Decide whether the equation 10 = 5 cos(2 pi) is true.

Step-by-step solution

1. 1

   Evaluate the cosine at the full turn. An angle of $2\pi$ radians is one complete revolution, returning to the point $(1,0)$ on the unit circle, and cosine reads the $x$-coordinate:

   $\cos(2\pi)=1$

2. 2

   Compute the right-hand side.

   $5\cos(2\pi)=5\cdot 1=5$

3. 3

   Compare the two sides. The claim reduces to

   $10=5$

   which is false. There is no variable to solve for, so the statement is simply untrue.

4. 4

   Note the structural reason it could never hold. Because $-1\le\cos\theta\le 1$ for every angle, the expression $5\cos\theta$ is confined to $[-5,5]$:

   $|5\cos\theta|\le 5<10$

   So even if $2\pi$ were replaced by any angle, the equation $10=5\cos\theta$ would still have no solution — the amplitude $5$ is simply too small.

5. 5

   State the conclusion. The equation is false; had it been written as $10=5\cos\theta$ with $\theta$ unknown, the solution set would be empty $\checkmark$.