created Jan 14, 2020

Most programming exercises from other chapters could be improved by using `printf()`

in place of the usual `println()`

.
See also the exercises for chapter 14 which can be done without reading that chapter.

1. A fact from calculus is that

(1 + 1/x)^{x}

gets closer and closer the the value 2.71828... as `x`

gets larger and larger.
This value is so useful in math that it has a name: `e`

.
It is the base for natural logarithms.
(But don't worry: you don't need to do math to write this program.)

Write a program that calculates this value for an `x`

entered by the user.

Enter x: 40000 Approximation to e: 2.7182479

`x`

gets very large before `e`

gets close to its true value.
To raise `val`

to a power, import `java.lang.Math`

and use

Math.pow( val, power )

which takes double precision arguments and returns a double precision value

2. Everyone's favorite trig identity is

sin( θ )^{2}+ cos( θ )^{2}= 1

In math books this is usually written

sin^{2}( θ ) + cos^{2}( θ ) = 1

Write a program that demonstrates this. Prompt the user for an angle in degrees, then print out the sum of the two squares.

Input an angle: 37.5 sin(37.50) is: 0.61 cos(37.50) is: 0.79 sin(37.50)^2 is: 0.37 cos(37.50)^2 is: 0.63 sum is: 1.00

A problem is that `Math.sin( rad )`

and `Math.cos( rad )`

expect angles in radians.
There are 360 degrees in a circle and 2π radians in a circle.
To convert an angle from degrees to radians, multiply degrees by 2π/360.

Another way to do this is with:

Math.toRadians(deg)

To square a value, multiply it by itself (using *). Using Math.pow() is unneeded complication for squaring.

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