Shapes and Designs
Extensions 3
1. Write a clear explanation of the measurement of angles in radians and why this is a
more natural notion than measurement in degrees.
2. Propose a definition of the measure of a solid angle where three, four, or more planes
meet at common vertex of a polyhedron, and explain why your definition is reasonable.
In particular, your definition should be compatible with a three-dimensional analog of
the Angle Addition Postulate (CliffsQuickReview Geometry, p. 12).
3. Describe all possible cases when two sets (r, θ), (r0, θ0) of polar coordinates actually
correspond to the same point.
4. Review the definitions of trigonometric functions from the unit circle.
(a) Drawing on this, sketch the graphs of the functions f(θ) = sin θand 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. sin2(θ) + cos2(θ) =
5. Describe a procedure to determine the rectangular coordinates (x, y ) of a point from
its polar coordinates (r, θ) and justify why it works.
6. Cylindrical and Spherical Coordinates
1
