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sec^2(x)=4tan^2(x)

الحلّ

sec^{2} (x) = 4 tan^{2} (x)

الحلّ

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

درجات

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sec^{2} (x) = 4 tan^{2} (x)

من الطرفين

4 tan^{2} (x)

اطرح

sec^{2} (x) - 4 tan^{2} (x) = 0

sec^{2} (x) - 4 tan^{2} (x)

حلل إلى عوامل:

(sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

sec^{2} (x) - 4 tan^{2} (x)

sec^{2} (x) - (2 tan (x))^{2}

كـ

sec^{2} (x) - 4 tan^{2} (x)

اكتب مجددًا

sec^{2} (x) - 4 tan^{2} (x)

2^{2}

كـ

4

اكتب مجددًا

= sec^{2} (x) - 2^{2} tan^{2} (x)

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

2^{2} tan^{2} (x) = (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

sec^{2} (x) - (2 tan (x))^{2} = (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

= (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

(sec (x) + 2 tan (x)) (sec (x) - 2 tan (x)) = 0

حلّ كل جزء على حدة

sec (x) + 2 tan (x) = 0 or sec (x) - 2 tan (x) = 0

sec (x) + 2 tan (x) = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) + 2 tan (x) = 0

sin, cos :

عبّر بواسطة

sec (x) + 2 tan (x)

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

:Use the basic trigonometric identity

= \frac{1}{cos ( x )} + 2 tan (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{1 + 2 sin ( x )}{cos ( x )}

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

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

اضرب بـ:

\frac{2 sin ( x )}{cos ( x )}

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

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

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

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

sec (x) - 2 tan (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sec (x) - 2 tan (x) = 0

sin, cos :

عبّر بواسطة

sec (x) - 2 tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

اضرب بـ:

\frac{2 sin ( x )}{cos ( x )}

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

\frac{1 - 2 sin ( x )}{cos ( x )} = 0

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

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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

وحّد الحلول

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

أمثلة شائعة

5sin(x)=-4

5 sin (x) = - 4

-pisin(pix)=0

- π sin (π x) = 0

7tan(4x+199)=8sin(4x+199)

7 tan (4 x + 199) = 8 sin (4 x + 199)

4sin^2(x)+4cos(x)-5=0,0<= x<= 2pi

4 sin^{2} (x) + 4 cos (x) - 5 = 0, 0 \leq x \leq 2 π

sqrt(2)sin(2θ)=1

2 sin (2 θ) = 1