Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

How do you verify sin(2πθ)=sin θ ?

Answer
VerifiedVerified
474.6k+ views
like imagedislike image
Hint: In this question, we need to prove a trigonometric identity that is sin(2πθ)=sin θ . We will do this by using the properties of trigonometric functions and its identity. First, we will use this sin(xy)=sinxcos ysinycosx identity and then, find the values and prove the required identity.

Formula used: sin(xy)=sinxcos ysinycosx
cos360=1
sin360=0

Complete step-by-step solution:
We need to prove a trigonometric identity that is sin(2πθ)=sin θ .
First of all, these identities are always proved or derived from the basic trigonometric identity or formula. Trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Therefore all the trigonometric functions and the properties are interlinked with each other.
Now to prove this identity, we can write it like, sin(2πθ)=sin(360θ)
We know that sin(xy)=sinxcos ysinycosx
Using this identity, in the previous one, that is sin(360θ) ,
sin(360θ)=sin360cosθsinθcos360
Since we know that, sin360=0 , and cos360=1 , applying this in the above
sin(360θ)=sin360cosθsinθcos360
sin(360θ)=(0)cosθsinθ(1)
And it becomes,
sin(360θ)=0sinθ
Ultimately, we get,
sin(360θ)=sinθ
Left hand side = right hand side
Hence the given identity is proved.

Note: Trigonometry is an important branch of Mathematics and the trigonometric functions and its identities are very important identities to learn as we can’t solve most of the trigonometric problems without using it.
Here are some of the important identities or formulas which will be frequently used and asked for.
sin(xy)=sinxcos ysinycosx
sin(x+y)=sinxcos y+sinycosx
cos (x  y) = cos xcos y + sinysinx
cos (x + y) = cos xcos y - sinysinx
cos(πθ)=cosθ
cos(π+θ)=cosθ
cos(2πθ)=cosθ
sin(πθ)=sinθ
sin(π+θ)=sinθ
sin(2πθ)=sinθ
These identities alone aren’t enough for a better preparation. You need to study the proof and memorise the identities to solve all kinds of trigonometric functions.