Question

How do you find the value of \[\sin 630\]?

Answer 1

Hint: We will write the value of the given angle by breaking it in terms of \[\pi \]. Use the concept of complementary angles to write the value of sine of the angle by converting it into cosine of angle. Use the graph of cosine to calculate the value of the cosine at that angle.
* Two angles are said to be complementary to each other if the sum of angles is \[{90^ \circ }\]. Sine and cosine are complementary angles, we write \[\cos x = \sin ({90^ \circ } - x)\]

Complete step-by-step answer:
We have to find the value of \[\sin {630^ \circ }\]
We can write \[{630^ \circ } = 4 \times {180^ \circ } - {90^ \circ }\]
Substitute the value of angle in the bracket of sine.
Then \[\sin {630^ \circ } = \sin (4 \times {180^ \circ } - {90^ \circ })\]
i.e. substitute the value of \[{180^ \circ } = \pi \]and \[{90^ \circ } = \dfrac{\pi }{2}\]in the equation
\[ \Rightarrow \sin {630^ \circ } = \sin (4 \times \pi - \dfrac{\pi }{2})\]
Multiply the values in the bracket
\[ \Rightarrow \sin {630^ \circ } = \sin (4\pi - \dfrac{\pi }{2})\]
Take negative sign outside the angle and write the angle inside according to it
\[ \Rightarrow \sin {630^ \circ } = \sin \left( { - \left( {\dfrac{\pi }{2} - 4\pi } \right)} \right)\]
Since sine is an odd function we can write \[\sin ( - \theta ) = - \sin \theta \], write the value of angle accordingly.
\[ \Rightarrow \sin {630^ \circ } = - \sin \left( {\dfrac{\pi }{2} - 4\pi } \right)\]
Now we know that sine and cosine are complementary angles, we write \[\cos x = \sin ({90^ \circ } - x)\]
Then we can write \[\sin \left( {\dfrac{\pi }{2} - 4\pi } \right) = \cos 4\pi \]
\[ \Rightarrow \sin {630^ \circ } = - \cos 4\pi \] … (1)
Now we draw the graph of cosine at angles from 0 to \[4\pi \]
Here we see that \[\cos 4\pi = 1\]
Substitute the value in equation (1)
\[ \Rightarrow \sin {630^ \circ } = - 1\]

\[\therefore \]The value of \[\sin {630^ \circ }\] is -1.

Note:
Many students make the mistake of writing the value of writing the value of \[\cos 4\pi \] as 1 , Keep in mind the odd multiples of \[\pi \] have cosine value as -1 and even multiples have value +1. Also, many students make the mistake of opening the angle as \[{630^ \circ } = 3 \times {180^ \circ } + {90^ \circ }\], but then we won’t be able to apply the complimentary angle concept. So, we break the angle such that we have \[{90^ \circ }\] in subtraction.