Question

How do you evaluate \[sin( - 3pi)\] ?

Answer 1

Hint: In this question, we need to find the value of \[sin(-3pi)\]. Mathematically, Pi \[(\pi)\] is Greek letter and is a mathematical constant. In trigonometry, the value of \[(\pi)\] is \[180^{o}\] . We can find the value of \[sin(-3pi)\] by using trigonometric identities and values. Sine is nothing but a ratio of the opposite side of a right angle to the hypotenuse of the right angle. The basic trigonometric functions are sine , cosine and tangent. The value of \[cos 90^{o}\] is used to find the value. With the help of the Trigonometric functions , we can find the value of \[sin(-3pi)\]
Trigonometry table:

Angle	\[0^{0}\]	\[30^{o}\]	\[45^{o}\]	\[60^{o}\]	\[90^{o}\]
Cosine	\[1\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{2}\]	\[0\]

Identity used :
1.\[sin(360^{o} + \ \theta)\ = \ sin\ \theta\]
2.\[sin\ ( - \theta)\ = \ - \ sin\ \theta\ \](since sine is an odd function )
3.\[sin(90 ^{o}+ \theta)\ = \ cos\ \theta\]

Complete step-by-step solution:
Given,
\[sin( - 3pi)\]
Here we need to find the value of \[sin( - 3pi)\]
We know that the value of \[\pi\] is \[180^{o}\] , by substituting the value of \[\pi\] in \[sin( - 3pi)\]
We get,
\[\Rightarrow \ sin( - 3pi) = sin( - 3 \times 180^{o})\]
By multiplying,
We get,
\[\Rightarrow \ sin( - 540^{o})\]
Since sine is an odd function.
\[\sin\left( - 540^{o} \right) = - \sin\left( 540^{o} \right)\]
Now we need to find the value of \[- sin(540)^{o}\] .
We can find the value of \[- sin(540^{o})\ \] by using other angles of sine functions.
We can rewrite \[540^{o}\] as \[360^{o} + 180^{o}\]
Thus we get,
\[- sin(540^{o})\ = - sin(360^{o} + 180^{o})\]
We know that, \[sin(360^{o} + \theta)\ = \ sin\ \theta\]
Thus we get,
\[- sin(360^{o} + 180^{o}) = - sin(180^{o})\]
Now we need to find the value of \[sin(180^{o})\] .
We can rewrite \[180^{o}\] as \[(90^{o} + 90^{o})\]
Therefore we get,
\[- sin\left( 180^{o} \right) = - sin(90^{o} + 90^{o})\]
We know that \[sin (90^{o} + \theta) = cos\theta\]
\[\Rightarrow - cos(90^{o})\]
Since cosine is an even function. We get,
\[\Rightarrow - cos(90^{o}) = cos(90^{o})\]
From the trigonometry table, we know that the value of \[cos\ 90^{o}\] is \[0\] .
\[\Rightarrow\cos(90^{o})\ = 0\]
Therefore \[- sin(180^{o}) = 0\]
Thus we get the value of \[- sin(540^{o})\] is \[0\]
That is the value of \[sin( - 3pi) = 0\]
Final answer :
The value of \[sin( - 3pi) = 0\]

Note: The concept used in this problem is trigonometric identities and ratios. Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. The common technique used in this problem is the use of trigonometric functions. For the sine of \[540\] degrees , we use the abbreviation of the sine as sin of the trigonometric function together with the degree symbol \[^{o}\], and write it as \[sin\ 540^{o}\]. Geometrically, \[sin(540^{o})\] lies in the second quadrant. Hence the value of \[sin(540^{o})\] is non-negative.