Solve for x: sqrt(x^2 + 125.4^2 + 100^2) = x / cos(20 degrees).

Step-by-step solution

1. 1

   Note the domain before squaring. The left side is a square root, so it is non-negative; the right side is $x/\cos 20^\circ$ with $\cos 20^\circ>0$. Hence any solution must satisfy $x\ge 0$, and a negative root produced later would have to be discarded as extraneous.

2. 2

   Square both sides and collect the constant. Squaring is legitimate here because both sides are non-negative:

   $x^2+125.4^2+100^2=\frac{x^2}{\cos^2 20^\circ}\;\Longrightarrow\;125.4^2+100^2=x^2\left(\frac{1}{\cos^2 20^\circ}-1\right)$

   The constant is $K=125.4^2+10000=25725.16$.

3. 3

   Use the Pythagorean identity to simplify the bracket. Since $\sec^2\theta-1=\tan^2\theta$:

   $\frac{1}{\cos^2 20^\circ}-1=\tan^2 20^\circ$

   so the equation becomes $x^2\tan^2 20^\circ=25725.16$. Recognising this identity is what turns a messy trigonometric equation into a one-line solve.

4. 4

   Solve for $x$, keeping only the positive root.

   $x=\frac{\sqrt{25725.16}}{\tan 20^\circ}=\frac{160.3907}{0.3639702}\approx 440.67$

   The negative root $-x$ satisfies the squared equation but not the original one, since it would make the right-hand side negative while the left side stays positive.

5. 5

   Verify by substituting back. With $x\approx 440.67$ the left side is $\sqrt{440.67^2+25725.16}$ and the right side is $440.67/\cos 20^\circ$; both evaluate to the same value to seven significant figures, confirming the root.

   $\sqrt{x^2+25725.16}=\frac{x}{\cos 20^\circ}\;\checkmark$