Getal & Ruimte (13e 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

\(\sin(4 x) = 0\)

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

a

(Exacte waardencirkel)
\(4 x = k ⋅ \pi \)

1p

○

\(4 x = k ⋅ \pi \)
\(x = k ⋅ \frac{1}{4} \pi \)

1p

○

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

1p

4p

b

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

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

b

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

1p

○

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

1p

○

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

1p

○

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

1p

4p

c

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

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

c

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

1p

○

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

1p

○

\(\frac{2}{3} \pi x = \frac{11}{12} \pi + k ⋅ 2 \pi ∨ \frac{2}{3} \pi x = 1\frac{5}{12} \pi + k ⋅ 2 \pi \)
\(x = 1\frac{3}{8} + k ⋅ 3 ∨ x = 2\frac{1}{8} + k ⋅ 3\)

1p

○

\(x\) in \([0 , 2 \pi ]\) geeft \(x = 1\frac{3}{8} ∨ x = 4\frac{3}{8} ∨ x = 2\frac{1}{8} ∨ x = 5\frac{1}{8}\)

1p

4p

d

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

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

d

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

1p

○

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

1p

○

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

1p

○

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

1p

opgave 2

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

4p

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

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

○

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

1p

○

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

1p

○

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

1p

○

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

1p

opgave 3

Los exact op.

3p

a

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

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

a

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

1p

○

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

1p

○

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

1p

3p

b

\(\frac{5}{9} \sin(\frac{3}{4} x + \frac{2}{3} \pi ) \sin(\frac{3}{4} x - \frac{1}{6} \pi ) = 0\)

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

b

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

1p

○

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

1p

○

\(\frac{3}{4} x = -\frac{2}{3} \pi + k ⋅ \pi ∨ \frac{3}{4} x = \frac{1}{6} \pi + k ⋅ \pi \)
\(x = -\frac{8}{9} \pi + k ⋅ 1\frac{1}{3} \pi ∨ x = \frac{2}{9} \pi + k ⋅ 1\frac{1}{3} \pi \)

1p
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