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شائع علم المثلثات >

tan(x)+tan(x+45)=-2

الحلّ

tan (x) + tan (x + 4 5^{\circ}) = - 2

الحلّ

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

راديان

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

خطوات الحلّ

tan (x) + tan (x + 4 5^{\circ}) = - 2

Rewrite using trig identities

tan (x) + tan (x + 4 5^{\circ}) = - 2

Rewrite using trig identities

tan (x + 4 5^{\circ})

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

:Use the basic trigonometric identity

= \frac{sin ( x + 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

:فعّل متطابقة الجمع لزوايا

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

:فعّل متطابقة الجمع لزوايا

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

بسّط:

\frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ}) = \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ})

cos (4 5^{\circ})

بسّط:

\frac{2}{2}

cos (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} sin (x) + sin (4 5^{\circ}) cos (x)

sin (4 5^{\circ})

بسّط:

\frac{2}{2}

sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ}) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ})

cos (4 5^{\circ})

بسّط:

\frac{2}{2}

cos (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{2}{2} cos (x) - sin (4 5^{\circ}) sin (x)

sin (4 5^{\circ})

بسّط:

\frac{2}{2}

sin (4 5^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (4 5^{\circ}) = \frac{2}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

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

اضرب بـ:

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

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

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

:اضرب كسور

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

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

اضرب بـ:

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

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

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

:اضرب كسور

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

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

اضرب بـ:

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

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

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

:اضرب كسور

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

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

وحّد الكسور:

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

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

فعّل القانون

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

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

وحّد الكسور:

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

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

فعّل القانون

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

= \frac{\frac{2 s i n ( x ) + 2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

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

:اقسم الكسور

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

2 :

إلغ العوامل المشتركة

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

2

قم باخراج العامل المشترك

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

2

قم باخراج العامل المشترك

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

2 :

إلغ العوامل المشتركة

= \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

= \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

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

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

من الطرفين

- 2

اطرح

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

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} + 2

بسّط:

\frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )}

tan (x) + \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} + 2

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

tan (x) (cos (x) - sin (x)) + sin (x) + cos (x) + 2 (cos (x) - sin (x))

وسٌع:

tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

tan (x) (cos (x) - sin (x)) + sin (x) + cos (x) + 2 (cos (x) - sin (x))

tan (x) (cos (x) - sin (x))

وسٌع:

tan (x) cos (x) - tan (x) sin (x)

tan (x) (cos (x) - sin (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = tan (x), b = cos (x), c = sin (x)

= tan (x) cos (x) - tan (x) sin (x)

= tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 (cos (x) - sin (x))

2 (cos (x) - sin (x))

وسٌع:

2 cos (x) - 2 sin (x)

2 (cos (x) - sin (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 2, b = cos (x), c = sin (x)

= 2 cos (x) - 2 sin (x)

= tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 cos (x) - 2 sin (x)

tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 cos (x) - 2 sin (x)

بسّط:

tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

tan (x) cos (x) - tan (x) sin (x) + sin (x) + cos (x) + 2 cos (x) - 2 sin (x)

cos (x) + 2 cos (x) = 3 cos (x) :

اجمع العناصر المتشابهة

= tan (x) cos (x) - tan (x) sin (x) + sin (x) + 3 cos (x) - 2 sin (x)

sin (x) - 2 sin (x) = - sin (x) :

اجمع العناصر المتشابهة

= tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

= tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x)

= \frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )}

\frac{tan ( x ) cos ( x ) - tan ( x ) sin ( x ) - sin ( x ) + 3 cos ( x )}{cos ( x ) - sin ( x )} = 0

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

tan (x) cos (x) - tan (x) sin (x) - sin (x) + 3 cos (x) = 0

sin, cos :

عبّر بواسطة

- sin (x) + 3 cos (x) + cos (x) tan (x) - sin (x) tan (x)

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

:Use the basic trigonometric identity

= - sin (x) + 3 cos (x) + cos (x) \frac{sin ( x )}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

- sin (x) + 3 cos (x) + cos (x) \frac{sin ( x )}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

بسّط:

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

- sin (x) + 3 cos (x) + cos (x) \frac{sin ( x )}{cos ( x )} - sin (x) \frac{sin ( x )}{cos ( x )}

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

= sin (x)

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

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

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

:اضرب كسور

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

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

sin (x) sin (x)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (x)

