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

\(\cos(\frac{2}{3}x-\frac{1}{2}\pi )=0\)

ExacteWaarde (0)
004f - Goniometrische vergelijkingen - basis - 41ms - dynamic variables

a

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

1p

○

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

1p

○

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

1p

4p

b

\(4\sin(2t+\frac{5}{6}\pi )=-2\)

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

b

Balansmethode geeft \(\sin(2t+\frac{5}{6}\pi )=-\frac{1}{2}\text{.}\)

1p

○

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

1p

○

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

1p

○

\(t\) in \([0, 2\pi ]\) geeft \(t=\frac{1}{2}\pi ∨t=1\frac{1}{2}\pi ∨t=\frac{1}{6}\pi ∨t=1\frac{1}{6}\pi \)

1p

4p

c

\(5\cos(\frac{1}{5}\pi t-\frac{1}{3}\pi )=2\frac{1}{2}\sqrt{2}\)

ExacteWaarde (2)
004h - Goniometrische vergelijkingen - basis - 0ms - dynamic variables

c

Balansmethode geeft \(\cos(\frac{1}{5}\pi t-\frac{1}{3}\pi )=\frac{1}{2}\sqrt{2}\text{.}\)

1p

○

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

1p

○

\(\frac{1}{5}\pi t=\frac{7}{12}\pi +k⋅2\pi ∨\frac{1}{5}\pi t=2\frac{1}{12}\pi +k⋅2\pi \)
\(t=2\frac{11}{12}+k⋅10∨t=10\frac{5}{12}+k⋅10\)

1p

○

\(t\) in \([0, 2\pi ]\) geeft \(t=2\frac{11}{12}∨t=\frac{5}{12}\)

1p

4p

d

\(3\sin(\frac{3}{4}x)=1\frac{1}{2}\sqrt{3}\)

ExacteWaarde (3)
006x - Goniometrische vergelijkingen - basis - 0ms - dynamic variables

d

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

1p

○

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

1p

○

\(\frac{3}{4}x=\frac{1}{3}\pi +k⋅2\pi ∨\frac{3}{4}x=\frac{2}{3}\pi +k⋅2\pi \)
\(x=\frac{4}{9}\pi +k⋅2\frac{2}{3}\pi ∨x=\frac{8}{9}\pi +k⋅2\frac{2}{3}\pi \)

1p

○

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{4}{9}\pi ∨x=\frac{8}{9}\pi \)

1p

opgave 2

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

4p

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

ExacteWaarde (4)
006y - Goniometrische vergelijkingen - basis - 1ms - dynamic variables

○

Balansmethode geeft \(5\sin(3x+\frac{1}{4}\pi )=-5\) dus \(\sin(3x+\frac{1}{4}\pi )=-1\text{.}\)

1p

○

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

1p

○

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

1p

○

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

1p

opgave 3

Los exact op.

3p

a

\(\cos^2(1\frac{1}{2}x-\frac{1}{6}\pi )=1\)

Kwadraat
006z - Goniometrische vergelijkingen - basis - 0ms - dynamic variables

a

\(\cos(1\frac{1}{2}x-\frac{1}{6}\pi )=1∨\cos(1\frac{1}{2}x-\frac{1}{6}\pi )=-1\)

1p

○

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

1p

○

\(1\frac{1}{2}x=\frac{1}{6}\pi +k⋅2\pi ∨1\frac{1}{2}x=1\frac{1}{6}\pi +k⋅2\pi \)
\(x=\frac{1}{9}\pi +k⋅1\frac{1}{3}\pi ∨x=\frac{7}{9}\pi +k⋅1\frac{1}{3}\pi \)

1p

3p

b

\(\frac{5}{6}\cos(3x-\frac{1}{2}\pi )\cos(\frac{4}{5}x-\frac{5}{6}\pi )=0\)

ProductIsNul
0070 - Goniometrische vergelijkingen - basis - 1ms - dynamic variables

b

\(\cos(3x-\frac{1}{2}\pi )=0∨\cos(\frac{4}{5}x-\frac{5}{6}\pi )=0\)

1p

○

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

1p

○

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

1p
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