English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(x)=8sin^2(x)-6sin(x)

Solution

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

Solution

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

Degrees

x = - 7.93624 \dots^{\circ} + 36 0^{\circ} n, x = 187.93624 \dots^{\circ} + 36 0^{\circ} n, x = 53.58494 \dots^{\circ} + 36 0^{\circ} n, x = 126.41505 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

Subtract

8 sin^{2} (x) - 6 sin (x)

from both sides

cos^{2} (x) - 8 sin^{2} (x) + 6 sin (x) = 0

Rewrite using trig identities

cos^{2} (x) + 6 sin (x) - 8 sin^{2} (x)

Use the Pythagorean identity:

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

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

= 1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x)

Simplify

1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x) : - 9 sin^{2} (x) + 6 sin (x) + 1

1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x)

Group like terms

= - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x) + 1

Add similar elements:

- sin^{2} (x) - 8 sin^{2} (x) = - 9 sin^{2} (x)

= - 9 sin^{2} (x) + 6 sin (x) + 1

= - 9 sin^{2} (x) + 6 sin (x) + 1

1 + 6 sin (x) - 9 sin^{2} (x) = 0

Solve by substitution

1 + 6 sin (x) - 9 sin^{2} (x) = 0

Let:

sin (x) = u

1 + 6 u - 9 u^{2} = 0

1 + 6 u - 9 u^{2} = 0 : u = - \frac{- 1 + 2}{3}, u = \frac{1 + 2}{3}

1 + 6 u - 9 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 9 u^{2} + 6 u + 1 = 0

Solve with the quadratic formula

- 9 u^{2} + 6 u + 1 = 0

Quadratic Equation Formula:

For

a = - 9, b = 6, c = 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 9 ) \cdot 1}{2 ( - 9 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 9 ) \cdot 1}{2 ( - 9 )}

6^{2} - 4 (- 9) \cdot 1 = 62

6^{2} - 4 (- 9) \cdot 1

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 9 \cdot 1

Multiply the numbers:

4 \cdot 9 \cdot 1 = 36

= 6^{2} + 36

6^{2} = 36

= 36 + 36

Add the numbers:

36 + 36 = 72

= 72

Prime factorization of

72 : 2^{3} \cdot 3^{2}

72

72

divides by

272 = 36 \cdot 2

= 2 \cdot 36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

= 2^{3} \cdot 3^{2}

= 2^{3} \cdot 3^{2}

Apply exponent rule:

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

= 2^{2} \cdot 3^{2} \cdot 2

Apply radical rule:

= 2 2^{2} 3^{2}

Apply radical rule:

2^{2} = 2

= 22 3^{2}

Apply radical rule:

3^{2} = 3

= 2 \cdot 32

Refine

= 62

u_{1, 2} = \frac{- 6 \pm 6 2}{2 ( - 9 )}

Separate the solutions

u_{1} = \frac{- 6 + 6 2}{2 ( - 9 )}, u_{2} = \frac{- 6 - 6 2}{2 ( - 9 )}

u = \frac{- 6 + 6 2}{2 ( - 9 )} : - \frac{- 1 + 2}{3}

\frac{- 6 + 6 2}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 6 2}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 6 + 6 2}{- 18}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 6 + 6 2}{18}

Cancel

\frac{- 6 + 6 2}{18} : \frac{2 - 1}{3}

\frac{- 6 + 6 2}{18}

Factor

- 6 + 62 : 6 (- 1 + 2)

- 6 + 62

Rewrite as

= - 6 \cdot 1 + 62

Factor out common term

6

= 6 (- 1 + 2)

= \frac{6 ( - 1 + 2 )}{18}

Cancel the common factor:

6

= \frac{- 1 + 2}{3}

= - \frac{2 - 1}{3}

= - \frac{- 1 + 2}{3}

u = \frac{- 6 - 6 2}{2 ( - 9 )} : \frac{1 + 2}{3}

\frac{- 6 - 6 2}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 6 2}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 6 - 6 2}{- 18}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 6 - 62 = - (6 + 62)

= \frac{6 + 6 2}{18}

Factor

6 + 62 : 6 (1 + 2)

6 + 62

Rewrite as

= 6 \cdot 1 + 62

Factor out common term

6

= 6 (1 + 2)

= \frac{6 ( 1 + 2 )}{18}

Cancel the common factor:

6

= \frac{1 + 2}{3}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 2}{3}, u = \frac{1 + 2}{3}

Substitute back

u = sin (x)

sin (x) = - \frac{- 1 + 2}{3}, sin (x) = \frac{1 + 2}{3}

sin (x) = - \frac{- 1 + 2}{3}, sin (x) = \frac{1 + 2}{3}

sin (x) = - \frac{- 1 + 2}{3} : x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn

sin (x) = - \frac{- 1 + 2}{3}

Apply trig inverse properties

sin (x) = - \frac{- 1 + 2}{3}

General solutions for

sin (x) = - \frac{- 1 + 2}{3}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn

x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn

sin (x) = \frac{1 + 2}{3} : x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

sin (x) = \frac{1 + 2}{3}

Apply trig inverse properties

sin (x) = \frac{1 + 2}{3}

General solutions for

sin (x) = \frac{1 + 2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn, x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

Show solutions in decimal form

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

Popular Examples

sqrt(3)sin(x)-cos(x)=1

3 sin (x) - cos (x) = 1

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

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

(csc(θ)-2)(cot(θ)+1)=0

(csc (θ) - 2) (cot (θ) + 1) = 0

sin(x)=csc(x)

sin (x) = csc (x)

9sin(x)tan(x)=10tan(x)

9 sin (x) tan (x) = 10 tan (x)