English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

8cos(2x)=2sin(x)+7

Solution

8 cos (2 x) = 2 sin (x) + 7

Solution

x = - 0.32593 \dots + 2 πn, x = π + 0.32593 \dots + 2 πn, x = 0.19645 \dots + 2 πn, x = π - 0.19645 \dots + 2 πn

Degrees

x = - 18.67466 \dots^{\circ} + 36 0^{\circ} n, x = 198.67466 \dots^{\circ} + 36 0^{\circ} n, x = 11.25606 \dots^{\circ} + 36 0^{\circ} n, x = 168.74393 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 cos (2 x) = 2 sin (x) + 7

Subtract

2 sin (x) + 7

from both sides

8 cos (2 x) - 2 sin (x) - 7 = 0

Rewrite using trig identities

- 7 - 2 sin (x) + 8 cos (2 x)

Use the Double Angle identity:

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

= - 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x))

Simplify

- 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x)) : - 16 sin^{2} (x) - 2 sin (x) + 1

- 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x))

Expand

8 (1 - 2 sin^{2} (x)) : 8 - 16 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 8, b = 1, c = 2 sin^{2} (x)

= 8 \cdot 1 - 8 \cdot 2 sin^{2} (x)

Simplify

8 \cdot 1 - 8 \cdot 2 sin^{2} (x) : 8 - 16 sin^{2} (x)

8 \cdot 1 - 8 \cdot 2 sin^{2} (x)

Multiply the numbers:

8 \cdot 1 = 8

= 8 - 8 \cdot 2 sin^{2} (x)

Multiply the numbers:

8 \cdot 2 = 16

= 8 - 16 sin^{2} (x)

= 8 - 16 sin^{2} (x)

= - 7 - 2 sin (x) + 8 - 16 sin^{2} (x)

Simplify

- 7 - 2 sin (x) + 8 - 16 sin^{2} (x) : - 16 sin^{2} (x) - 2 sin (x) + 1

- 7 - 2 sin (x) + 8 - 16 sin^{2} (x)

Group like terms

= - 2 sin (x) - 16 sin^{2} (x) - 7 + 8

Add/Subtract the numbers:

- 7 + 8 = 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

1 - 16 sin^{2} (x) - 2 sin (x) = 0

Solve by substitution

1 - 16 sin^{2} (x) - 2 sin (x) = 0

Let:

sin (x) = u

1 - 16 u^{2} - 2 u = 0

1 - 16 u^{2} - 2 u = 0 : u = - \frac{1 + 17}{16}, u = \frac{17 - 1}{16}

1 - 16 u^{2} - 2 u = 0

Write in the standard form

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

- 16 u^{2} - 2 u + 1 = 0

Solve with the quadratic formula

- 16 u^{2} - 2 u + 1 = 0

Quadratic Equation Formula:

For

a = - 16, b = - 2, c = 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 16 ) \cdot 1}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 16 ) \cdot 1}{2 ( - 16 )}

(- 2)^{2} - 4 (- 16) \cdot 1 = 217

(- 2)^{2} - 4 (- 16) \cdot 1

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 16 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 2^{2} + 4 \cdot 16 \cdot 1

Multiply the numbers:

4 \cdot 16 \cdot 1 = 64

= 2^{2} + 64

2^{2} = 4

= 4 + 64

Add the numbers:

4 + 64 = 68

= 68

Prime factorization of

68 : 2^{2} \cdot 17

68

68

divides by

268 = 34 \cdot 2

= 2 \cdot 34

34

divides by

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 17

2, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

Apply radical rule:

= 17 2^{2}

Apply radical rule:

2^{2} = 2

= 217

u_{1, 2} = \frac{- ( - 2 ) \pm 2 17}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 17}{2 ( - 16 )}, u_{2} = \frac{- ( - 2 ) - 2 17}{2 ( - 16 )}

u = \frac{- ( - 2 ) + 2 17}{2 ( - 16 )} : - \frac{1 + 17}{16}

\frac{- ( - 2 ) + 2 17}{2 ( - 16 )}

Remove parentheses:

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

= \frac{2 + 2 17}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{2 + 2 17}{- 32}

Apply the fraction rule:

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

= - \frac{2 + 2 17}{32}

Cancel

\frac{2 + 2 17}{32} : \frac{1 + 17}{16}

\frac{2 + 2 17}{32}

Factor

2 + 217 : 2 (1 + 17)

2 + 217

Rewrite as

= 2 \cdot 1 + 217

Factor out common term

2

= 2 (1 + 17)

= \frac{2 ( 1 + 17 )}{32}

Cancel the common factor:

2

= \frac{1 + 17}{16}

= - \frac{1 + 17}{16}

u = \frac{- ( - 2 ) - 2 17}{2 ( - 16 )} : \frac{17 - 1}{16}

\frac{- ( - 2 ) - 2 17}{2 ( - 16 )}

Remove parentheses:

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

= \frac{2 - 2 17}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{2 - 2 17}{- 32}

Apply the fraction rule:

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

2 - 217 = - (217 - 2)

= \frac{2 17 - 2}{32}

Factor

217 - 2 : 2 (17 - 1)

217 - 2

Rewrite as

= 217 - 2 \cdot 1

Factor out common term

2

= 2 (17 - 1)

= \frac{2 ( 17 - 1 )}{32}

Cancel the common factor:

2

= \frac{17 - 1}{16}

The solutions to the quadratic equation are:

u = - \frac{1 + 17}{16}, u = \frac{17 - 1}{16}

Substitute back

u = sin (x)

sin (x) = - \frac{1 + 17}{16}, sin (x) = \frac{17 - 1}{16}

sin (x) = - \frac{1 + 17}{16}, sin (x) = \frac{17 - 1}{16}

sin (x) = - \frac{1 + 17}{16} : x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

sin (x) = - \frac{1 + 17}{16}

Apply trig inverse properties

sin (x) = - \frac{1 + 17}{16}

General solutions for

sin (x) = - \frac{1 + 17}{16}

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

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

sin (x) = \frac{17 - 1}{16} : x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

sin (x) = \frac{17 - 1}{16}

Apply trig inverse properties

sin (x) = \frac{17 - 1}{16}

General solutions for

sin (x) = \frac{17 - 1}{16}

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

x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn, x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

Show solutions in decimal form

x = - 0.32593 \dots + 2 πn, x = π + 0.32593 \dots + 2 πn, x = 0.19645 \dots + 2 πn, x = π - 0.19645 \dots + 2 πn

Popular Examples

cos(θ)=0.5332,θ

cos (θ) = 0.5332, θ

sin(k)=0.5

sin (k) = 0.5

4sin(x)+4cos(x)=2

4 sin (x) + 4 cos (x) = 2

cos(5a+40)= 3/5

cos (5 a + 4 0^{\circ}) = \frac{3}{5}

2cos(x^2)-sqrt(2)=0

2 cos (x^{2}) - 2 = 0