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شائع علم المثلثات >

sin^2(x)csc(x)=-1/2

الحلّ

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

الحلّ

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

درجات

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

- \frac{1}{2}

اطرح

sin^{2} (x) csc (x) + \frac{1}{2} = 0

sin^{2} (x) csc (x) + \frac{1}{2}

بسّط:

\frac{2 sin ^{2} ( x ) csc ( x ) + 1}{2}

sin^{2} (x) csc (x) + \frac{1}{2}

sin^{2} (x) csc (x) = \frac{sin ^{2} ( x ) csc ( x ) 2}{2}

:حوّل الأعداد لكسور

= \frac{sin ^{2} ( x ) csc ( x ) \cdot 2}{2} + \frac{1}{2}

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ^{2} ( x ) csc ( x ) \cdot 2 + 1}{2}

\frac{2 sin ^{2} ( x ) csc ( x ) + 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin^{2} (x) csc (x) + 1 = 0

Rewrite using trig identities

1 + 2 csc (x) sin^{2} (x)

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= 1 + 2 \cdot \frac{1}{sin ( x )} sin^{2} (x)

2 \cdot \frac{1}{sin ( x )} sin^{2} (x) = 2 sin (x)

2 \cdot \frac{1}{sin ( x )} sin^{2} (x)

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2 sin ^{2} ( x )}{sin ( x )}

1 \cdot 2 = 2 :

اضرب الأعداد

= \frac{2 sin ^{2} ( x )}{sin ( x )}

sin (x) :

إلغ العوامل المشتركة

= 2 sin (x)

= 1 + 2 sin (x)

1 + 2 sin (x) = 0

انقل

1

إلى الجانب الأيمن

1 + 2 sin (x) = 0

من الطرفين

1

اطرح

1 + 2 sin (x) - 1 = 0 - 1

بسّط

2 sin (x) = - 1

2 sin (x) = - 1

2

اقسم الطرفين على

2 sin (x) = - 1

2

اقسم الطرفين على

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

بسّط

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

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

sin (x) = - \frac{1}{2} :

حلول عامّة لـ

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

رسم

أمثلة شائعة

sec^{2} (2 x) - 1 = 0

1 = 2 cos (x)

solvefor x,y=arcsin(x/(11))

so l v e f or x, y = arcsin (\frac{x}{11})

(sin(48))/(13)=(sin(b))/(16)

\frac{sin ( 4 8 ^{\circ} )}{13} = \frac{sin ( b )}{16}

9 cos (2 θ) = 0