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פּוֹפּוּלָרִי טריגונומטריה >

(1+tan^2(x))cos^2(x)=sec(x)

פתרון

(1 + tan^{2} (x)) cos^{2} (x) = sec (x)

פתרון

x = 2 πn

+1

מעלות

x = 0^{\circ} + 36 0^{\circ} n

צעדי פתרון

(1 + tan^{2} (x)) cos^{2} (x) = sec (x)

משני האגפים

sec (x)

החסר

cos^{2} (x) (1 + tan^{2} (x)) - sec (x) = 0

Rewrite using trig identities

- sec (x) + (1 + tan^{2} (x)) cos^{2} (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:הפעל את חוק החזקות

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

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

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

הרחב את:

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

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

a (b + c) = ab + a c

: פתח סוגריים בעזרת

a = cos^{2} (x), b = 1, c = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

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

פשט את:

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

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

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

1 \cdot cos^{2} (x)

1 \cdot cos^{2} (x) = cos^{2} (x) :

הכפל

= cos^{2} (x)

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

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

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

:הכפל שברים

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

cos^{2} (x) :

בטל את הגורמים המשותפים

= sin^{2} (x)

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

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

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

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

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

:הפעל זהות פיטגורית

= - sec (x) + 1

- sec (x) + 1 = 0

לצד ימין

1

העבר

- sec (x) + 1 = 0

משני האגפים

1

החסר

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

פשט

- sec (x) = - 1

- sec (x) = - 1

- 1

חלק את שני האגפים ב

- sec (x) = - 1

- 1

חלק את שני האגפים ב

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

פשט

sec (x) = 1

sec (x) = 1

sec (x) = 1 :

פתרונות כלליים עבור

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

x = 0 + 2 πn

פתור את:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

גרף

דוגמאות פופולריות

(cos^2(x)-1)/(sin^2(x))=-tan(x)

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

cos^6(x)=-cos^2(x)

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

2sin^3(x)-5sin^2(x)+2sin(x)=0

2 sin^{3} (x) - 5 sin^{2} (x) + 2 sin (x) = 0

(cos^2(a)-3cos(a)+2)/(sin^2(a))=1

\frac{cos ^{2} ( a ) - 3 cos ( a ) + 2}{sin ^{2} ( a )} = 1

(sin(x)-(sqrt(2)))/2 =0

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