What is the general solution for cos(105)cos(x)=-1/4 ?

The general solution for cos(105)cos(x)=-1/4 is No Solution for x\in\mathbb{R}

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cos(105)cos(x)=-1/4

Solution

cos (10 5^{\circ}) cos (x^{\circ}) = - \frac{1}{4}

Solution

No Solution for x \in R

Solution steps

cos (10 5^{\circ}) cos (x^{\circ}) = - \frac{1}{4}

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

Rewrite using trig identities:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

Write

cos (10 5^{\circ})

cos (6 0^{\circ} + 4 5^{\circ})

= cos (6 0^{\circ} + 4 5^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

Simplify

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( 1 - 3 )}{4}

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

Factor out common term

\frac{2}{2}

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

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

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

Apply rule

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

= \frac{1 - 3}{2}

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

Multiply fractions:

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

= \frac{( 1 - 3 ) 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( 1 - 3 )}{4}

= \frac{2 ( 1 - 3 )}{4}

\frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = - \frac{1}{4}

Multiply both sides by

4

\frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = - \frac{1}{4}

Multiply both sides by

4

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = 4 (- \frac{1}{4})

Simplify

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) = 4 (- \frac{1}{4})

Simplify

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ}) : 2 (1 - 3) cos (x^{\circ})

4 \cdot \frac{2 ( 1 - 3 )}{4} cos (x^{\circ})

Multiply fractions:

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

= \frac{2 ( 1 - 3 ) \cdot 4 cos ( x ^{\circ} )}{4}

Cancel the common factor:

4

= 2 (1 - 3) cos (x^{\circ})

Simplify

4 (- \frac{1}{4}) : - 1

4 (- \frac{1}{4})

Remove parentheses:

(- a) = - a

= - 4 \cdot \frac{1}{4}

Multiply fractions:

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

= - \frac{1 \cdot 4}{4}

Cancel the common factor:

4

= - 1

2 (1 - 3) cos (x^{\circ}) = - 1

2 (1 - 3) cos (x^{\circ}) = - 1

2 (1 - 3) cos (x^{\circ}) = - 1

Divide both sides by

2 (1 - 3)

2 (1 - 3) cos (x^{\circ}) = - 1

Divide both sides by

2 (1 - 3)

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} = \frac{- 1}{2 ( 1 - 3 )}

Simplify

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} = \frac{- 1}{2 ( 1 - 3 )}

Simplify

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )} : cos (x^{\circ})

\frac{2 ( 1 - 3 ) cos ( x ^{\circ} )}{2 ( 1 - 3 )}

Cancel the common factor:

2

= \frac{( 1 - 3 ) cos ( x ^{\circ} )}{1 - 3}

Cancel the common factor:

1 - 3

= cos (x^{\circ})

Simplify

\frac{- 1}{2 ( 1 - 3 )} : \frac{2 ( 1 + 3 )}{4}

\frac{- 1}{2 ( 1 - 3 )}

Apply the fraction rule:

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

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

Rationalize

- \frac{1}{2 ( 1 - 3 )} : \frac{2 ( 1 + 3 )}{4}

- \frac{1}{2 ( 1 - 3 )}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 ( 1 - 3 ) 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{( 1 - 3 ) \cdot 2}

Multiply by the conjugate

\frac{1 + 3}{1 + 3}

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

(1 - 3) \cdot 2 (1 + 3) = - 4

(1 - 3) \cdot 2 (1 + 3)

= 2 (1 - 3) (1 + 3)

Expand

(1 - 3) (1 + 3) : - 2

(1 - 3) (1 + 3)

Apply Difference of Two Squares Formula:

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

a = 1, b = 3

= 1^{2} - (3)^{2}

Simplify

1^{2} - (3)^{2} : - 2

1^{2} - (3)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (3)^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 1 - 3

Subtract the numbers:

1 - 3 = - 2

= - 2

= - 2

= 2 (- 2)

Expand

2 (- 2) : - 4

2 (- 2)

Distribute parentheses

= 2 (- 2)

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= - 4

= - 4

= - \frac{2 ( 1 + 3 )}{- 4}

Apply the fraction rule:

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

= - (- \frac{2 ( 1 + 3 )}{4})

Apply rule

- (- a) = a

= \frac{2 ( 1 + 3 )}{4}

= \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

Apply trig inverse properties

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

General solutions for

cos (x^{\circ}) = \frac{2 ( 1 + 3 )}{4}

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x^{\circ} = arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n, x^{\circ} = 36 0^{\circ} - arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n

x^{\circ} = arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n, x^{\circ} = 36 0^{\circ} - arccos (\frac{2 ( 1 + 3 )}{4}) + 36 0^{\circ} n

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for cos(105)cos(x)=-1/4 ?
The general solution for cos(105)cos(x)=-1/4 is No Solution for x\in\mathbb{R}