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شائع علم المثلثات >

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

الحلّ

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

الحلّ

x = 2 πn, x = π + 2 πn

+1

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

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

اطرح

sin^{2} (x) - \frac{1 - cos ^{2} ( x )}{2} = 0

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

وسٌع:

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

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

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

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

افتح أقواس

= - (1) - (- cos^{2} (x))

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

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

\frac{2 sin ^{2} ( x ) - 1 + cos ^{2} ( x )}{2} = 0

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

بسّط

= sin^{2} (x)

sin^{2} (x) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

sin (x) = 0

sin (x) = 0 :

حلول عامّة لـ

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

رسم

أمثلة شائعة

cot(θ)=cos(θ)

cot (θ) = cos (θ)

3csc^3(x)-4csc(x)=0

3 csc^{3} (x) - 4 csc (x) = 0

tan(θ)=(6.06)/(-33.5)

tan (θ) = \frac{6.06}{- 33.5}

csc(θ)=(sqrt(13))/2 ,sec(θ)=(sqrt(13))/3

csc (θ) = \frac{13}{2}, sec (θ) = \frac{13}{3}

4cos^2(x)+cos(2x)-7cos(x)=-2,0<= x<= 360

4 cos^{2} (x) + cos (2 x) - 7 cos (x) = - 2, 0^{\circ} \leq x \leq 36 0^{\circ}