Question

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

Answer 1

See below

Explanation:

Given that:

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

Where
`\displaystyle{DC = √(12)}` and
`\displaystyle{a = 2}`. Substitute these values in the expression:

`\displaystyle{\left(√(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{(12)/(8)= 1+\cos 2\theta}\\\\\displaystyle{(3)/(2)=1+\cos 2\theta}`

Substitute
`\displaystyle{\theta = 30^(\circ)}`

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

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

Hence, proved.