Equal sign or approximation sign?

Question

I want to solve the equation $2\enspace \sin^2\enspace \theta+2\enspace \sin\enspace \theta-1=0$, $0\leq\theta<2\pi$. Using the quadratic formula, I found one of the solution to be $\theta=\sin^{-1}( \frac{-1+\sqrt{3}}{2})$. Which of the following ways of writing the numerical value of $\theta$ is correct? (i) or (ii)?

(i) $\theta=0.3747$, (ii) $\theta\approx 0.3747$

Are both of them acceptable? I noticed that, frequently, the numerical solution of a trigonometric equation is written using the equal sign (as in (ii)) even the numerical answer has been rounded. Why equal sign is used instead of approximation sign? In this case, does the equal sign have a different meaning?

Answer 1

It is correct to use the $\approx$ sign for rounded answers, and the $=$ sign for closed form answers. For example, if you are solving $\sin x=\cos x$, one solution would be $x=\frac{\pi}4\approx 0.866$. The use of an equal sign is probably because of a bad textbook, or that the writers used it because the rounded answer is very close to the real answer. This is usually the case when the answers are off by less than $\pm \epsilon$, which is determined by the writers (usually $10^{-4}$)

Answer 2

The equal sign $=$ means that the LHS and the RHS indicate the same number so, if we write $\theta=\sin^{-1} \left(\frac {\sqrt{3}-1}{2} \right)$, this means that $\theta$ is exactly the real number indicated by the symbols in the RHS.

The symbol of approximation $\approx$ means that the LHS and the RHS can differ by a quantity that is less than a value, usually little with respect to the values of the two quantities an that can be explicitely specified or can be deduced from the contest.

This is the case when we write: $\theta \approx 0.3747$, where the approximation is suggested by the number of decimals.