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

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

- sin (x) + sin (x) = 0 :

اجمع العناصر المتشابهة

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

3 cos (x) = \frac{3 cos ( x ) cos ( x )}{cos ( x )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

3 cos (x) cos (x) - sin^{2} (x) = 3 cos^{2} (x) - sin^{2} (x)

3 cos (x) cos (x) - sin^{2} (x)

3 cos (x) cos (x) = 3 cos^{2} (x)

3 cos (x) cos (x)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos (x) cos (x) = cos^{1 + 1} (x)

= 3 cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= 3 cos^{2} (x)

= 3 cos^{2} (x) - sin^{2} (x)

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

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

\frac{- sin ^{2} ( x ) + 3 cos ^{2} ( x )}{cos ( x )} = 0

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

- sin^{2} (x) + 3 cos^{2} (x) = 0

- sin^{2} (x) + 3 cos^{2} (x)

حلل إلى عوامل:

(3 cos (x) + sin (x)) (3 cos (x) - sin (x))

- sin^{2} (x) + 3 cos^{2} (x)

(3 cos (x))^{2} - sin^{2} (x)

كـ

3 cos^{2} (x) - sin^{2} (x)

اكتب مجددًا

3 cos^{2} (x) - sin^{2} (x)

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

= (3)^{2} cos^{2} (x) - sin^{2} (x)

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

:فعّل قانون القوى

(3)^{2} cos^{2} (x) = (3 cos (x))^{2}

= (3 cos (x))^{2} - sin^{2} (x)

= (3 cos (x))^{2} - sin^{2} (x)

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

فعّل قانون فرق المربّعات

(3 cos (x))^{2} - sin^{2} (x) = (3 cos (x) + sin (x)) (3 cos (x) - sin (x))

= (3 cos (x) + sin (x)) (3 cos (x) - sin (x))

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

حلّ كل جزء على حدة

3 cos (x) + sin (x) = 0 or 3 cos (x) - sin (x) = 0

3 cos (x) + sin (x) = 0 : x = 12 0^{\circ} + 18 0^{\circ} n

3 cos (x) + sin (x) = 0

Rewrite using trig identities

3 cos (x) + sin (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

\frac{3 cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

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

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

:Use the basic trigonometric identity

3 + tan (x) = 0

3 + tan (x) = 0

انقل

3

إلى الجانب الأيمن

3 + tan (x) = 0

من الطرفين

3

اطرح

3 + tan (x) - 3 = 0 - 3

بسّط

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3 :

حلول عامّة لـ

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 12 0^{\circ} + 18 0^{\circ} n

x = 12 0^{\circ} + 18 0^{\circ} n

3 cos (x) - sin (x) = 0 : x = 6 0^{\circ} + 18 0^{\circ} n

3 cos (x) - sin (x) = 0

Rewrite using trig identities

3 cos (x) - sin (x) = 0

cos (x) \neq = 0, cos (x)

اقسم الطرفين على

\frac{3 cos ( x ) - sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

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

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

:Use the basic trigonometric identity

3 - tan (x) = 0

3 - tan (x) = 0

انقل

3

إلى الجانب الأيمن

3 - tan (x) = 0

من الطرفين

3

اطرح

3 - tan (x) - 3 = 0 - 3

بسّط

- tan (x) = - 3

- tan (x) = - 3

- 1

اقسم الطرفين على

- tan (x) = - 3

- 1

اقسم الطرفين على

\frac{- tan ( x )}{- 1} = \frac{- 3}{- 1}

بسّط

tan (x) = 3

tan (x) = 3

tan (x) = 3 :

حلول عامّة لـ

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 6 0^{\circ} + 18 0^{\circ} n

x = 6 0^{\circ} + 18 0^{\circ} n

وحّد الحلول

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

رسم

أمثلة شائعة

tan(x)=(22.5)/(34)

tan (x) = \frac{22.5}{34}

tan^2(4x)=0

tan^{2} (4 x) = 0

tan(x-45)-tan(x+45)=4

tan (x - 4 5^{\circ}) - tan (x + 4 5^{\circ}) = 4

sin(2x)=10cos(x)

sin (2 x) = 10 cos (x)

3-4sin(θ)=4-6sin(θ)

3 - 4 sin (θ) = 4 - 6 sin (θ)