Question

Prove that: DC² = 2a²(1 + cos 20) If DC = √12 and a = 2, show that 0 = 30°.​

Answer 1

Answer:

See below

Step-by-step explanation:

Given that:

$\displaystyle{DC^2 = 2a^2(1+\cos 2\theta)}$

Where $\displaystyle{DC = \sqrt{12}}$ and $\displaystyle{a = 2}$. Substitute these values in the expression:

$\displaystyle{\left(\sqrt{12}\right)^2 = 2\cdot 2^2(1+\cos 2\theta)}\\\\\displaystyle{12= 2\cdot 4(1+\cos 2\theta)}\\\\\displaystyle{12= 8(1+\cos 2\theta)}\\\\\displaystyle{\dfrac{12}{8}= 1+\cos 2\theta}\\\\\displaystyle{\dfrac{3}{2}=1+\cos 2\theta}$

Substitute $\displaystyle{\theta = 30^{\circ}}$

$\displaystyle{\dfrac{3}{2}= 1+\cos (2 \cdot 30^{\circ})}\\\\\displaystyle{\dfrac{3}{2}= 1+\cos 60^{\circ}}\\\\\displaystyle{\dfrac{3}{2}= 1+\dfrac{1}{2}}\\\\\displaystyle{\dfrac{3}{2}= \dfrac{2}{2}+\dfrac{1}{2}}\\\\\displaystyle{\dfrac{3}{2}= \dfrac{3}{2}}$

Therefore, since both sides are the same after substituting theta = 30 degrees.

Hence, proved.