In a given circle a central angle of 40° intercepts a chord of 6 cm. What is the circumference of the circle?

## Author: huggins.5@osu.edu

## Illusion of Wrong Areas

## 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?

## Use cos 30 & cos 36 to Get cos 3

Given that cos 30° = √3/2 and cos 36° = (1 + √5)/4. Use formulas for the cosines of sums of angles, differences of angles, etc., to come up with an exact expression (involving radicals) for cos 3°.

## 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.)

## cos2 1° + . . . + cos2 90°

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

## Complex Fraction of sin, cos, etc.

## 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*.

## 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.

## Angles of a Triangle

It so happens that the triangle whose sides are 4, 5, and 6 has a special property: Its largest angle is twice its smallest angle. Make calculations that support this assertion.