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# How do you find the value of $\sin 50\cos 25 - \cos 50\sin 25$ using the sum and the difference, double angle or half-angle formulas?

Last updated date: 13th Jun 2024
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Hint: In this question we need to find the method to find the value of $\sin 50\cos 25 - \cos 50\sin 25$. Trigonometry is a part of calculus and the basic ratios of trigonometric are sine and cosine which have their application in sound and lightwave theories. The trigonometric have vast applications in naval engineering such as determining the height of the wave and the tide in the ocean.

Complete step by step solution:
In this question we have given the trigonometric ratio as $\sin 50\cos 25 - \cos 50\sin 25$
Now we will consider the trigonometric identity for difference as,$\sin \left( {a - b} \right) = \sin a\cos b - \sin b\cos a$.
Now we will consider the trigonometric identity for the sum as$\sin \left( {a + b} \right) = \sin a\cos b + \sin b\cos a$
Now we will consider the trigonometric identity for half angle as,
$\sin \left( {\dfrac{A}{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos A}}{2}}$
Consider the trigonometric identity for full angle $\sin \left( {2A} \right) = 2\sin A\cos A$
From the above identities the correct formula to solve $\sin 50\cos 25 - \cos 50\sin 25$ is$\sin \left( {a - b} \right) = \sin a\cos b - \sin b\cos a$.
Now we will substitute $50$ for angle $a$ and $25$ for angle $b$ in the given above formula.
That is given by,
$\Rightarrow \sin 50\cos 25 - \cos 50\sin 25 = \sin \left( {50 - 25} \right)$
After simplification we will get,
$\therefore \sin 50\cos 25 - \cos 50\sin 25 = \sin 25$

Thus, the formula to find the correct value of $\sin 50\cos 25 - \cos 50\sin 25$ is the sum formula.

Note:
As we know that the sine angle formula is used to determine the ratio of perpendicular to height in a right-angle triangle. It is also used to determine the missing sides and the angles in other types of triangles.