Computer Science 1620

Math Library

Remember this program?suppose that I invest $25000 into a mutual

fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

#include <iostream> using namespace std;

int main() {

cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

#include <iostream> using namespace std;

int main() {

double balance = 25000.0;

balance = balance * 1.08; cout << "Year 1: $" << balance << endl;

balance = balance * 1.08; cout << "Year 2: $" << balance << endl;

balance = balance * 1.08; cout << "Year 3: $" << balance << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

#include <iostream>#include <iomanip> using namespace std;

int main() {

double balance = 25000.0; cout << fixed << showpoint << setprecision(2);

balance = balance * 1.08; cout << "Year 1: $" << balance << endl;

balance = balance * 1.08; cout << "Year 2: $" << balance << endl;

balance = balance * 1.08; cout << "Year 3: $" << balance << endl;

return 0;}

Suppose we alter the program?suppose that I invest $25000 into a mutual

fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 20: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 30: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

double balance = 25000.0; cout << fixed << showpoint << setprecision(2);

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 10: $" << balance << endl;

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 20: $" << balance << endl;

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 30: $" << balance << endl;

return 0;}

The previous example worked, but … to calculate 1.0830 required that we manually

type in thirty multiplicationssuppose we had wanted to know the

balance after 100 years?

pow(b,e) to compute b raised to the power of e, one can use

the pow function syntax:

the pow statement itself is an expression value: the first expression raised to the power of the

second expression

pow( , )numericexpression

numeric expression

Example:#include <iostream>#include <cmath> using namespace std; int main() {

cout << pow(3.0, 3.0) << endl; cout << pow(3, 4.0) << endl; cout << pow(4.0, 3) << endl; int x = 5; double y = 6.0; cout << pow(x, y) << endl; cout << pow(x, y) + pow(3.0, 3.0) << endl; cout << pow( pow(x, y), pow(3.0, 3.0) ) << endl; return 0;

}

A note about powyou cannot send it two integer expressions

cout << pow(3, 4) << endl; // compiler error

this has to do with function overloadingwe will examine functions later

for now, just know that one expression has to be a floating point type

The cmath library

Function Description

abs(x) Returns the absolute value of x

pow(x, y) Returns xy

sqrt(x) Returns the square root of x

sin(x) Returns the sine of x

cos(x) Returns the cosine of x

tan(x) Returns the tangent of x

log(x) Returns the natural logarithm of x ln x

log10(x) Returns log10 x

exp(x) Returns ex

floor(x) Returns the closest whole number < x

ceil(x) Returns the closest whole number > x

A Note about Types when using a math function (from previous slide), the type of

the expression is the same as the type of the input expression, with some exceptions:

an integer expression is converted to a double for the pow function, the type of the expression is the largest of

the base and exponent expressions for abs

integers are not converted to doubles shorts and chars are converted to ints

Expression Type

sqrt(2.0) double

sin(2.0f) float

cos(2) double

log(3 + 4.0f) float

exp(2) + exp(2.0) double

abs(75); int

floor(3.5); double

A Note about the Trig Functions the input expression is considered to be

radians

hence, sin(30.0) = -0.988032, not 0.5

to convert radians to degrees, multiply the value by (π / 180)

Example:#include <iostream>#include <cmath> using namespace std; int main() {

const double PI = 3.14159;

cout << sin(30) << endl;

cout << sin(30 * PI / 180) << endl; return 0;

}

Back to our previous examplesuppose that I invest $25000 into a mutual

fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 20: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 30: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * pow(1.08, 10) << endl; cout << "Year 10: $" << 25000 * pow(1.08, 20) << endl; cout << "Year 10: $" << 25000 * pow(1.08, 30) << endl;

return 0;

}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <iomanip> using namespace std;

int main() {

double balance = 25000.0; cout << fixed << showpoint << setprecision(2);

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 10: $" << balance << endl;

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 20: $" << balance << endl;

balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 30: $" << balance << endl;

return 0;}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

double balance = 25000.0; cout << fixed << showpoint << setprecision(2);

balance = balance * pow(1.08, 10); cout << "Year 10: $" << balance << endl;

balance = balance * pow(1.08, 10); cout << "Year 20: $" << balance << endl;

balance = balance * pow(1.08, 10); cout << "Year 30: $" << balance << endl;

return 0;}

Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

double balance = 25000.0; double increase = pow(1.08, 10); cout << fixed << showpoint << setprecision(2);

balance = balance * increase; cout << "Year 10: $" << balance << endl;

balance = balance * increase; cout << "Year 20: $" << balance << endl;

balance = balance * increase; cout << "Year 30: $" << balance << endl; return 0;}

Example 2: Suppose you have a cannon. Given an initial velocity of a cannonball and the angle of the cannon θ, output (1) the time that the ball remains in the air and (2) the maximum height of its trajectory

Arithmetic Data and Operators

Given a value θ and initial velocity v0, we can calculate the time that the ball stays in the air:

and its maximum height:

secsin2 0

g

vt

θ−=

metersg

vh

2

)sin( 20 θ−

=

• g is approximately -9.81 m/s at sea level

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / -9.81; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

cmathcall

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / -9.81; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

MagicNumber

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

Expects radians

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * 3.14159 / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

MagicNumber

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

secsin2 0

g

vt

θ−=

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

metersg

vh

2

)sin( 20 θ−

=

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

metersg

vh

2

)sin( 20 θ−

=

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}

Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places.#include <iostream>#include <cmath>#include <iomanip> using namespace std;

int main() {

const double PI = 3.14159; const double G = -9.81;

double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(3); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0;}