Trigonometric Identities and Proofs

The video presents an elegant semicircle proof of the double angle formulas for sine and cosine. It begins by constructing a semicircle with a radius of one and analyzing the properties of triangles formed within it. By using triangle similarity, the video demonstrates the double angle identities for sine and cosine. The proof is attributed to Roger B Nelson, and additional resources are provided for further exploration.

What is the initial setup for the semicircle proof?

• A semicircle with radius 1 and points O, B, and C.

• A full circle with radius 1 and points O, A, and B.

• A semicircle with radius 1 and points O, A, and B.

• A semicircle with radius 2 and points O, A, and B.

What is the significance of point D in the semicircle proof?

• It is the endpoint of the semicircle.

• It is the intersection of the perpendicular from C to the x-axis.

• It is the midpoint of the semicircle.

• It is the center of the semicircle.

How is the angle B in triangle ACB related to angle 2T?

• Angle B is supplementary to angle 2T.

• Angle B is twice angle 2T.

• Angle B is half of angle 2T.

• Angle B is equal to angle 2T.

What is the length of the hypotenuse in triangle ACB?

• 3

• 4

• 2

• 1

What is the relationship between the short leg and hypotenuse in the similar triangles?

• The ratio is equal to the cotangent of angle T.

• The ratio is equal to the tangent of angle T.

• The ratio is equal to the cosine of angle T.

• The ratio is equal to the sine of angle T.

What is the double angle identity for sine derived in the proof?

• sin(2T) = 2 * sin(T) * cos(T)

• sin(2T) = sin(T) + cos(T)

• sin(2T) = sin^2(T) + cos^2(T)

• sin(2T) = 2 * cos(T) * sin(T)

How is the cosine of 2T expressed in terms of cosine T?

• cos(2T) = 2 * sin^2(T) - 1

• cos(2T) = 1 - 2 * sin^2(T)

• cos(2T) = cos^2(T) + sin^2(T)

• cos(2T) = 2 * cos^2(T) - 1

What is the final conclusion of the semicircle proof?

• The proof derives the Pythagorean identity.

• The proof explains the law of cosines.

• The proof shows the relationship between sine and tangent.

• The proof demonstrates the double angle identities for sine and cosine.

Who is credited with this semicircle proof?

• Pythagoras

• Roger B. Nelson

• Isaac Newton

• Albert Einstein

What additional resources are suggested for further understanding?

• A link to a video on the law of cosines.

• A link to a proof using the law of sines and cosines.

• A link to a proof using trigonometric identities.

• A link to a video on the Pythagorean theorem.