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

\(\sin(\frac{3}{5} x - \frac{1}{5} \pi ) = 0\)

ExacteWaarde (0)
004f - Goniometrische vergelijkingen - basis - basis - 45ms - dynamic variables

a

(Exacte waardencirkel)
\(\frac{3}{5} x - \frac{1}{5} \pi = k ⋅ \pi \)

1p

○

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

1p

○

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

1p

4p

b

\(5 \cos(2 x + \frac{5}{6} \pi ) = -2\frac{1}{2}\)

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

b

(Balansmethode)
\(\cos(2 x + \frac{5}{6} \pi ) = -\frac{1}{2} \text{.}\)

1p

○

(Exacte waardencirkel)
\(2 x + \frac{5}{6} \pi = \frac{2}{3} \pi + k ⋅ 2 \pi ∨ 2 x + \frac{5}{6} \pi = -\frac{2}{3} \pi + k ⋅ 2 \pi \)

1p

○

\(2 x = -\frac{1}{6} \pi + k ⋅ 2 \pi ∨ 2 x = -1\frac{1}{2} \pi + k ⋅ 2 \pi \)
\(x = -\frac{1}{12} \pi + k ⋅ \pi ∨ x = -\frac{3}{4} \pi + k ⋅ \pi \)

1p

○

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

1p

4p

c

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

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

c

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

1p

○

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

1p

○

\(\frac{2}{5} \pi x = -\frac{1}{4} \pi + k ⋅ 2 \pi ∨ \frac{2}{5} \pi x = 1\frac{1}{4} \pi + k ⋅ 2 \pi \)
\(x = -\frac{5}{8} + k ⋅ 5 ∨ x = 3\frac{1}{8} + k ⋅ 5\)

1p

○

\(x\) in \([0 , 2 \pi ]\) geeft \(x = 4\frac{3}{8} ∨ x = 3\frac{1}{8}\)

1p

4p

d

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

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

d

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

1p

○

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

1p

○

\(\frac{3}{4} x = \frac{5}{12} \pi + k ⋅ 2 \pi ∨ \frac{3}{4} x = \frac{1}{12} \pi + k ⋅ 2 \pi \)
\(x = \frac{5}{9} \pi + k ⋅ 2\frac{2}{3} \pi ∨ x = \frac{1}{9} \pi + k ⋅ 2\frac{2}{3} \pi \)

1p

○

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

1p

opgave 2

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

4p

\(3 - 5 \sin(3 x + \frac{1}{4} \pi ) = 8\)

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

○

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

1p

○

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

1p

○

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

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

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

Substitutie (1)
006z - Goniometrische vergelijkingen - basis - midden - 0ms - dynamic variables

a

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

1p

○

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

1p

○

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

1p

3p

b

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

Product
0070 - Goniometrische vergelijkingen - basis - midden - 1ms - dynamic variables

b

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

1p

○

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

1p

○

\(\frac{4}{5} x = 1\frac{1}{10} \pi + k ⋅ \pi ∨ 4 x = \pi + k ⋅ \pi \)
\(x = 1\frac{3}{8} \pi + k ⋅ 1\frac{1}{4} \pi ∨ x = \frac{1}{4} \pi + k ⋅ \frac{1}{4} \pi \)

1p
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