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شائع علم المثلثات >

csc(x)=sec(x)

الحلّ

csc (x) = sec (x)

الحلّ

x = \frac{π}{4} + πn

درجات

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

خطوات الحلّ

csc (x) = sec (x)

من الطرفين

sec (x)

اطرح

csc (x) - sec (x) = 0

sin, cos :

عبّر بواسطة

csc (x) - sec (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

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

اقسم الطرفين على

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

بسّط

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

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

:Use the basic trigonometric identity

1 - tan (x) = 0

1 - tan (x) = 0

انقل

1

إلى الجانب الأيمن

1 - tan (x) = 0

من الطرفين

1

اطرح

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

بسّط

- tan (x) = - 1

- tan (x) = - 1

- 1

اقسم الطرفين على

- tan (x) = - 1

- 1

اقسم الطرفين على

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

رسم

أمثلة شائعة

sec(x)=3

sec (x) = 3

-sin(x)=cos(x)

- sin (x) = cos (x)

sin(3x)+sin(x)=0,0<= x<= 2pi

sin (3 x) + sin (x) = 0, 0 \leq x \leq 2 π

4sin(θ)+3=1

4 sin (θ) + 3 = 1

25cot^2(θ)-1=0

25 cot^{2} (θ) - 1 = 0