Goniometrische vergelijkingen

a

De exacte waardencirkel geeft
\(\frac{3}{5}x-\frac{1}{2}\pi =\frac{1}{2}\pi +k⋅\pi \)

1p

\(\frac{3}{5}x=\pi +k⋅\pi \)
\(x=1\frac{2}{3}\pi +k⋅1\frac{2}{3}\pi \)

1p

\(x\) in \([0, 2\pi ]\) geeft \(x=1\frac{2}{3}\pi ∨x=0\)

1p

a

Balansmethode geeft \(\cos(\frac{3}{4}x-\frac{1}{6}\pi )=-\frac{1}{2}\text{.}\)

1p

De exacte waardencirkel geeft
\(\frac{3}{4}x-\frac{1}{6}\pi =\frac{2}{3}\pi +k⋅2\pi ∨\frac{3}{4}x-\frac{1}{6}\pi =-\frac{2}{3}\pi +k⋅2\pi \)

1p

\(\frac{3}{4}x=\frac{5}{6}\pi +k⋅2\pi ∨\frac{3}{4}x=-\frac{1}{2}\pi +k⋅2\pi \)
\(x=1\frac{1}{9}\pi +k⋅2\frac{2}{3}\pi ∨x=-\frac{2}{3}\pi +k⋅2\frac{2}{3}\pi \)

1p

\(x\) in \([0, 2\pi ]\) geeft \(x=1\frac{1}{9}\pi ∨x=2\pi \)

1p

a

Balansmethode geeft \(\sin(2x+\frac{1}{4}\pi )=-\frac{1}{2}\sqrt{2}\text{.}\)

1p

De exacte waardencirkel geeft
\(2x+\frac{1}{4}\pi =1\frac{1}{4}\pi +k⋅2\pi ∨2x+\frac{1}{4}\pi =1\frac{3}{4}\pi +k⋅2\pi \)

1p

\(2x=\pi +k⋅2\pi ∨2x=1\frac{1}{2}\pi +k⋅2\pi \)
\(x=\frac{1}{2}\pi +k⋅\pi ∨x=\frac{3}{4}\pi +k⋅\pi \)

1p

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{1}{2}\pi ∨x=1\frac{1}{2}\pi ∨x=\frac{3}{4}\pi ∨x=1\frac{3}{4}\pi \)

1p

a

Balansmethode geeft \(\sin(\frac{3}{4}\pi x+\frac{1}{6}\pi )=\frac{1}{2}\sqrt{3}\text{.}\)

1p

De exacte waardencirkel geeft
\(\frac{3}{4}\pi x+\frac{1}{6}\pi =\frac{1}{3}\pi +k⋅2\pi ∨\frac{3}{4}\pi x+\frac{1}{6}\pi =\frac{2}{3}\pi +k⋅2\pi \)

1p

\(\frac{3}{4}\pi x=\frac{1}{6}\pi +k⋅2\pi ∨\frac{3}{4}\pi x=\frac{1}{2}\pi +k⋅2\pi \)
\(x=\frac{2}{9}+k⋅2\frac{2}{3}∨x=\frac{2}{3}+k⋅2\frac{2}{3}\)

1p

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{2}{9}∨x=2\frac{8}{9}∨x=5\frac{5}{9}∨x=\frac{2}{3}∨x=3\frac{1}{3}∨x=6\)

1p

a

Balansmethode geeft \(-4\cos(3x+\frac{2}{3}\pi )=4\) dus \(\cos(3x+\frac{2}{3}\pi )=-1\text{.}\)

1p

De exacte waardencirkel geeft
\(3x+\frac{2}{3}\pi =\pi +k⋅2\pi \)

1p

\(3x=\frac{1}{3}\pi +k⋅2\pi \)
\(x=\frac{1}{9}\pi +k⋅\frac{2}{3}\pi \)

1p

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{1}{9}\pi ∨x=\frac{7}{9}\pi ∨x=1\frac{4}{9}\pi \)

1p

a

\(\sin(2x+\frac{2}{3}\pi )=1∨\sin(2x+\frac{2}{3}\pi )=-1\)

1p

De exacte waardencirkel geeft
\(2x+\frac{2}{3}\pi =\frac{1}{2}\pi +k⋅2\pi ∨2x+\frac{2}{3}\pi =1\frac{1}{2}\pi +k⋅2\pi \)

1p

\(2x=-\frac{1}{6}\pi +k⋅2\pi ∨2x=\frac{5}{6}\pi +k⋅2\pi \)
\(x=-\frac{1}{12}\pi +k⋅\pi ∨x=\frac{5}{12}\pi +k⋅\pi \)

1p

a

\(\sin(\frac{3}{4}x)=0∨\cos(2x+\frac{1}{6}\pi )=0\)

1p

De exacte waardencirkel geeft
\(\frac{3}{4}x=k⋅\pi ∨2x+\frac{1}{6}\pi =\frac{1}{2}\pi +k⋅\pi \)

1p

\(\frac{3}{4}x=k⋅\pi ∨2x=\frac{1}{3}\pi +k⋅\pi \)
\(x=k⋅1\frac{1}{3}\pi ∨x=\frac{1}{6}\pi +k⋅\frac{1}{2}\pi \)

1p