cos^{2}1° + cos^{2}2° + cos^{2}3° + . . . + cos^{2}89° + cos^{2}90° = ?

## 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 2*r*. 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* = (2*ab*)/(*a*^{2} – *b*^{2}), 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?