English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove csc(pi/2-x)=sec(x)

الحلّ

prove

csc (\frac{π}{2} - x) = sec (x)

صحيح

خطوات الحلّ

csc (\frac{π}{2} - x) = sec (x)

قم بالعمل على الطرف الأيسر

csc (\frac{π}{2} - x)

Rewrite using trig identities

csc (\frac{π}{2} - x)

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{1}{sin ( \frac{π}{2} - x )}

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

:فعّل متطابقة الطرح لزوايا

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

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

بسّط:

\frac{1}{cos ( x )}

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

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

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

sin (\frac{π}{2}) cos (x) = cos (x)

sin (\frac{π}{2}) cos (x)

sin (\frac{π}{2})

بسّط:

1

sin (\frac{π}{2})

استخدم المتطابقة الأساسيّة التالية:

sin (\frac{π}{2}) = 1

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}

= 1

= 1 \cdot cos (x)

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

اضرب

= cos (x)

cos (\frac{π}{2}) sin (x) = 0

cos (\frac{π}{2}) sin (x)

cos (\frac{π}{2})

بسّط:

0

cos (\frac{π}{2})

استخدم المتطابقة الأساسيّة التالية:

cos (\frac{π}{2}) = 0

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}

= 0

= 0 \cdot sin (x)

0 \cdot a = 0

فعّل القانون

= 0

= cos (x) - 0

cos (x) - 0 = cos (x)

= cos (x)

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

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

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

Rewrite using trig identities

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

:Use the basic trigonometric identity

\frac{1}{\frac{1}{s e c ( x )}}

بسّط

\frac{1}{\frac{1}{s e c ( x )}}

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: استخدم ميزات الكسور التالية

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

\frac{a}{1} = a

فعّل القانون

= sec (x)

sec (x)

sec (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e cos (x) cot (x) = \frac{1 - sin ^{2} ( x )}{sin ( x )}

p ro v e tan (\frac{π}{2} - x) sec (x) = csc (x)

p ro v e \frac{1}{cos ( x )} - cos (x) = sin (x) tan (x)

p ro v e sec (u) cos (u) - sin^{2} (u) = cos^{2} (u)

p ro v e sin (6 0^{\circ}) = 2 sin (3 0^{\circ}) cos (3 0^{\circ})