

Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Various topics in geometry, including the explanation of radian measurement and its advantages over degrees, definition and reasoning behind solid angle measure, review of trigonometric functions from the unit circle, derivation of their graphs and formulas, conversion of polar to rectangular coordinates, and proof of the angle sum theorem without assuming postulates 11 and 12.
Typology: Study notes
1 / 2
This page cannot be seen from the preview
Don't miss anything!
Shapes and Designs Extensions 3
(a) Drawing on this, sketch the graphs of the functions f (θ) = sin θ and f (θ) = cos θ, and explain how you can deduce these naturally from the unit circle definition, (b) Continuing to think about the unit circle definition, complete the following for- mulas and give brief explanations for each. i. sin(−θ) = − sin(θ). ii. cos(−θ) = iii. sin(π + θ) = iv. cos(π + θ) = v. sin(π − θ) = vi. cos(π − θ) = vii. sin(π/2 + θ) = viii. cos(π/2 + θ) = ix. sin(π/ 2 − θ) = x. cos(π/ 2 − θ) = xi. sin^2 (θ) + cos^2 (θ) =
(a) Justify the following conversion from cylindrical coordinates (r, θ, z) to rectangular coordinates (x, y, z). x = r cos θ y = r sin θ z = z
(b) Justify the following conversion from spherical coordinates (r, θ, φ) to rectangular coordinates (x, y, z). x = r cos θ sin φ y = r sin θ sin φ z = r cos φ
(a) Learn the proof of this angle sum theorem. See, for example, the proof of the Saccheri-Legendre Theorem in Kay, College Geometry: A Discovery Approach. (b) Use this result to prove Postulate 12, thereby showing that it was not necessary to assume this as a postulate after all.