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Question: How do you prove \(\sin ( - a) = \sin ({360^ \circ } - a) = - \sin a\)?...

How do you prove sin⁡(−a)=sin⁡(360∘−a)=−sin⁡a\sin ( - a) = \sin ({360^ \circ } - a) = - \sin a?

Explanation

Solution

In this question we will use the trigonometric identity formula of sin⁡(a+b)\sin (a + b) for equating the given term. On doing some simplification we get the final answer.

Formula used: sin⁡(a+b)=sin⁡acos⁡b−cos⁡asin⁡b\sin (a + b) = \sin a\cos b - \cos a\sin b

Complete step-by-step solution:
We have the term given to us as:
⇒sin⁡(−a)\Rightarrow \sin ( - a)
Now we know that sin⁡(−a)\sin ( - a) can also be written as sin⁡(360∘−a)\sin ({360^ \circ } - a) because addition of 360∘{360^ \circ } doesn’t change the final value of the term therefore we can write is as:
⇒sin⁡(360∘−a)\Rightarrow \sin ({360^ \circ } - a)
Now since the above expression is in the format of sin⁡(a+b)\sin (a + b) we can expand it using the formula and write it as:
⇒sin⁡(360∘)cos⁡(a)−sin⁡(a)cos⁡(360∘)\Rightarrow \sin ({360^ \circ })\cos (a) - \sin (a)\cos ({360^ \circ })
Now we know that the value of sin⁡(360∘)=0\sin ({360^ \circ }) = 0 and the value of cos⁡(360∘)=1\cos ({360^ \circ }) = 1
On substituting both the values in the expression we get:
⇒0×cos⁡(a)−sin⁡(a)×1\Rightarrow 0 \times \cos (a) - \sin (a) \times 1
On simplifying we get:
⇒−sin⁡(a)\Rightarrow - \sin (a)
Hence Proved.

Note: It is to be remembered which trigonometric functions are positive and negative in what quadrants.
The formula used over here is for sin⁡(a+b)\sin (a + b), the other formulas for the sine and cosine should be remembered.
The other identity formula for cosine and tangent should be remembered too and whenever there is a trigonometric proof required, all the terms in the equation should be converted to the basic trigonometric identities of sine and cosine.
There also exist half angle formulas which are an addition to the general angle’s addition-subtraction formulas.
In this question all the value of the angle is given to us in degrees, the symbol of degrees is ∘\circ and angle can also be represented in radians where π\pi is used which is equal to 180180 degrees.
When you add 180∘{180^ \circ } to any angle, its position on the graph reverses, and whenever you add 360∘{360^ \circ } to any angle, it reaches the same point after a complete rotation.