Evaluate y = 2 cos(20 degrees) - 1 to four decimal places.

Step-by-step solution

1. 1

   Settle the units before computing. The angle is $20$ degrees, not radians. This matters enormously: $\cos 20^\circ\approx 0.93969$ whereas $\cos(20\ \text{rad})\approx 0.40808$, which would give a completely different answer.

2. 2

   Evaluate the cosine. Converting, $20^\circ=\tfrac{\pi}{9}\approx 0.349066$ rad, and

   $\cos 20^\circ\approx 0.9396926$

3. 3

   Multiply by $2$. Order of operations does the multiplication before the subtraction:

   $2\cos 20^\circ\approx 1.8793852$

4. 4

   Subtract $1$.

   $y\approx 1.8793852-1=0.8793852$

5. 5

   Round and sanity-check the range. To four decimals $y\approx 0.8794$. As a check on plausibility, $2\cos\theta-1$ must lie in $[-3,1]$ for any $\theta$, and $0.8794$ is just under the maximum $1$ — appropriate for an angle close to $0^\circ$ $\checkmark$. Note also that $2\cos 20^\circ-1$ is not $\cos 40^\circ$; the double-angle identity is $2\cos^2\theta-1$, with the square on the cosine.