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شائع علم المثلثات >

sin(2x)=1.47(sin(x))^2

الحلّ

sin (2 x) = 1.47 (sin (x))^{2}

الحلّ

x = 2 πn, x = π + 2 πn, x = 0.93696 \dots + πn

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 53.68411 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

sin (2 x) = 1.47 (sin (x))^{2}

من الطرفين

1.47 (sin (x))^{2}

اطرح

sin (2 x) - 1.47 sin^{2} (x) = 0

Rewrite using trig identities

sin (2 x) - 1.47 sin^{2} (x)

sin (2 x) = 2 sin (x) cos (x)

:فعّل متطابقة الزاوية المضاعفة

= 2 sin (x) cos (x) - 1.47 sin^{2} (x)

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

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

حلل إلى عوامل:

sin (x) (- 1.47 sin (x) + 2 cos (x))

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

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

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

= - 1.47 sin (x) sin (x) + 2 sin (x) cos (x)

sin (x)

قم باخراج العامل المشترك

= sin (x) (- 1.47 sin (x) + 2 cos (x))

sin (x) (- 1.47 sin (x) + 2 cos (x)) = 0

حلّ كل جزء على حدة

sin (x) = 0 or - 1.47 sin (x) + 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

- 1.47 sin (x) + 2 cos (x) = 0 : x = arctan (\frac{200}{147}) + πn

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

Rewrite using trig identities

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

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

اقسم الطرفين على

\frac{- 1.47 sin ( x ) + 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

بسّط

- \frac{1.47 sin ( x )}{cos ( x )} + 2 = 0

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

:Use the basic trigonometric identity

- 1.47 tan (x) + 2 = 0

- 1.47 tan (x) + 2 = 0

100

اضرب الطرفين بـ

- 1.47 tan (x) + 2 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 1.47 tan (x) \cdot 100 + 2 \cdot 100 = 0 \cdot 100

بسّط

- 147 tan (x) + 200 = 0

- 147 tan (x) + 200 = 0

انقل

200

إلى الجانب الأيمن

- 147 tan (x) + 200 = 0

من الطرفين

200

اطرح

- 147 tan (x) + 200 - 200 = 0 - 200

بسّط

- 147 tan (x) = - 200

- 147 tan (x) = - 200

- 147

اقسم الطرفين على

- 147 tan (x) = - 200

- 147

اقسم الطرفين على

\frac{- 147 tan ( x )}{- 147} = \frac{- 200}{- 147}

بسّط

tan (x) = \frac{200}{147}

tan (x) = \frac{200}{147}

Apply trig inverse properties

tan (x) = \frac{200}{147}

tan (x) = \frac{200}{147} :

حلول عامّة لـ

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{200}{147}) + πn

x = arctan (\frac{200}{147}) + πn

وحّد الحلول

x = 2 πn, x = π + 2 πn, x = arctan (\frac{200}{147}) + πn

أظهر الحلّ بالتمثيل العشريّ

x = 2 πn, x = π + 2 πn, x = 0.93696 \dots + πn

رسم

أمثلة شائعة

2sin^2(u)=1+,u

2 sin^{2} (u) = 1 +, u

8.5=12sin(a)

8.5 = 12 sin (a)

cos(x)= 1/15

cos (x) = \frac{1}{15}

cot(θ)=-sqrt(3),0<= θ<2pi

cot (θ) = - 3, 0 \leq θ < 2 π

(sin(90))/(458.912)=(sin(x))/(450)

\frac{sin ( 9 0 ^{\circ} )}{458.912} = \frac{sin ( x )}{450}