What is the general solution for cos(x+60)*cos(x-60)= 1/2 ?

The general solution for cos(x+60)*cos(x-60)= 1/2 is x=(1.38389…)/2+pin,x=pi-(1.38389…)/2+pin

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cos(x+60)*cos(x-60)= 1/2

Solution

cos (x + 60) \cdot cos (x - 60) = \frac{1}{2}

Solution

x = \frac{1.38389 \dots}{2} + πn, x = π - \frac{1.38389 \dots}{2} + πn

Degrees

x = 39.64555 \dots^{\circ} + 18 0^{\circ} n, x = 140.35444 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cos (x + 60) cos (x - 60) = \frac{1}{2}

Subtract

\frac{1}{2}

from both sides

cos (x + 60) cos (x - 60) - \frac{1}{2} = 0

Simplify

cos (x + 60) cos (x - 60) - \frac{1}{2} : \frac{2 cos ( x + 60 ) cos ( x - 60 ) - 1}{2}

cos (x + 60) cos (x - 60) - \frac{1}{2}

Convert element to fraction:

cos (x + 60) cos (x - 60) = \frac{cos ( x + 60 ) cos ( x - 60 ) 2}{2}

= \frac{cos ( x + 60 ) cos ( x - 60 ) \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( x + 60 ) cos ( x - 60 ) \cdot 2 - 1}{2}

\frac{2 cos ( x + 60 ) cos ( x - 60 ) - 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cos (x + 60) cos (x - 60) - 1 = 0

Rewrite using trig identities

- 1 + 2 cos (- 60 + x) cos (60 + x)

Use the Product to Sum identity:

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

= - 1 + 2 \cdot \frac{1}{2} (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

2 \cdot \frac{1}{2} (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x)) = cos (120) + cos (2 x)

2 \cdot \frac{1}{2} (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

Multiply fractions:

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

= \frac{1 \cdot 2 ( cos ( - 60 + x - ( 60 + x ) ) + cos ( - 60 + x + 60 + x ) )}{2}

Cancel the common factor:

2

= 1 \cdot (cos (- 60 + x - (60 + x)) + cos (- 60 + x + 60 + x))

cos (- 60 + x - (60 + x)) = cos (120)

cos (- 60 + x - (60 + x))

Expand

- 60 + x - (60 + x) : - 120

- 60 + x - (60 + x)

- (60 + x) : - 60 - x

- (60 + x)

Distribute parentheses

= - (60) - (x)

Apply minus-plus rules

+ (- a) = - a

= - 60 - x

= - 60 + x - 60 - x

Simplify

- 60 + x - 60 - x : - 120

- 60 + x - 60 - x

Group like terms

= x - x - 60 - 60

Add similar elements:

x - x = 0

= - 60 - 60

Subtract the numbers:

- 60 - 60 = - 120

= - 120

= - 120

= cos (- 120)

Use the following property:

cos (- x) = cos (x)

cos (- 120) = cos (120)

= cos (120)

cos (- 60 + x + 60 + x) = cos (2 x)

cos (- 60 + x + 60 + x)

Simplify

- 60 + x + 60 + x : 2 x

- 60 + x + 60 + x

Group like terms

= x + x - 60 + 60

Add similar elements:

x + x = 2 x

= 2 x - 60 + 60

- 60 + 60 = 0

= 2 x

= cos (2 x)

= 1 \cdot (cos (2 x) + cos (120))

Refine

= cos (120) + cos (2 x)

= - 1 + cos (120) + cos (2 x)

- 1 + cos (120) + cos (2 x) = 0

Move

1

to the right side

- 1 + cos (120) + cos (2 x) = 0

Add

1

to both sides

- 1 + cos (120) + cos (2 x) + 1 = 0 + 1

Simplify

cos (120) + cos (2 x) = 1

cos (120) + cos (2 x) = 1

Move

cos (120)

to the right side

cos (120) + cos (2 x) = 1

Subtract

cos (120)

from both sides

cos (120) + cos (2 x) - cos (120) = 1 - cos (120)

Simplify

cos (2 x) = 1 - cos (120)

cos (2 x) = 1 - cos (120)

Apply trig inverse properties

cos (2 x) = 1 - cos (120)

General solutions for

cos (2 x) = 1 - cos (120)

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

2 x = arccos (1 - cos (120)) + 2 πn, 2 x = 2 π - arccos (1 - cos (120)) + 2 πn

2 x = arccos (1 - cos (120)) + 2 πn, 2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Solve

2 x = arccos (1 - cos (120)) + 2 πn : x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

2 x = arccos (1 - cos (120)) + 2 πn

Divide both sides by

2

2 x = arccos (1 - cos (120)) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( 1 - cos ( 120 ) )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

Solve

2 x = 2 π - arccos (1 - cos (120)) + 2 πn : x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (1 - cos (120)) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 1 - cos ( 120 ) )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

x = \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn, x = π - \frac{arccos ( 1 - cos ( 120 ) )}{2} + πn

Show solutions in decimal form

x = \frac{1.38389 \dots}{2} + πn, x = π - \frac{1.38389 \dots}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+60)*cos(x-60)= 1/2 ?
The general solution for cos(x+60)*cos(x-60)= 1/2 is x=(1.38389…)/2+pin,x=pi-(1.38389…)/2+pin