# Find angle from its sine?

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 $$\sin(\theta)=0.719$$?

• There are infinitely many solutions without further restrictions. Nov 25 '19 at 16:32
• So I saw somewhere that for sin(θ)=0.719, θ is 46 degrees. How did they find that? @AndrewChin Nov 25 '19 at 16:35
• 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. Nov 25 '19 at 16:41

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 $$\theta = n\pi + (-1)^n\sin^{-1}(0.695), \space n \in Z$$
• BUT: the periodicity of $\sin$ doesn't explain why $136^\circ$ is another solution between $0^\circ$ and $360^\circ$. Nov 25 '19 at 17:12
• Oops, my fault. The other solution for $\sin(x)=0.719$ is $180-46=134$ instead, which isn't explained by periodicity. Nov 25 '19 at 17:56