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Parents Pricing Home NSC Mathematics Trigonometry 5.1 Gegee: sin 2x = \frac{\sqrt{15}}{8} en 0° ≤ 2x ≤ 90°
5.1 Gegee: sin 2x = \frac{\sqrt{15}}{8} en 0° ≤ 2x ≤ 90° - NSC Mathematics - Question 5 - 2017 - Paper 2 Question 5
View full question 5.1 Gegee: sin 2x = \frac{\sqrt{15}}{8} en 0° ≤ 2x ≤ 90°.
Bepaal met behulp van ‘n diagram en sonder die gebruik van ‘n sakrekenaar die waarde van cos x.
5.2 Veree... show full transcript
View marking scheme Worked Solution & Example Answer:5.1 Gegee: sin 2x = \frac{\sqrt{15}}{8} en 0° ≤ 2x ≤ 90° - NSC Mathematics - Question 5 - 2017 - Paper 2
5.1 Bepaal met behulp van ‘n diagram en sonder die gebruik van ‘n sakrekenaar die waarde van cos x. Only available for registered users.
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Om cos x te bepaal, kan ons gebruik maak van die gespesifiseerde waarde van sin 2x:
Gegee:
e x t s i n 2 x = 15 8 ext{sin } 2x = \frac{\sqrt{15}}{8} e x t s in 2 x = 8 15 Gebruik die verhouding e x t s i n 2 + e x t c o s 2 = 1 ext{sin}^2 + ext{cos}^2 = 1 e x t s in 2 + e x t cos 2 = 1
Bereken sin² 2x:
e x t s i n 2 2 x = ( 15 8 ) 2 = 15 64 ext{sin}^2 2x = \left(\frac{\sqrt{15}}{8}\right)^2 = \frac{15}{64} e x t s in 2 2 x = ( 8 15 ) 2 = 64 15
Bereken cos² 2x:
e x t c o s 2 2 x = 1 − e x t s i n 2 2 x = 1 − 15 64 = 49 64 ext{cos}^2 2x = 1 - ext{sin}^2 2x = 1 - \frac{15}{64} = \frac{49}{64} e x t cos 2 2 x = 1 − e x t s in 2 2 x = 1 − 64 15 = 64 49
Neem die vierkantswortel:
e x t c o s 2 x = 49 64 = 7 8 ext{cos } 2x = \sqrt{\frac{49}{64}} = \frac{7}{8} e x t cos 2 x = 64 49 = 8 7
Daarom is cos x gelyk aan:
e x t c o s x = cos ( 2 x 2 ) = 7 8 . ext{cos } x = \text{cos }\left(\frac{2x}{2}\right) = \frac{7}{8}. e x t cos x = cos ( 2 2 x ) = 8 7 .
5.2 Vereenvoudig die volgende uitdrukking tot een trigonometriese verhouding van θ. Only available for registered users.
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Die stelling is:
sin ( 180 ° − θ ) . sin ( 540 ° − θ ) . cos ( θ − 90 ° ) \text{sin}(180° - θ) . \text{sin}(540° - θ) . \text{cos}(θ - 90°) sin ( 180° − θ ) . sin ( 540° − θ ) . cos ( θ − 90° ) en
tan ( − θ ) . sin ( 360 ° − θ ) . \text{tan}(−θ) . \text{sin}(360° - θ). tan ( − θ ) . sin ( 360° − θ ) . Begin met die vereenvoudiging:
\text{sin}(180° - θ) = \text{sin} θ,
\text{sin}(540° - θ) = \text{sin}(180° + 360° - θ) = -\text{sin} θ,
\text{cos}(θ - 90°) = \text{sin} θ,
Из terwyl,
\text{tan}(−θ) = -\text{tan } θ,
\text{sin}(360° - θ) = -\text{sin}(θ).
Daarom:
sin ( θ ) . ( − sin θ ) . sin θ = − sin 2 θ . \text{sin}(θ) . (-\text{sin} θ) . \text{sin} θ = -\text{sin}^2 θ. sin ( θ ) . ( − sin θ ) . sin θ = − sin 2 θ . Dit vereenvoudig na:
− sin 2 θ = − tan θ ⋅ sin θ . - \text{sin}^2 θ = -\text{tan} θ\cdot\text{sin} θ. − sin 2 θ = − tan θ ⋅ sin θ .
5.3.1 Bewys die bestaande identiteit. Only available for registered users.
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Begin met die linkerhandse sy:
L H S = sin ( 5 x ) cos ( 3 x ) − cos ( 5 x ) sin ( 3 x ) tan ( 2 x ) − 1 . LHS = \frac{\text{sin}(5x)\text{cos}(3x) - \text{cos}(5x)\text{sin}(3x)}{\text{tan}(2x) - 1}. L H S = tan ( 2 x ) − 1 sin ( 5 x ) cos ( 3 x ) − cos ( 5 x ) sin ( 3 x ) . Plaas die identiteite in die uitdrukking:
Gebruik die identiteit e x t s i n ( A − B ) = e x t s i n ( A ) e x t c o s ( B ) − e x t c o s ( A ) e x t s i n ( B ) ext{sin}(A - B) = ext{sin}(A) ext{cos}(B) - ext{cos}(A) ext{sin}(B) e x t s in ( A − B ) = e x t s in ( A ) e x t cos ( B ) − e x t cos ( A ) e x t s in ( B ) e x t s i n ( 5 x − 3 x ) = e x t s i n ( 2 x ) ; ext{sin}(5x - 3x) = ext{sin}(2x); e x t s in ( 5 x − 3 x ) = e x t s in ( 2 x ) ;
Simplifiseer en herskryf:
= e x t s i n ( 2 x ) ; = ext{sin}(2x); = e x t s in ( 2 x ) ;
5.3.2 Vir watter waarde(s) van x sal die bestaande identiteit ongedefinieerd wees vir 0° ≤ x ≤ 180°. Only available for registered users.
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Die identiteit kan ongedefinieerd wees wanneer die noemer van e x t t a n ( 2 x ) ext{tan}(2x) e x t t an ( 2 x )
a n ( 2 x ) = 0 e x t w a n n e e r 2 x = 0 ° e x t o f 180 ° . an(2x) = 0 ext{ wanneer } 2x = 0° ext{ of } 180°. an ( 2 x ) = 0 e x t w ann eer 2 x = 0° e x t o f 180°. Dus:
2 x = 0 ° 2x = 0° 2 x = 0° x = 0 ° . x = 0°. x = 0°. 2 x = 180 ° 2x = 180° 2 x = 180° x = 90 ° . x = 90°. x = 90°.
Die waardes van x wat die identiteit ongedefinieerd maak is 0 ° 0° 0° 90 ° . 90°. 90°.
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