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How do you solve $\sin ({180^0} - a) = \sin (a)$ ?

Last updated date: 20th Jun 2024
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Hint: Take out all the like terms to one side and all the alike terms to the other side. Take out all the common terms. Reduce the terms on the both sides until they cannot be reduced any further if possible. Then finally evaluate the value of the unknown variable.

Complete step-by-step solution:
First we will start off by applying the trigonometric identity.
$\sin (a - b) = \sin a\cos b - \sin b\cos a$
Then next we will substitute the values in the identity.
  \sin (a - b) = \sin a\cos b - \sin b\cos a \\
  \sin (180 - a) = \sin 180\cos a - \cos 180\sin a \\
Now we know that the value of $\sin {180^0}$ is $0$ and $\cos {180^0}$ is $ - 1$. Hence, now we substitute the value of $\sin {180^0}$ and $\cos {180^0}$.
  \sin (180 - a) = \sin 180\cos a - \cos 180\sin a \\
  \sin (180 - a) = 0.\cos a - \sin a( - 1) \\
  \sin (180 - a) = \sin a \\
Hence, the value of $\sin ({180^0} - a)$ is $\sin (a)$.

Additional Information: To cross multiply terms, you will multiply the numerator in the first fraction times the denominator in the second fraction, then you write that number down. Then you multiply the numerator of the second fraction times the number in the denominator of your first fraction, and then you write that number down. By Cross multiplication of fractions, we get to know if two fractions are equal or which one is greater. This is especially useful when you are working with larger fractions that you are not sure how to reduce. Cross multiplication also helps us to solve for unknown variables in fractions.

Note: While cross multiplying the terms, multiply the terms step-by-step to avoid any mistakes. Always take the variables to one side and integer type of terms to the other side. Also remember that $\sin {180^0}$ is $0$ and $\cos {180^0}$ is $ - 1$.