Getal & Ruimte (12e editie) - havo wiskunde B

'Goniometrische vergelijkingen'.

havo wiskunde B 8.4 Goniometrische vergelijkingen

Goniometrische vergelijkingen (7)

opgave 1

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

3p

a

\(\sin(4t-\frac{1}{2}\pi )=0\)

ExacteWaarde (0)
004f - Goniometrische vergelijkingen - basis - 52ms - dynamic variables

a

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

1p

○

\(4t=\frac{1}{2}\pi +k⋅\pi \)
\(t=\frac{1}{8}\pi +k⋅\frac{1}{4}\pi \)

1p

○

\(t\) in \([0, 2\pi ]\) geeft \(t=\frac{1}{8}\pi ∨t=\frac{3}{8}\pi ∨t=\frac{5}{8}\pi ∨t=\frac{7}{8}\pi ∨t=1\frac{1}{8}\pi ∨t=1\frac{3}{8}\pi ∨t=1\frac{5}{8}\pi ∨t=1\frac{7}{8}\pi \)

1p

4p

b

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

ExacteWaarde (1)
004g - Goniometrische vergelijkingen - basis - 0ms - dynamic variables

b

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

1p

○

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

1p

○

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

1p

○

\(q\) in \([0, 2\pi ]\) geeft \(q=1∨q=2\frac{1}{3}∨q=3\frac{2}{3}∨q=5∨q=\frac{1}{9}∨q=1\frac{4}{9}∨q=2\frac{7}{9}∨q=4\frac{1}{9}∨q=5\frac{4}{9}\)

1p

4p

c

\(4\sin(2x)=2\sqrt{2}\)

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

c

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

1p

○

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

1p

○

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

1p

○

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

1p

4p

d

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

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

d

Balansmethode geeft \(\cos(4x+\frac{5}{6}\pi )=\frac{1}{2}\sqrt{3}\text{.}\)

1p

○

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

1p

○

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

1p

○

\(x\) in \([0, 2\pi ]\) geeft \(x=\frac{1}{3}\pi ∨x=\frac{5}{6}\pi ∨x=1\frac{1}{3}\pi ∨x=1\frac{5}{6}\pi ∨x=\frac{1}{4}\pi ∨x=\frac{3}{4}\pi ∨x=1\frac{1}{4}\pi ∨x=1\frac{3}{4}\pi \)

1p

opgave 2

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

4p

\(-5-3\cos(2t-\frac{1}{6}\pi )=-2\)

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

○

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

1p

○

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

1p

○

\(2t=1\frac{1}{6}\pi +k⋅2\pi \)
\(t=\frac{7}{12}\pi +k⋅\pi \)

1p

○

\(t\) in \([0, 2\pi ]\) geeft \(t=\frac{7}{12}\pi ∨t=1\frac{7}{12}\pi \)

1p

opgave 3

Los exact op.

3p

a

\(\cos^2(3x+\frac{1}{2}\pi )=1\)

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

a

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

1p

○

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

1p

○

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

1p

3p

b

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

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

b

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

1p

○

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

1p

○

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

1p
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