Question

How do you find the value of \[\sin (225)\]?

Answer 1

Hint: We can write \[{{225}^{\circ }}\] as \[{{\left( 180+45 \right)}^{\circ }}\]. From the basic concepts of trigonometry, \[{{\left( 180+\theta \right)}^{\circ }}\] must lie in \[{{3}^{rd}}\]quadrant. Since \[\sin \]is negative in the third quadrant, we can write\[\sin {{\left( 180+\theta \right)}^{\circ }}\] as\[-\sin \theta \]. After that we have to know the value of \[\sin {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}\].

Complete step by step answer:
From the question it is clear that we have to find the value of \[\sin ({{225}^{\circ }})\]
Let us take the value of \[\sin ({{225}^{\circ }})\] is equal to \[x\]
So,
\[\Rightarrow x=\sin ({{225}^{\circ }})\]…………….(1)
Now we can write \[{{225}^{\circ }}\] as \[{{\left( 180+45 \right)}^{\circ }}\]
So, equation (1) becomes as follows,
\[\Rightarrow x=\sin {{\left( 180+45 \right)}^{\circ }}\]
\[\sin {{\left( 180+45 \right)}^{\circ }}\]is in the form of \[\sin {{\left( 180+\theta \right)}^{\circ }}\].
From the basic concepts of trigonometry, \[{{\left( 180+\theta \right)}^{\circ }}\] must lie in \[{{3}^{rd}}\]quadrant.
So, \[{{\left( 180+45 \right)}^{\circ }}\] lies in the third quadrant.
From the basic concepts of trigonometry, we know that the value of sine is negative in the third quadrant.
Hence, we can write \[\sin {{\left( 180+\theta \right)}^{\circ }}=-\sin \left( \theta \right)\]
\[\Rightarrow x=\sin {{\left( 180+45 \right)}^{\circ }}\]
\[\Rightarrow x=-\sin \left( {{45}^{\circ }} \right)\]…………………(2)
from the standard values of \[\sin e\]. We know that the values of \[\sin ({{45}^{\circ }})=\dfrac{1}{\sqrt{2}}\].
Now put \[\sin \theta =\dfrac{1}{\sqrt{2}}\] in equation (2).
\[\Rightarrow x=-\sin \left( {{45}^{\circ }} \right)\]
\[\Rightarrow x=-\left( \dfrac{1}{\sqrt{2}} \right)\]
\[\Rightarrow x=-\dfrac{1}{\sqrt{2}}\].
Since we have taken that \[x=\sin ({{225}^{\circ }})\] replace \[x\] as \[\sin ({{225}^{\circ }})\]
\[\Rightarrow \sin ({{225}^{\circ }})=-\dfrac{1}{\sqrt{2}}\]

Now, according to the question we have found the value of \[\sin ({{225}^{\circ }})\] as \[-\dfrac{1}{\sqrt{2}}\].

Note: Students have to be clear when and where the value of sine is positive and negative.
Students should know all the standard values if sine function. In this type of question students should split the given angle as the sum of two angles (one angle should be either of \[{{90}^{\circ }},{{180}^{\circ }},{{270}^{\circ }},{{360}^{\circ }}\]) this makes to solve the question easy.