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Units and Measurements — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsUnits and Measurements3 Marks
Question
A function $\text{f}(\theta)$ is defined as: $\text{f}(\theta)=1-\theta+\frac{\theta^2}{2!}-\frac{\theta^3}{3!}+\frac{\theta^4}{4!}$ Why is it necessary for q to be a dimensionless quantity?
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Answer

$\theta$ is represented by angle which is equal to $\frac{\text{arc}}{\text{radius}}$ so angle $\theta$ is dimensionless physical quantity. First term is 1 which is dimensionless, next term contain only powers of $\theta$, as $\theta$ is dimensionless so their powers will also be dimensionless. Hence, each term in R.H.S. expression are dimensionless so left hand side $\text{f}(\theta)$ must be dimensionless.

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