CPM Homework Help : APCALC Problem 9-53

No calculator! Determine all point(s) of intersection of $y = 3\cos(x)$ and $y = 2\cos^2(x) + 1$ over the interval $0 ≤ x ≤ 2π$ .

Compute without a calculator

Step 1:

$3\cos(x)=2\cos^2(x)+1$

Step 2:

$0=2\cos^2(x)-3\cos(x)+1$

Step 3:

$0=(2\cos(x)-1)(\cos(x)-1)$

Step 4:

$2\cos(x)-1=0\text{ or }\cos(x)-1=0$

Step 5:

Solve the equations in Step 4. Note that these solutions only give the $x$ -values. The problem asks for the "points", so you need to calculate the corresponding $y$ -values.