Question

Square $\mathbf{(}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{B}\mathbf{)}$; interpret your result geometrically. Hint: Your answer is a law which you learned in trigonometry.

Short answer

The law of cosine $|A{|}^2+|B{|}^2+2A\times B$.

Step-by-step solution

Concept and formula used

Using the formula,

$\mathbf{(}\mathbf{A}\mathbf{+}\mathbf{B}{\mathbf{)}}^{\mathbf{2}}\mathbf{=}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}\mathbf{\times }\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}$

The Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

Step 2: Using the trigonometric law

Squaring $(A+B)$,

$$\begin{gathered} (A+B{)}^2=(A+B)\times (A+B) \\ =|A{|}^2+A\times B+B\times A+|B{|}^2 \\ =|A{|}^2+2A\times B+|B{|}^2 \end{gathered}$$

This is law of cosine.