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In the following work, two reasons are not given (in steps 1 and 5). What are these reasons?
[math] ||vec{FC}||^2 [/math] | [math] = [/math] | [math] FD^2 + CD^2[/math] | [math]"Step 1 "[/math] |
[math] [/math] | [math] = [/math] | [math] (||vec{FE}|| + ED)^2 + CD^2[/math] | [math] "Info from previous questions"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| ED + ED^2 + CD^2 [/math] | [math] "Expanding the square"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| \ ||vec{EC}||cos theta + ||vec{EC}||^2 cos^2 theta + ||vec{EC}||^2sin^2theta [/math] | [math]"Info from previous questions"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| \ ||vec{EC}||cos theta + ||vec{EC}||^2 [/math] | [math]"Step 5"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + ||vec{FB}||^2 + 2||vec{FE}|| \ ||vec{FB}||cos theta [/math] | [math]"Info from previous questions"[/math] |
Taking the square root of both sides gives the formula for the magnitude of the sum of the vectors [math]vec{FB}[/math] and [math]vec{FE}[/math]:
[math] ||vec{FC}|| = sqrt(||vec{FE}||^2 + ||vec{FB}||^2 + 2||vec{FE}|| \ ||vec{FB}||cos theta)[/math].
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Pythagorean Theorem; [math]sin^2x + cos^2x = 1[/math]
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Pythagorean Theorem; Law of Sines
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Law of Vector Addition; Law of Cosines
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Parallelogram Law; [math]cos2x = cos^2x - sin^2x[/math]