$ between the horizon and a star, and $\beta$ is the star’s declination, the identity helps compute the star’s altitude.

4. Computer Graphics (Rotation Matrices)

In 2D graphics, rotating a point $(x,y)$ by an angle $\theta$ uses the sine addition formula implicitly: $$x^{\prime }=x\cos \theta -y\sin \theta ,y^{\prime }=x\sin \theta +y\cos \theta$$ This is derived from the sine and cosine of sum/difference identities.

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(Note: If the original question referred to a specific identity like $\sin (n\theta )$ or another form, the answer would adjust accordingly. The above assumes the standard sine addition formula.)