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How do you evaluate \[\sin \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{5}} \right)} \right)\] ?

Last updated date: 01st Mar 2024
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IVSAT 2024
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Hint: In the above question, is based on the inverse trigonometry concept. The trigonometric functions are the relationship between the angles and the sides of the triangle. Since measure is given in the function, we need to find the angle of that particular measure of trigonometric function.

Complete step by step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle. \[{\cos ^{ - 1}}\] is an inverse trigonometric function -1 here is just the way of showing that it is inverse of cosx. Inverse cosine does the opposite of cosine. Cosine function gives the angle which is calculated by dividing the adjacent side and hypotenuse in a right-angle triangle, but the inverse of it gives the measure of an angle. The given expression is
\[\sin \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{5}} \right)} \right)\]

So now the angle \[\theta \]
\[\cos \theta = \left( {\dfrac{{Adjacent}}{{Hypotenuse}}} \right)\]
\[\Rightarrow\theta = {\cos ^{ - 1}}\left( {\dfrac{{Adjacent}}{{Hypotenuse}}} \right)\]
In the above cosine function, we have to find the inverse of cosine function. Given,
\[\cos \theta = \dfrac{3}{5}\]
Since 3 is the adjacent side in a right-angle triangle therefore the opposite side will be 4.
The formula for sine function is
\[\therefore\sin \theta = \left( {\dfrac{{Opposite}}{{Hypotenuse}}} \right) = \dfrac{4}{5}\]
Since the formula is the opposite side divided by hypotenuse of a right-angle triangle, we therefore get the above value.

Note: An important thing to note is that the opposite side is calculated so we calculate the value of outer function i.e., sine function. The opposite side can be calculated by applying Pythagoras theorem as
\[{\left( {Opposite} \right)^2} = {\left( {Hypotenuse} \right)^2} - {\left( {Adjacent} \right)^2} = 25 - 9 = 16\] and then taking square root we get the value of opposite side as 4 .
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