Question

Find the value of \[\sin {{135}^{\circ }}\].

Answer 1

Hint: In this question, we first need to write \[{{135}^{\circ }}\] as the sum of the known angles and convert it accordingly by using the trigonometric ratios of compound angles formula. Then we can get the value from the trigonometric ratios of some standard angles.

Complete step-by-step answer:
Trigonometric Ratios:
Relations between different sides and angles of a right angles triangle are called trigonometric ratios.
Trigonometric Ratios of Allied Angles:
Two angles are said to be allied when their sum or difference is either zero or a multiple of \[{{90}^{\circ }}\]. The angles \[-\theta ,{{90}^{\circ }}\pm \theta ,{{180}^{\circ }}\pm \theta ,{{270}^{\circ }}\pm \theta ,{{360}^{\circ }}-\theta \] etc. are angles allied to the angle \[\theta \]
if \[\theta \] is measured in degrees.
Trigonometric Ratios of Compound Angles:
The algebraic sum of two or more angles are generally called compound angles.
Some standard formula of compound angles can be given by
\[\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B\]
Trigonometric ratios of some standard angles are
\[\begin{align}
  & \sin {{45}^{\circ }}=\dfrac{1}{\sqrt{2}} \\
 & \cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}} \\
 & \sin {{90}^{\circ }}=1 \\
 & \cos {{90}^{\circ }}=0 \\
\end{align}\]
Now, from the given question we have
\[\Rightarrow \sin {{135}^{\circ }}\]
Now, this can be further written in terms of compound angles as
\[\Rightarrow \sin \left( {{90}^{\circ }}+{{45}^{\circ }} \right)\]
Now, by using the compound angles formula mentioned above we can further expand it as
\[\Rightarrow \sin {{90}^{\circ }}\times \cos {{45}^{\circ }}+\cos {{90}^{\circ }}\times \sin {{45}^{\circ }}\]
Now, by substituting the corresponding values obtained from the trigonometric ratios of some standard angles in the above equation we get,
\[\Rightarrow 1\times \dfrac{1}{\sqrt{2}}+0\times \dfrac{1}{\sqrt{2}}\]
Now, this can be further simplified and written as
\[\Rightarrow \dfrac{1}{\sqrt{2}}\]
Hence, the value of \[\sin {{135}^{\circ }}\] is \[\dfrac{1}{\sqrt{2}}\].

Note: Instead of using the formula of trigonometric ratios of compound angles and expanding it we can also solve it by using the trigonometric ratios of allied angles and convert it according to the result. Both the methods give the same result.
It is important to note that while expanding or when substituting the respective values we should not neglect any of the terms or should do wrong calculation because it changes the corresponding equation and so the final result.