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شائع علم المثلثات >

tan(pi/4-x)+cot(pi/4-x)=4

الحلّ

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

الحلّ

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

درجات

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

خطوات الحلّ

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

Rewrite using trig identities

tan (\frac{π}{4} - x) + cot (\frac{π}{4} - x) = 4

Rewrite using trig identities

cot (\frac{π}{4} - x)

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

:Use the basic trigonometric identity

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

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

:فعّل متطابقة الطرح لزوايا

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

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

:فعّل متطابقة الطرح لزوايا

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

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

بسّط:

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

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

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

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

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) + sin (\frac{π}{4}) sin (x)

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

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

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

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

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) - cos (\frac{π}{4}) sin (x)

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

2 :

إلغ العوامل المشتركة

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

2

قم باخراج العامل المشترك

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

2

قم باخراج العامل المشترك

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

2 :

إلغ العوامل المشتركة

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

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

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

:Use the basic trigonometric identity

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

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

:فعّل متطابقة الطرح لزوايا

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

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

:فعّل متطابقة الطرح لزوايا

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

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

بسّط:

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

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

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

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

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) - cos (\frac{π}{4}) sin (x)

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

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

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

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

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) + sin (\frac{π}{4}) sin (x)

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

2 :

إلغ العوامل المشتركة

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

2

قم باخراج العامل المشترك

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

2

قم باخراج العامل المشترك

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

2 :

إلغ العوامل المشتركة

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

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

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

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

بسّط:

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

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

cos (x) + sin (x), cos (x) - sin (x)

المضاعف المشترك الأصغر لـ:

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

cos (x) + sin (x), cos (x) - sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x) + sin (x)

cos (x) - sin (x)

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

cos (x) - sin (x)

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

For

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

multiply the denominator and numerator by

cos (x) + sin (x)

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

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

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

وسٌع:

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

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

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

(a - b)^{2} = a^{2} - 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

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

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

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

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

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

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

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

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

بسّط:

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

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

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

اجمع العناصر المتشابهة

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

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

اجمع العناصر المتشابهة

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

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

اجمع العناصر المتشابهة

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

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

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

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

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

من الطرفين

4

اطرح

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - 4 = 0

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

بسّط:

\frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

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

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

:حوّل الأعداد لكسور

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

2 cos^{2} (x) + 2 sin^{2} (x) - 4 (cos (x) + sin (x)) (cos (x) - sin (x))

وسٌع:

- 2 cos^{2} (x) + 6 sin^{2} (x)

2 cos^{2} (x) + 2 sin^{2} (x) - 4 (cos (x) + sin (x)) (cos (x) - sin (x))

- 4 (cos (x) + sin (x)) (cos (x) - sin (x))

وسٌع:

- 4 cos^{2} (x) + 4 sin^{2} (x)

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

وسٌع:

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

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

(a + b) (a - b) = a^{2} - b^{2}

فعّل قانون فرق المربّعات

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

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

= - 4 (cos^{2} (x) - sin^{2} (x))

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

وسٌع:

- 4 cos^{2} (x) + 4 sin^{2} (x)

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

= - 4 cos^{2} (x) - (- 4) sin^{2} (x)

فعّل قوانين سالب-موجب

- (- a) = a

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= 2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

بسّط:

- 2 cos^{2} (x) + 6 sin^{2} (x)

2 cos^{2} (x) + 2 sin^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

2 cos^{2} (x) - 4 cos^{2} (x) = - 2 cos^{2} (x) :

اجمع العناصر المتشابهة

= - 2 cos^{2} (x) + 2 sin^{2} (x) + 4 sin^{2} (x)

2 sin^{2} (x) + 4 sin^{2} (x) = 6 sin^{2} (x) :

اجمع العناصر المتشابهة

= - 2 cos^{2} (x) + 6 sin^{2} (x)

= - 2 cos^{2} (x) + 6 sin^{2} (x)

= \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = 0

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

- 2 cos^{2} (x) + 6 sin^{2} (x) = 0

- 2 cos^{2} (x) + 6 sin^{2} (x)

حلل إلى عوامل:

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

- 2 cos^{2} (x) + 6 sin^{2} (x)

3 \cdot 2

كـ

6

اكتب مجددًا

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

2

قم باخراج العامل المشترك

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

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

حلل إلى عوامل:

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

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

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

كـ

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

اكتب مجددًا

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

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

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

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

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

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

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

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

حلّ كل جزء على حدة

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

3 sin (x) + cos (x) = 0 : x = \frac{5 π}{6} + πn

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

من الطرفين

1

اطرح

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{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{1}{3}

- \frac{1}{3}

حوّل لصيغة عدد كسريّ:

- \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}

= - \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

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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

3 sin (x) - cos (x) = 0 : x = \frac{π}{6} + πn

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

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}{6} + πn

x = \frac{π}{6} + πn

وحّد الحلول

x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

رسم

أمثلة شائعة

7sin(θ)-1=5sin(θ)

7 sin (θ) - 1 = 5 sin (θ)

csc(θ)=-7/3

csc (θ) = - \frac{7}{3}

sin(θ)= 5/8

sin (θ) = \frac{5}{8}

arctan(2x)+arctan(x)= pi/4

arctan (2 x) + arctan (x) = \frac{π}{4}

tan(a)=1

tan (a) = 1