cos21° + cos22° + cos23° + . . . + cos289° + cos290° = ?
Algebra II / Trigonometry
Find sin A in Isosceles Triangle
Find a formula for sin A in terms of b and c only. (Note: △ABC is an isosceles triangle.)
Tape Around a Pipe
A long strip of insulating tape is to be wrapped around a hot water pipe. If the width of the strip is w and the diameter of the pipe is d, at what angle must the strip be placed at the edge of the pipe so that no overlapping and no spaces occur?
Illusion of Wrong Areas
Triangle Rotating in Square
Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:
- 20
/3 - 32/3
- 12
- 40/3
- 15
Tangent Circles, Area of Shaded Region
A circle centered at A of radius r is externally tangent to a circle centered at B of radius 2r. Tangent lines are drawn from A to the larger circle with points of tangency at D and E respectively. Find the area of the shaded portion of the diagram.
Find Sine, Given Tangent
Let tan x = (2ab)/(a2 – b2), where a > b > 0 and 0 < x < 90. Draw a right triangle with angle x and label two sides of the triangle to show tan x. Now find an expression for sin x.
Yaks, Blats, and Blobs
Given the two hypotheses:
1. Some yaks are not blats and
2. No blats are blobs,
if “some” means “at least one,” we can conclude that:
- Some yaks are not blobs.
- Some blobs are not yaks.
- No yak is a blob.
- Some yaks are blobs.
- I, II, III, and IV are not deducible from the given statements.
Wind Farm
Denmark is a major manufacturer of energy-producing wind turbines. Suppose both the following statements are true:
1. Some wind turbines in Ohio are not from Denmark, but are American-built.
2. All of the wind turbines in Lorain County are indeed from Denmark.
From these two statements, which of the following can be concluded?
- Some American-built wind turbines are in Lorain County.
- All Lorain county wind turbines are not American-built.
- Some American-built wind turbines are not in Lorain County.
- No Lorain County wind turbine is American-built.
- Some wind turbines in Ohio, but not in Lorain County, were built in Denmark.
Sphere Packed in Box
A spherical basketball is packed in a cubical box into which it fits exactly. What percent of the volume of the box is used up by the basketball?