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How can I find the actual value of an angle if I know the value of its sine?

Supposing that the angle I am looking for is named as $\theta$, what's it's value if $\sin(\theta)=0.719$?

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  • $\begingroup$ There are infinitely many solutions without further restrictions. $\endgroup$ Nov 25 '19 at 16:32
  • $\begingroup$ So I saw somewhere that for sin(θ)=0.719, θ is 46 degrees. How did they find that? @AndrewChin $\endgroup$
    – Themelis
    Nov 25 '19 at 16:35
  • $\begingroup$ Seems like there's an answer below for you. $\sin^{-1} x$ or $\arcsin x$ is the inverse function of $\sin x$ and is used to solve for $x$ in a trigonometric equation. $\endgroup$ Nov 25 '19 at 16:41
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The function you're looking for is called $\sin^{-1}(x)$. Answering your question, $$\theta = \sin^{-1}(0.695) \approx 0.768 \space rad$$

since $\sin(x)$ is a periodic function, the value will repeat itself. In these cases, it is solved by using a trigonometric equation. The general solution of the above equation is $$\theta = n\pi + (-1)^n\sin^{-1}(0.695), \space n \in Z$$

These can be found in any high school maths textbook as well.

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  • $\begingroup$ BUT: the periodicity of $\sin$ doesn't explain why $136^\circ$ is another solution between $0^\circ$ and $360^\circ$. $\endgroup$ Nov 25 '19 at 17:12
  • $\begingroup$ @MichaelHoppe I'm unable to understand.. 136° is a solution for what? $\endgroup$ Nov 25 '19 at 17:16
  • $\begingroup$ Oops, my fault. The other solution for $\sin(x)=0.719$ is $180-46=134$ instead, which isn't explained by periodicity. $\endgroup$ Nov 25 '19 at 17:56

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