Question

How do you simplify the expression \[\sin \left( {\arctan \left( {\dfrac{{ - 4}}{3}} \right)} \right)\]?

Answer 1

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 answer:
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.
arctan is an inverse trigonometric function which can also be written as \[{\tan ^{ - 1}}\]. -1 here is just the way of showing that it is the inverse of tanx . Inverse tangent does the opposite of tangent. Tangent function gives the angle which is calculated by dividing the opposite side and adjacent side in a right-angle triangle, but the inverse of it gives the measure of an angle.
The cosine of the angle $ \theta $ is:
 $ \tan \theta = \dfrac{{Opposite}}{{Adjacent}} $
Therefore, inverse of cosine is
 $ \theta = {\tan ^{ - 1}}\left( {\dfrac{{opposite}}{{Adjacent}}} \right) $
In the above tangent function, we have to find inverse of tangent function.Given,
\[\arctan \left( {\dfrac{{ - 4}}{3}} \right)\] which can also be written as \[{\tan ^{ - 1}}\left( {\dfrac{{ - 4}}{3}} \right)\].
Since the argument is negative which means it lies in the fourth quadrant with opposite side 4 and adjacent side 3 and therefore the hypotenuse is 5.
Hence the ratio of sine from the triangle is \[ - \dfrac{4}{5}\].
So, the correct answer is “ \[ - \dfrac{4}{5}\]”.

Note: An important thing to note is that the formula for sine function is the opposite side divided by hypotenuse. Here the opposite side is 4 and hypotenuse is 5 so by substituting in the sine function formula we get \[ - \dfrac{4}{5}\].