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פּוֹפּוּלָרִי טריגונומטריה >

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

פתרון

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

פתרון

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

מעלות

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

צעדי פתרון

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

משני האגפים

1 - sin^{2} (x)

החסר

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

Rewrite using trig identities

- 1 + sin^{2} (x) + sin^{2} (x) tan^{2} (x)

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

:הפעל זהות פיטגורית

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

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

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

פשט את:

sin^{2} (x) sec^{2} (x) - 1

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

sin^{2} (x) (sec^{2} (x) - 1)

הרחב את:

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

sin^{2} (x) (sec^{2} (x) - 1)

a (b - c) = ab - a c

: פתח סוגריים בעזרת

a = sin^{2} (x), b = sec^{2} (x), c = 1

= sin^{2} (x) sec^{2} (x) - sin^{2} (x) \cdot 1

= sin^{2} (x) sec^{2} (x) - 1 \cdot sin^{2} (x)

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

הכפל

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

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

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

פשט את:

sin^{2} (x) sec^{2} (x) - 1

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

קבץ ביטויים דומים יחד

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

sin^{2} (x) - sin^{2} (x) = 0 :

חבר איברים דומים

= sin^{2} (x) sec^{2} (x) - 1

= sin^{2} (x) sec^{2} (x) - 1

= sin^{2} (x) sec^{2} (x) - 1

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

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

פרק לגורמים את:

(sec (x) sin (x) + 1) (sec (x) sin (x) - 1)

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

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

:הפעל את חוק החזקות

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

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

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

הפעל את חוק הפרש הריבועים

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

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

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

פתור כל חלק בנפרד

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

sec (x) sin (x) + 1 = 0 : x = \frac{3 π}{4} + πn

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

Rewrite using trig identities

sec (x) sin (x) + 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

sin, cos :

בטא באמצאות

sec (x) sin (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

:הכפל שברים

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

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

הכפל

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

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

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

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

:Use the basic trigonometric identity

= tan (x)

= 1 + tan (x)

1 + tan (x) = 0

לצד ימין

1

העבר

1 + tan (x) = 0

משני האגפים

1

החסר

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

פשט

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1 :

פתרונות כלליים עבור

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

sec (x) sin (x) - 1 = 0 : x = \frac{π}{4} + πn

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

Rewrite using trig identities

sec (x) sin (x) - 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

sin, cos :

בטא באמצאות

sec (x) sin (x)

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

:Use the basic trigonometric identity

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

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

פשט את:

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

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

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

:הכפל שברים

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

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

הכפל

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

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

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

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

:Use the basic trigonometric identity

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

לצד ימין

1

העבר

- 1 + tan (x) = 0

לשני האגפים

1

הוסף

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

פשט

tan (x) = 1

tan (x) = 1

tan (x) = 1 :

פתרונות כלליים עבור

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

אחד את הפתרונות

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

גרף

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