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شائع علم المثلثات >

sin^2(x)sec(x)cot(x)=(sqrt(3))/2

الحلّ

sin^{2} (x) sec (x) cot (x) = \frac{3}{2}

الحلّ

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

+1

درجات

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sin^{2} (x) sec (x) cot (x) = \frac{3}{2}

من الطرفين

\frac{3}{2}

اطرح

sin^{2} (x) sec (x) cot (x) - \frac{3}{2} = 0

sin^{2} (x) sec (x) cot (x) - \frac{3}{2}

بسّط:

\frac{2 sin ^{2} ( x ) sec ( x ) cot ( x ) - 3}{2}

sin^{2} (x) sec (x) cot (x) - \frac{3}{2}

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

:حوّل الأعداد لكسور

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) \cdot 2}{2} - \frac{3}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) \cdot 2 - 3}{2}

\frac{2 sin ^{2} ( x ) sec ( x ) cot ( x ) - 3}{2} = 0

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

2 sin^{2} (x) sec (x) cot (x) - 3 = 0

sin, cos :

عبّر بواسطة

- 3 + 2 cot (x) sec (x) sin^{2} (x)

cot (x) = \frac{cos ( x )}{sin ( x )}

:Use the basic trigonometric identity

= - 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} sec (x) sin^{2} (x)

sec (x) = \frac{1}{cos ( x )}

:Use the basic trigonometric identity

= - 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

- 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

بسّط:

- 3 + 2 sin (x)

- 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

= \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)

= - 3 + 2 sin (x)

= - 3 + 2 sin (x)

- 3 + 2 sin (x) = 0

انقل

3

إلى الجانب الأيمن

- 3 + 2 sin (x) = 0

للطرفين

3

أضف

- 3 + 2 sin (x) + 3 = 0 + 3

بسّط

2 sin (x) = 3

2 sin (x) = 3

2

اقسم الطرفين على

2 sin (x) = 3

2

اقسم الطرفين على

\frac{2 sin ( x )}{2} = \frac{3}{2}

بسّط

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{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{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

رسم

أمثلة شائعة

solvefor x,sin(y)=ycos(2x)

so l v e f or x, sin (y) = y cos (2 x)

2sin(pi/2-θ)=sin(-θ)

2 sin (\frac{π}{2} - θ) = sin (- θ)

tan (x) = \frac{50}{36}

solvefor θ,ρ=sin(φ)sin(θ)

so l v e f or θ, ρ = sin (φ) sin (θ)

3cos^2(x)-6cos(x)=-3,0<= x<= 2pi

3 cos^{2} (x) - 6 cos (x) = - 3, 0 \leq x \leq 2 π