Getal & Ruimte (13e editie) - vwo wiskunde B

'Goniometrische vergelijkingen'.

vwo wiskunde B 8.3 Goniometrische vergelijkingen

Goniometrische vergelijkingen (7)

opgave 1

Bereken zo mogelijk exact de oplossingen in \([0, 2\pi ]\text{.}\)

3p

a

\(\cos(1\frac{1}{2}x-\frac{1}{5}\pi )=0\)

ExacteWaarde (0)
004f - Goniometrische vergelijkingen - basis - dynamic variables

a

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

1p

○

\(1\frac{1}{2}x=\frac{7}{10}\pi +k⋅\pi \)
\(x=\frac{7}{15}\pi +k⋅\frac{2}{3}\pi \)

1p

○

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{7}{15}\pi ∨x=1\frac{2}{15}\pi ∨x=1\frac{4}{5}\pi \)

1p

4p

b

\(5\sin(\frac{3}{4}\pi x-\frac{2}{3}\pi )=2\frac{1}{2}\)

ExacteWaarde (1)
004g - Goniometrische vergelijkingen - basis - dynamic variables

b

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

1p

○

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

1p

○

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

1p

○

\(x\) in \([0, 2\pi ]\) geeft \(x=1\frac{1}{9}∨x=3\frac{7}{9}∨x=2∨x=4\frac{2}{3}\)

1p

4p

c

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

ExacteWaarde (2)
004h - Goniometrische vergelijkingen - basis - dynamic variables

c

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

1p

○

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

1p

○

\(3x=\frac{3}{4}\pi +k⋅2\pi ∨3x=1\frac{1}{4}\pi +k⋅2\pi \)
\(x=\frac{1}{4}\pi +k⋅\frac{2}{3}\pi ∨x=\frac{5}{12}\pi +k⋅\frac{2}{3}\pi \)

1p

○

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{1}{4}\pi ∨x=\frac{11}{12}\pi ∨x=1\frac{7}{12}\pi ∨x=\frac{5}{12}\pi ∨x=1\frac{1}{12}\pi ∨x=1\frac{3}{4}\pi \)

1p

4p

d

\(-4\cos(\frac{3}{4}x-\frac{1}{3}\pi )=-2\sqrt{3}\)

ExacteWaarde (3)
006x - Goniometrische vergelijkingen - basis - dynamic variables

d

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

1p

○

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

1p

○

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

1p

○

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

1p

opgave 2

Bereken zo mogelijk exact de oplossingen in \([0, 2\pi ]\text{.}\)

4p

\(-1-5\cos(2x+\frac{1}{4}\pi )=4\)

ExacteWaarde (4)
006y - Goniometrische vergelijkingen - basis - dynamic variables

○

Balansmethode geeft \(-5\cos(2x+\frac{1}{4}\pi )=5\) dus \(\cos(2x+\frac{1}{4}\pi )=-1\text{.}\)

1p

○

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

1p

○

\(2x=\frac{3}{4}\pi +k⋅2\pi \)
\(x=\frac{3}{8}\pi +k⋅\pi \)

1p

○

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

1p

opgave 3

Los exact op.

3p

a

\(\sin^2(1\frac{1}{2}q+\frac{1}{4}\pi )=1\)

Kwadraat
006z - Goniometrische vergelijkingen - basis - dynamic variables

a

\(\sin(1\frac{1}{2}q+\frac{1}{4}\pi )=1∨\sin(1\frac{1}{2}q+\frac{1}{4}\pi )=-1\)

1p

○

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

1p

○

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

1p

3p

b

\(\frac{7}{9}\cos(\frac{3}{4}q-\frac{4}{5}\pi )\sin(\frac{3}{4}q-\frac{1}{6}\pi )=0\)

ProductIsNul
0070 - Goniometrische vergelijkingen - basis - dynamic variables

b

\(\cos(\frac{3}{4}q-\frac{4}{5}\pi )=0∨\sin(\frac{3}{4}q-\frac{1}{6}\pi )=0\)

1p

○

De exacte waardencirkel geeft
\(\frac{3}{4}q-\frac{4}{5}\pi =\frac{1}{2}\pi +k⋅\pi ∨\frac{3}{4}q-\frac{1}{6}\pi =k⋅\pi \)

1p

○

\(\frac{3}{4}q=1\frac{3}{10}\pi +k⋅\pi ∨\frac{3}{4}q=\frac{1}{6}\pi +k⋅\pi \)
\(q=1\frac{11}{15}\pi +k⋅1\frac{1}{3}\pi ∨q=\frac{2}{9}\pi +k⋅1\frac{1}{3}\pi \)

1p
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