What is the general solution for sqrt(8)cos(θ)+7=-6cos(θ)+sqrt(5) ?

The general solution for sqrt(8)cos(θ)+7=-6cos(θ)+sqrt(5) is θ=2.14077…+2pin,θ=-2.14077…+2pin

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Popular Trigonometry >

sqrt(8)cos(θ)+7=-6cos(θ)+sqrt(5)

Solution

8 cos (θ) + 7 = - 6 cos (θ) + 5

Solution

θ = 2.14077 \dots + 2 πn, θ = - 2.14077 \dots + 2 πn

Degrees

θ = 122.65728 \dots^{\circ} + 36 0^{\circ} n, θ = - 122.65728 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 cos (θ) + 7 = - 6 cos (θ) + 5

Solve by substitution

8 cos (θ) + 7 = - 6 cos (θ) + 5

Let:

cos (θ) = u

8 u + 7 = - 6 u + 5

8 u + 7 = - 6 u + 5 : u = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

8 u + 7 = - 6 u + 5

Simplify

8 u + 7 : 22 u + 7

8 u + 7

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= 22 u + 7

22 u + 7 = - 6 u + 5

Move

7

to the right side

22 u + 7 = - 6 u + 5

Subtract

7

from both sides

22 u + 7 - 7 = - 6 u + 5 - 7

Simplify

22 u = - 6 u + 5 - 7

22 u = - 6 u + 5 - 7

Move

6 u

to the left side

22 u = - 6 u + 5 - 7

Add

6 u

to both sides

22 u + 6 u = - 6 u + 5 - 7 + 6 u

Simplify

22 u + 6 u = 5 - 7

22 u + 6 u = 5 - 7

Factor

22 u + 6 u : 2 (2 + 3) u

22 u + 6 u

Rewrite as

= 2 u 2 + 3 \cdot 2 u

Factor out common term

2 u

= 2 u (2 + 3)

2 (2 + 3) u = 5 - 7

Divide both sides by

2 (2 + 3)

2 (2 + 3) u = 5 - 7

Divide both sides by

2 (2 + 3)

\frac{2 ( 2 + 3 ) u}{2 ( 2 + 3 )} = \frac{5}{2 ( 2 + 3 )} - \frac{7}{2 ( 2 + 3 )}

Simplify

\frac{2 ( 2 + 3 ) u}{2 ( 2 + 3 )} = \frac{5}{2 ( 2 + 3 )} - \frac{7}{2 ( 2 + 3 )}

Simplify

\frac{2 ( 2 + 3 ) u}{2 ( 2 + 3 )} : u

\frac{2 ( 2 + 3 ) u}{2 ( 2 + 3 )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{( 3 + 2 ) u}{( 2 + 3 )}

Cancel the common factor:

2 + 3

= u

Simplify

\frac{5}{2 ( 2 + 3 )} - \frac{7}{2 ( 2 + 3 )} : \frac{( 5 - 7 ) ( 3 - 2 )}{14}

\frac{5}{2 ( 2 + 3 )} - \frac{7}{2 ( 2 + 3 )}

Apply rule

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

= \frac{5 - 7}{2 ( 3 + 2 )}

Multiply by the conjugate

\frac{3 - 2}{3 - 2}

= \frac{( 5 - 7 ) ( 3 - 2 )}{2 ( 3 + 2 ) ( 3 - 2 )}

2 (3 + 2) (3 - 2) = 14

2 (3 + 2) (3 - 2)

Expand

(3 + 2) (3 - 2) : 7

(3 + 2) (3 - 2)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 3, b = 2

= 3^{2} - (2)^{2}

Simplify

3^{2} - (2)^{2} : 7

3^{2} - (2)^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 9 - 2

Subtract the numbers:

9 - 2 = 7

= 7

= 7

= 2 \cdot 7

Expand

2 \cdot 7 : 14

2 \cdot 7

Distribute parentheses

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

= 14

= \frac{( 5 - 7 ) ( 3 - 2 )}{14}

u = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

u = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

u = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

Substitute back

u = cos (θ)

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14} : θ = arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn, θ = - arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

Apply trig inverse properties

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

General solutions for

cos (θ) = \frac{( 5 - 7 ) ( 3 - 2 )}{14}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

θ = arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn, θ = - arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn

θ = arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn, θ = - arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn

Combine all the solutions

θ = arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn, θ = - arccos (\frac{( 5 - 7 ) ( 3 - 2 )}{14}) + 2 πn

Show solutions in decimal form

θ = 2.14077 \dots + 2 πn, θ = - 2.14077 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(8)cos(θ)+7=-6cos(θ)+sqrt(5) ?
The general solution for sqrt(8)cos(θ)+7=-6cos(θ)+sqrt(5) is θ=2.14077…+2pin,θ=-2.14077…+2pin