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شائع علم المثلثات >

sin(x+30)=cos(x)

الحلّ

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

الحلّ

x = 3 0^{\circ} + 18 0^{\circ} n

راديان

x = \frac{π}{6} + πn

خطوات الحلّ

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

Rewrite using trig identities

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

Rewrite using trig identities

sin (x + 3 0^{\circ})

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

:فعّل متطابقة الجمع لزوايا

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

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ})

بسّط:

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

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ})

cos (3 0^{\circ})

بسّط:

\frac{3}{2}

cos (3 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (3 0^{\circ}) = \frac{3}{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{3}{2}

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

sin (3 0^{\circ})

بسّط:

\frac{1}{2}

sin (3 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

sin (3 0^{\circ}) = \frac{1}{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{1}{2}

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

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

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

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

من الطرفين

cos (x)

اطرح

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

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

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

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

:اضرب كسور

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

1 \cdot cos (x) = cos (x) :

اضرب

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

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

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

فعّل القانون

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

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

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

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

Rewrite using trig identities

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

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

3 tan (x) - 1 = 0

للطرفين

1

أضف

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

بسّط

3 tan (x) = 1

3 tan (x) = 1

3

اقسم الطرفين على

3 tan (x) = 1

3

اقسم الطرفين على

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

بسّط

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

\frac{3 tan ( x )}{3}

بسّط:

tan (x)

\frac{3 tan ( x )}{3}

3 :

إلغ العوامل المشتركة

= tan (x)

\frac{1}{3}

بسّط:

\frac{3}{3}

\frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= \frac{3}{3}

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

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

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

tan (x) = \frac{3}{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 = 3 0^{\circ} + 18 0^{\circ} n

x = 3 0^{\circ} + 18 0^{\circ} n

رسم

أمثلة شائعة

sin(pi/6 x)=0

sin (\frac{π}{6} x) = 0

sec(x)-2cos(x)=1

sec (x) - 2 cos (x) = 1

2sin(x)+3=0

2 sin (x) + 3 = 0

4cos(2θ)-10cos(θ)+14=7

4 cos (2 θ) - 10 cos (θ) + 14 = 7

2cos^2(2x)=cos(2x)

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