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sin(x+pi/4)=sqrt(2)cos(x)

الحلّ

sin (x + \frac{π}{4}) = 2 cos (x)

الحلّ

x = \frac{π}{4} + πn

درجات

x = 4 5^{\circ} + 18 0^{\circ} n

خطوات الحلّ

sin (x + \frac{π}{4}) = 2 cos (x)

Rewrite using trig identities

sin (x + \frac{π}{4}) = 2 cos (x)

Rewrite using trig identities

sin (x + \frac{π}{4})

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

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

بسّط:

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

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

2

قم باخراج العامل المشترك

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

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

اختزل:

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

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

:فعْل قانون الجذور

2 = 2^{\frac{1}{2}}

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

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

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

1 - \frac{1}{2} = \frac{1}{2} :

اطرح الأعداد

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

:فعْل قانون الجذور

2^{\frac{1}{2}} = 2

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

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

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

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

من الطرفين

2 cos (x)

اطرح

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

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

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

tan (x) - 1 = 0

tan (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

tan (x) - 1 = 0

للطرفين

1

أضف

tan (x) - 1 + 1 = 0 + 1

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{π}{4} + πn

x = \frac{π}{4} + πn

أمثلة شائعة

5cos(2x)-cos(x)+3=0

5 cos (2 x) - cos (x) + 3 = 0

cos(x)-2=-8+5

cos (x) - 2 = - 8 + 5

sin(2x-1)=(sqrt(3))/2

sin (2 x - 1) = \frac{3}{2}

arctan(x-1)=0

arctan (x - 1) = 0

sin(2x-arctan(1))= 1/2

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