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

How do you find the value of sin⁡630\sin 630?

Explanation

Solution

We will write the value of the given angle by breaking it in terms of π\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∘{90^ \circ }. Sine and cosine are complementary angles, we write cos⁡x=sin⁡(90∘−x)\cos x = \sin ({90^ \circ } - x)

Complete step-by-step answer:
We have to find the value of sin⁡630∘\sin {630^ \circ }
We can write 630∘=4×180∘−90∘{630^ \circ } = 4 \times {180^ \circ } - {90^ \circ }
Substitute the value of angle in the bracket of sine.
Then sin⁡630∘=sin⁡(4×180∘−90∘)\sin {630^ \circ } = \sin (4 \times {180^ \circ } - {90^ \circ })
i.e. substitute the value of 180∘=π{180^ \circ } = \pi and 90∘=π2{90^ \circ } = \dfrac{\pi }{2}in the equation
⇒sin⁡630∘=sin⁡(4×π−π2)\Rightarrow \sin {630^ \circ } = \sin (4 \times \pi - \dfrac{\pi }{2})
Multiply the values in the bracket
⇒sin⁡630∘=sin⁡(4π−π2)\Rightarrow \sin {630^ \circ } = \sin (4\pi - \dfrac{\pi }{2})
Take negative sign outside the angle and write the angle inside according to it
⇒sin⁡630∘=sin⁡(−(π2−4π))\Rightarrow \sin {630^ \circ } = \sin \left( { - \left( {\dfrac{\pi }{2} - 4\pi } \right)} \right)
Since sine is an odd function we can write sin⁡(−θ)=−sin⁡θ\sin ( - \theta ) = - \sin \theta , write the value of angle accordingly.
⇒sin⁡630∘=−sin⁡(π2−4π)\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∘−x)\cos x = \sin ({90^ \circ } - x)
Then we can write sin⁡(π2−4π)=cos⁡4π\sin \left( {\dfrac{\pi }{2} - 4\pi } \right) = \cos 4\pi
⇒sin⁡630∘=−cos⁡4π\Rightarrow \sin {630^ \circ } = - \cos 4\pi … (1)
Now we draw the graph of cosine at angles from 0 to 4π4\pi
Here we see that cos⁡4π=1\cos 4\pi = 1
Substitute the value in equation (1)
⇒sin⁡630∘=−1\Rightarrow \sin {630^ \circ } = - 1

∴\therefore The value of sin⁡630∘\sin {630^ \circ } is -1.

Note:
Many students make the mistake of writing the value of writing the value of cos⁡4π\cos 4\pi as 1 , Keep in mind the odd multiples of π\pi have cosine value as -1 and even multiples have value +1. Also, many students make the mistake of opening the angle as 630∘=3×180∘+90∘{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∘{90^ \circ } in subtraction.