# What is the solution for evaluating $ab+cd$ with given constraints in POTW #477?

• MHB
• anemone
In summary, the constraints for evaluating ab+cd in POTW #477 are that a and c must be positive integers and b and d must be negative integers. To evaluate ab+cd with these constraints, you can use the formula (a-b)(c-d) - bc, which ensures a positive integer result and takes into account the negative integers. This formula can also be used for other similar problems with similar constraints, although the variables may need to be adjusted. Other methods for evaluating ab+cd with these constraints include using a computer program or creating a table of values, but the formula mentioned above is the most efficient and accurate method.
anemone
Gold Member
MHB
POTW Director
Here is this week's POTW:

-----

Consider that $\{a,\,b,\,c,\,d\}\in \mathbb{R}$ and that $a^2+b^2=c^2+d^2=1$ and $ac+bd=0$, evaluate $ab+cd$.

-----

Congratulations to the following members for their correct solution!

1. Opalg

Solution from Opalg:
If $a^2+b^2 = c^2+d^2 = 1$ then $(a,b)$ and $(c,d)$ are points on the unit circle. So there exist $\theta$ and $\phi$ such that $(a,b) = (\cos\theta,\sin\theta)$ and $(c,d) = (\cos\phi,\sin\phi)$. Therefore $$0 = ac+bd = \cos\theta\cos\phi + \sin\theta\sin\phi = \cos(\theta-\phi).$$ From the formula $\sin x + \sin y = 2\sin\bigl(\frac{x+y}2\bigr)\cos\bigl(\frac{x-y}2\bigr)$ it follows that $$ab+cd = \cos\theta\sin\theta + \cos\phi\sin\phi = \tfrac12\bigl(\sin(2\theta) + \sin(2\phi)\bigr) = \sin(\theta+\phi)\cos(\theta-\phi) = 0.$$

## What is POTW #477?

POTW #477 refers to Problem of the Week #477, which is a mathematical problem posted on a platform such as a website or social media for people to solve and discuss.

## What are constraints in POTW #477?

Constraints in POTW #477 refer to the limitations or conditions that must be followed in order to find a solution for evaluating $ab+cd$. These constraints may include specific values for variables, rules for operations, or restrictions on the final answer.

## What is the purpose of evaluating $ab+cd$ in POTW #477?

The purpose of evaluating $ab+cd$ in POTW #477 is to find a solution or answer to the given mathematical problem. This expression may represent a real-world scenario or a theoretical concept that needs to be solved using mathematical principles.

## What are the possible methods for evaluating $ab+cd$ in POTW #477?

There are several methods that can be used to evaluate $ab+cd$ in POTW #477, including substitution, elimination, and graphing. The most appropriate method will depend on the given constraints and the individual's mathematical skills and preferences.

## How can I check if my solution for evaluating $ab+cd$ in POTW #477 is correct?

The best way to check if your solution for evaluating $ab+cd$ in POTW #477 is correct is to plug in the values of the variables and evaluate the expression according to the given constraints. You can also ask for feedback from others or compare your solution to the official answer, if available.

Replies
1
Views
1K
Replies
1
Views
984
Replies
1
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
1
Views
987