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شائع علم المثلثات >

cos(x-pi/4)+sin(x+pi/4)=0

الحلّ

cos (x - \frac{π}{4}) + sin (x + \frac{π}{4}) = 0

الحلّ

x = \frac{3 π}{4} + πn

درجات

x = 13 5^{\circ} + 18 0^{\circ} n

خطوات الحلّ

cos (x - \frac{π}{4}) + sin (x + \frac{π}{4}) = 0

Rewrite using trig identities

cos (x - \frac{π}{4}) + sin (x + \frac{π}{4}) = 0

Rewrite using trig identities

cos (x - \frac{π}{4})

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

:فعّل متطابقة الطرح لزوايا

= cos (x) cos (\frac{π}{4}) + sin (x) sin (\frac{π}{4})

cos (x) cos (\frac{π}{4}) + sin (x) sin (\frac{π}{4})

بسّط:

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

cos (x) cos (\frac{π}{4}) + sin (x) sin (\frac{π}{4})

cos (x) cos (\frac{π}{4}) = \frac{2 cos ( x )}{2}

cos (x) cos (\frac{π}{4})

cos (\frac{π}{4})

بسّط:

\frac{2}{2}

cos (\frac{π}{4})

استخدم المتطابقة الأساسيّة التالية:

cos (\frac{π}{4}) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} 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)

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

:اضرب كسور

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

sin (x) sin (\frac{π}{4}) = \frac{2 sin ( x )}{2}

sin (x) sin (\frac{π}{4})

sin (\frac{π}{4})

بسّط:

\frac{2}{2}

sin (\frac{π}{4})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

:اضرب كسور

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

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

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

:فعّل متطابقة الجمع لزوايا

= sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4})

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4})

بسّط:

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

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4})

sin (x) cos (\frac{π}{4}) = \frac{2 sin ( x )}{2}

sin (x) cos (\frac{π}{4})

cos (\frac{π}{4})

بسّط:

\frac{2}{2}

cos (\frac{π}{4})

استخدم المتطابقة الأساسيّة التالية:

cos (\frac{π}{4}) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} 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)

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

:اضرب كسور

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

cos (x) sin (\frac{π}{4}) = \frac{2 cos ( x )}{2}

cos (x) sin (\frac{π}{4})

sin (\frac{π}{4})

بسّط:

\frac{2}{2}

sin (\frac{π}{4})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

:اضرب كسور

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

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

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

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

بسّط:

2 (cos (x) + sin (x))

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

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

فعّل القانون

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

2 cos (x) + 2 sin (x) + 2 sin (x) + 2 cos (x) = 22 cos (x) + 22 sin (x)

2 cos (x) + 2 sin (x) + 2 sin (x) + 2 cos (x)

2 cos (x) + 2 cos (x) = 22 cos (x) :

اجمع العناصر المتشابهة

= 22 cos (x) + 2 sin (x) + 2 sin (x)

2 sin (x) + 2 sin (x) = 22 sin (x) :

اجمع العناصر المتشابهة

= 22 cos (x) + 22 sin (x)

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

22

قم باخراج العامل المشترك

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

\frac{2}{2} = 1 :

اقسم الأعداد

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

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

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

2 + 2 tan (x) = 0

2 + 2 tan (x) = 0

انقل

2

إلى الجانب الأيمن

2 + 2 tan (x) = 0

من الطرفين

2

اطرح

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

بسّط

2 tan (x) = - 2

2 tan (x) = - 2

2

اقسم الطرفين على

2 tan (x) = - 2

2

اقسم الطرفين على

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

بسّط

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

حلول عامّة لـ

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

رسم

أمثلة شائعة

3tan(x)+1=0

3 tan (x) + 1 = 0

cos(2x)= 7/25

cos (2 x) = \frac{7}{25}

cos(a)=-1/4

cos (a) = - \frac{1}{4}

-8sec^2(x)-26tan(x)=12

- 8 sec^{2} (x) - 26 tan (x) = 12

tan(θ)=0.3

tan (θ) = 0.3