Double Angle Formulas

[vanbeersj]

I have the following question. I always have difficulties getting started with word problems....

The cross section of a radio wave reflector is defined by x = cos2b, y=sinb. Find the relation between x and y by eliminating b.

I'm not sure where to start.

 

[galactus]

Solve \(\displaystyle x=cos(2b)\) for b and sub into \(\displaystyle y=sin(b)\)

y will then be in terms of x.

 

[soroban]

Hello, vanbeersj!

\(\displaystyle \text{The cross section of a radio wave reflector is defined by: }\;\begin{array}{cccc} x &=& \cos2\theta & [1] \\ y&=&\sin\theta & [2]\end{array}\)

\(\displaystyle \text{Find the relation between }x\text{ and }y\text{ by eliminating }\theta.\)

\(\displaystyle \text{We know that: }\;\sin^2\!\theta + \cos^2\!\theta \:=\:1\)

\(\displaystyle \text{Substitute [2]: }\;y^2 + \cos^2\!\theta \:=\:1 \quad\Rightarrow\quad \cos^2\!\theta \:=\:1-y^2 \quad\Rightarrow\quad \cos\theta \:=\:\sqrt{1-y^2}\;\;[3]\)


\(\displaystyle \text{From [1], we have: }\;x \;=\;\cos2\theta \quad\Rightarrow\quad x \;=\;\cos^2\!\theta - \sin^2\!\theta\)

\(\displaystyle \text{Substitute [2] and [3]: }\;x \;=\;(\sqrt{1-y^2})^2 - (y)^2 \;=\;1-y^2 - y^2\)

\(\displaystyle \text{Therefore: }\;\boxed{x \;=\;1-2y^2}\)