We have an answer to "How to evaluate sin2(π9)+cos2(π9)?"

Thaddeus Herman

Answered question

2023-01-07

How to evaluate

\sin^{2} (\frac{π}{9}) + \cos^{2} (\frac{π}{9})

halliekorinnbsn

Beginner2023-01-08Added 13 answers

Without even taking into account the sine and cosine arguments, there is an identity that for all

x

\sin^{2} (x) + \cos^{2} (x) = 1

.
One way of seeing that this is true is to consider the unit circle, and note that for any point on the circle with angle

θ

\cos (θ)

is the

x

-coordinate of that point, and

\sin (θ)

is the

y

-coordinate. Then, drawing a right triangle with the hypotenuse connecting the origin and the point, we find that the triangle has legs of length

\cos (θ)

and

\sin (θ)

, and a hypotenuse of length

1

. Applying the Pythagorean theorem gives us the desired result.

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